Review ArticleBilevel Programming and Applications
Vyacheslav V Kalashnikov123 Stephan Dempe4 Gerardo A Peacuterez-Valdeacutes5
Nataliya I Kalashnykova6 and Joseacute-Fernando Camacho-Vallejo6
1Departamento de Ingenierıa Industrial y de Sistemas Tecnologico de Monterrey Avenida Eugenio Garza Sada 2501 Sur64849 Monterrey NL Mexico2Department of Social Modeling Central Economics and Mathematics Institute (CEMI) of the Russian Academy of Sciences (RAS)Nakhimovsky Prospekt 17 Moscow 117418 Russia3Department of Electronics and Computing Sumy State University Rimsky-Korsakov Street 2 Sumy 40007 Ukraine4Institut fur Numerische Mathematik und Optimierung Fakultat fur Mathematik und Informatik TU Bergakademie FreibergAkademiestraszlige 6 09596 Freiberg Germany5SINTEF Box 4760 Sluppen 7465 Trondheim Norway6Facultad de Ciencias Fısico-Matematicas (FCFM) Universidad Autonoma de Nuevo Leon (UANL) Avenida Universidad SN66450 San Nicolas de los Garza NL Mexico
Correspondence should be addressed to Vyacheslav V Kalashnikov kalashitesmmx
Received 25 June 2014 Accepted 23 July 2014
Academic Editor Sergii V Kavun
Copyright copy 2015 Vyacheslav V Kalashnikov et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
A great amount of new applied problems in the area of energy networks has recently arisen that can be efficiently solved only asmixed-integer bilevel programs Among them are the natural gas cash-out problem the deregulated electricity market equilibriumproblem biofuel problems a problem of designing coupled energy carrier networks and so forth if we mention only part ofsuch applications Bilevel models to describe migration processes are also in the list of the most popular new themes of bilevelprogramming as well as allocation information protection and cybersecurity problems This survey provides a comprehensivereview of some of the above-mentioned new areas including both theoretical and applied results
1 Introduction
Although a wide range of applications fit the bilevel pro-gramming framework real-life implementations are scarcedue mainly to the lack of efficient algorithms for tacklingmedium- and large-scale bilevel programming problems(BLP) Solving a bilevel (more generally hierarchical) opti-mization problem even in its simplest form is a difficulttask A lot of different alternativemethodsmay be used basedon the structure of the problem analyzed but there is nogeneral method that guarantees convergence performanceor optimality for every type of problem
Many new ideas appeared and were discussed in worksof plenty of authors Among them we would name Dempe[1] Dempe et al [2] Dempe and Dutta [3] Dewez et al [4]Thi et al [5] and Vicente and Calamai [6] whose works
have developed various ways of reducing original bilevelprogramming problems to equivalent single-level ones thusmaking their solution somewhat easier task for conventionalmathematical programming software packages
Mixed-integer bilevel programming problems (with partof the variables at the upper andor lower level beingintegerBoolean ones) are even harder for the conventionaloptimization techniques For instance a usual replacement ofthe lower level optimization problem with a correspondingKKT condition may not work if some of the lower levelvariables are not continuous Therefore solid theoreticalbase is necessary to be found in order to propose efficientalgorithmic procedures aimed at finding local or globalsolutions of such a problem
A great amount of new applied problems in the area ofenergy networks has recently arisen that can be efficiently
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 310301 16 pageshttpdxdoiorg1011552015310301
2 Mathematical Problems in Engineering
solved only as mixed-integer bilevel programs Among themare the natural gas cash-out problem the deregulated electric-ity market equilibrium problem biofuel problems a problemof designing coupled energy carrier networks and so forth ifwe mention only part of such applications Bilevel models todescribe migration processes are also in the list of the mostpopular new themes of bilevel programming
This special volume of the Hindawi journalMathematicalProblems in Engineering comprises papers dealingwith threemain themes bilevel programming equilibriummodels andcombinatorial (integer programming) problems and theirapplications to engineering Because of that it opens withthis survey paper ldquoBilevel Programming and Applicationsrdquosumming up some recent and new directions and resultsof the development of the mathematical methods aimed atthe solution of bilevel programs of different types and theirapplications to real-life problems
The paper is organized as follows the survey of theliterature dealing with the formulation and history of bilevelprogramming problems is given in Section 2 Section 3describes the ways the linear bilevel programs are treatedwhile Section 4 surveys the recent results in an importantapplication of BLP to the well-known imbalance cash-outproblem arising in the natural gas industry Section 5 reviewsthe new methods of reducing the number of upper levelvariables which helps a lot in applying stochastic program-ming algorithms to solve the optimal cash-out problemsSection 6 describes various promising bilevel approachesto the mixed-integer allocation model Finally Section 7presents the latest bilevelmechanisms to solve very importantinformation protection and cybersecurity problems Theconclusion acknowledgements and the list of referencesfinish the survey
2 Bilevel Programs Statement and History
A bilevel program is an optimization problem where thefeasible set is partly determined through a solution setmapping of a second parametric optimization problem [1]The latter problem is given as
where 119891 119877119899 times 119877119898 rarr 119877 119892 119877119899 times 119877119898 rarr 119877119901 119879 sube 119877119898 is a(closed) set
Let 119884 119877119899 rarr 119877119898 denote the feasible set mapping let
119884 (119909) = 119910 119892 (119909 119910) le 0
where 119865 119877119899 times 119877119898 rarr 119877119866 119877119899 rarr 119877119902 119883 sube 119877119899 is a closedset
Problems (1) and (5) can be interpreted as an hierarchicalgame of two decision makers (or players) which make theirdecisions according to the hierarchical order The first player(called the leader)makes his selection first and communicatesit to the second player (the so-called follower)Then knowingthe choice of the leader the follower selects his responseas an optimal solution of problem (1) and gives this backto the leader Thus the leaderrsquos task is to determine a bestdecision that is a point 119909 which is feasible for problem (5)119866(119909) le 0 119909 isin 119883 minimizing together with the response119910 isin Ψ(119909) the function 119865(119909 119910) Therefore problem (1) iscalled the followerrsquos problem and (5) the leaderrsquos problemProblem (5) is the bilevel programming problem
21 Optimistic and Pessimistic Approaches Strictly speakingproblem (5) is ill-posed in the case when the set Ψ(119909) is nota singleton for some 119909 which means that the mapping 119909 997891rarr119865(119909 119910(119909)) is not a function but a multivalued mapping Thisis implied by an ambiguity in the computation of the upperlevel objective function value which is rather an element inthe set 119865(119909 119910) 119910 isin Ψ(119909) The quotation marks in (5) areused purely to indicate this ambiguity To cope with such anobstacle there are several ways out
(1) The leader can assume that the follower is willing(and able) to cooperate In this case the leader simplyselects the solution within the setΨ(119909) that is the bestone with respect to the upper level objective functionThis leads then to the function
to be minimized over the set 119909 119866(119909) le 0 119909 isin 119883This is the optimistic approach leading to the opti-mistic bilevel programming problem Roughly speak-ing this problem is closely related to the problem
If 119909 is a local minimum point of the function 120593119900(sdot) onthe set
119909 119866 (119909) le 0 119909 isin 119883 (8)
and 119910 isin Ψ(119909) then the point (119909 119910) is also a localminimum point of problem (7) The converse is ingeneral not true For more information about therelation between both problems the interested readeris referred to Dempe [1]
Mathematical Problems in Engineering 3
(2) The leader has no possibility to influence the fol-lowerrsquos selection neither has heshe any guess aboutthe followerrsquos choice In this case the leader has totake into account the followerrsquos ability to select theworst solution with respect to the leaderrsquos objectivefunction hence the leader has to diminish the damageresulting from such an unlucky selection This bringsup the function
to be minimized on the set 119909 119866(119909) le 0 119909 isin 119883
min 120593119901 (119909) 119866 (119909) le 0 119909 isin 119883 (10)
This is the pessimistic approach resulting in the pes-simistic bilevel programming problem This problemis often much more complicated than the optimisticbilevel programming problem see Dempe [1]There is also another pessimistic bilevel optimizationproblem in the literature To describe this problemconsider the bilevel optimization problem with con-necting upper level constraints and an upper levelobjective function depending only on the upper levelvariable 119909
This problem has been investigated in Wiesemann etal [7] The relations between (13) and (10) should bestudied in the future
(3) The leader is able to predict a selection of the follower119910(119909) isin Ψ(119909) for all 119909 If this function is inserted intothe upper level objective function this leads to theproblem
Such a function 119910(sdot) is called a selection function ofthe point-to-set mapping Ψ(sdot) Hence we call thisapproach the selection function approach One specialcase of this approach arises if the optimal solution ofthe lower level problem is unique for all values of 119909It is obvious that the optimistic and the pessimisticproblems are special cases of the selection functionapproach
Even under restrictive assumptions (as in the case oflinear bilevel optimization or if the followerrsquos problem hasa unique optimal solution for all 119909) the function 119910(sdot) is in
general nondifferentiable Hence the bilevel programmingproblem is a nonsmooth optimization problem
Various results and examplescounterexamples concern-ing the existence of solutions to different formulations ofbilevel programming problems can be found in [1 8ndash10] tomention only few
22 A Short History of Bilevel Programming The history ofbilevel programming dates back to von Stackelberg who (in1934 in monograph [11]) formulated a hierarchical game oftwo players now called Stackelberg game The formulationof the bilevel programming problem goes back to Brackenand McGill [12] the notion ldquobilevel programmingrdquo has beencoined probably by Candler and Norton [13] see also Vicenteet al [14] With the beginning of the 80rsquos of the last centurya very intensive investigation of bilevel programming startedA number of monographs for example Bard [15] Shimizuet al [16] and Dempe [1] edited volumes see Dempe andKalashnikov [17] and Migdalas et al [18] and (annotated)bibliographies for example Vicente and Calamai [6] andDempe [19] have been published in that field
One possibility to investigate bilevel programs is totransform them into single-level (or ordinary) optimizationproblems In the first years linear bilevel programmingproblems (where all the involved functions are affine (linear)and the sets 119883 and 119879 are whole spaces) were usuallytransformed making use of linear programming duality orequivalently the Karush-Kuhn-Tucker conditions for linearprogramming Applying this approach solution algorithmshave been developed compare for example Candler andTownsley [20] The transformed problem is a special caseof a mathematical program with equilibrium constraintsMPEC (now sometimes called mathematical program withcomplementarity constraints MPCC) We can call this theKKT transformation of the bilevel programming problemThis approach is also possible for convex parametric lowerlevel problems satisfying some regularity assumption
General MPCCs have been the topic of some mono-graphs see Luo et al [21] and Outrata et al [22] Solutionalgorithms for MPCCs (see for instance Outrata et al [22]Demiguel et al [23] Leyffer et al [24] and many others)have also been suggested for solving bilevel programmingproblems
Since MPCCs are nonconvex optimization problemssolution algorithms will hopefully compute local optimalsolutions of the MPCCs Thus it is interesting if a localoptimal solution of the KKT transformation of a bilevelprogramming problem is related to a local optimal solutionof the latter problem This has been the topic of the paper byDempe and Dutta [3]
Later on the selection function approach to bilevelprogramming has been investigated in the case when theoptimal solution of the lower level problem is uniquelydetermined and strongly stable in the sense of Kojima [25]Then under some assumptions the optimal solution of thelower level problem is a 1198751198621-function see Ralph and Dempe[26] and Scholtes [27] for the definition and properties of1198751198621-functionsThis can then be used to determine necessary
4 Mathematical Problems in Engineering
and sufficient optimality conditions for bilevel programmingcompare Dempe [28]
Using the optimal value function 120593(119909) of the lower levelproblem (1) the bilevel programming problem (7) can bereplaced with
min119909119910
119865 (119909 119910) 119866 (119909) le 0 119892 (119909 119910) le 0
119891 (119909 119910) le 120593 (119909) 119909 isin 119883 (15)
This is the so-called optimal value transformationSince the optimal value function is nonsmooth even
under very restrictive assumptions this is a nonsmooth non-convex optimization problem Using nonsmooth analysissee for example Mordukhovich [29 30] and Rockafellarand Wets [31] optimality conditions for the optimal valuetransformation can be obtained compareOutrata [32] Ye andZhu [33] and Dempe et al [34]
Nowadays a large number of PhD theses have beenwritten on bilevel programming problems very differenttypes of (necessary and sufficient) optimality conditions canbe found in the literature the number of applications is hugeand both exact and heuristic solution algorithms have beensuggested
3 Linear Bilevel Programming Problems
The linear bilevel program is the problem of the followingstructure
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (17)
Here 119860 119861 and 119862 are matrices of sizes 119901 times 119899 119901 times 119898 and119899 times 119898 respectively and all variables and vectors used areof appropriate dimensions Note that we have used here theso-called optimistic bilevel optimization problem which isrelated to problem (7)
The so-called connecting constraints 119860119909 + 119861119910 le 119888are included in the upper level problem Validity of suchconstraints is beyond the selection of the leader and can beverified only after the follower has selected hisher (possiblynot unique) optimal solution In the case especially whenΨ(119909) does not reduce to a singleton certain difficulties mayarise In order to examine the bilevel programming problemin the case that Ψ(119909) does not reduce to a singleton Ishizukaand Aiyoshi [35] introduced their double penalty method Ingeneral connecting constraints may imply that the feasibleset of the bilevel programming problem is disconnectedThissituation is illustrated by the following example
Example 1 (Mersha and Dempe [36]) Consider the problem
119911 = minus119909 minus 2119910 997888rarr min119909119910 (18)
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 1The problem with upper level connecting constraintsThefeasible set is depicted with bold lines The point 119862 is global optimalsolution and point 119860 is a local optimal solution
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 2The problem when the upper level connecting constraintsare shifted into the lower level problem The feasible set is depictedwith bold lines The global optimal solution is point 119861
subject to
2119909 minus 3119910 ge minus12
119909 + 119910 le 14
119910 isin Argmin119910
minus119910 minus3119909 + 119910 le minus3 3119909 + 119910 le 30
(19)
The optimal solution for this problem is the point 119862 at(119909 119910) = (8 6) (see Figure 1) But if we shift the two upperlevel constraints to the lower level we get the point 119861 at(119909 119910) = (6 8) as an optimal solution (see Figure 2) Fromthis example one can easily notice that if we shift constraintsfrom the upper level to the lower one the optimal solutionobtained prior to shifting is not optimal any more in generalHence ideas based on shifting constraints from one levelto another will lead to a solution which may not solve theoriginal problem
In Example 1 the optimal solution of the lower levelproblem was unique for all 119909 If this is not the case feasibilityof a selection of the upper level decision maker possibly
Mathematical Problems in Engineering 5
depends on the selection of the follower In the optimisticcase this means that the leader selects within the set ofoptimal solutions of the followerrsquos problem one point whichis at the same time feasible for the upper level connectingconstraints and gives the best objective function value for theupper level objective function
As we can see in Example 1 the existence of connectingupper level constraints might lead in general to a dis-connected feasible set in the bilevel programming problemTherefore solution algorithms will live in one of the con-nected components of the feasible set (ie a sequence offeasible points which all belong to one of the connected partsis computed) or they need to jump from one of the connectedparts of the feasible set to another one Some ideas of discreteoptimization are needed in such cases
In order to avoid the above-mentioned difficulties someresearchers restrict themselves to the cases when the upperlevel constraints depend on the upper level variables only (iematrix 119861 is zero-matrix 119861 = 0) Thus the bilevel problem(16)-(17) reduces to a simpler one
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (21)
In this problem parametric linear optimization (see forexample Nozicka et al [37]) can be used to show that thegraph of the mapping Ψ(sdot) equals the connected union offaces of the set (119909 119910)⊤ 119862119910 le 119909
4 Application of Bilevel Programming toImbalance Cash-Out Problem
In the early 1990s several regulations were passed in the USAand the European Union [38 39] changing the way naturalgas was marketed and traded Particularly this liberalization[40] effectively ended a period in which natural gas was astate-driven industry The liberalization has also created theemergent natural gas markets as well as a strong demandfor models to better tackle the new problems and profit fromthis new setting [41 42] It is possible to say that the above-mentioned processes formed the natural gas supply chainThe resulting market configuration demanded the indepen-dence of the transportation and commercialization processesAs a result of this paradigm shiftmdashand the accompanyingrestructurization of the marketmdasha systematic analysis ofseveral new features becomes indispensable
Of particular interest is a problem that arises in thenatural gas supply chain namely that of balancing the fuelvolumes over a distribution network Such a balancing pro-cedure directly concerns the Pipeline Operating Company(POC) since the correct operation of the pipeline means thewell controlled volumes of the transported gasMoreover anynatural gas shipping company (NGSC) is also concernedwiththe balancing of the volumes because it is often impossible
to avoid an imbalance justified by certain economic reasonsA natural gas shipping companyrsquos business is to sell the gasby moving it through the pipeline to its clients it has tofulfill signed contracts first and then market excesses of thegas to achieve the maximum profits In order to do that theNGSC has to manage the volumes at each selling point (so-called pipeline meters) taking into account the balance theselling prices and the total revenue The basic mathematicalframework of this problemrsquos modeling is found in [43]
Owing not only to this liberalization but also to the newlocal conditions that aremore open to competition new smallplayers entered the natural gas industry especially at the localscale Indeed the USA has over 80 interstate long-distancepipelines [44] serving different regions with various climaticdemographic economic andpolitical circumstancesNaturalgas usage in Alabama for example intuitively is not the sameas in Oregon thus the market dynamics of the fuel are alsodifferent and this we presume should be reflected in someway in the econometric data of the states
Not only macroeconomic trends however are affectedby this setting When doing cross-regions studies of variousaspects of the supply chain such as the forecasting of demand[45 46] the balancing of the pipelines after imbalances havebeen created by the natural gas shippers [43 47 48] or thedynamics of interstate-intrastate systems [49] one has to takeinto account the existence of different markets
The existence of a common relationship between priceand consumption of natural gas across several zones allowsfor strong claims of uniformity which are useful when forexample we are building scenarios for a stochastic problemIndeed if we manage to group the regions in clusters withsimilar price and consumption functions we can reduce thenumber of variables needed in a scenario tree formulation[42 50]
It must be emphasized that while natural gas pipelinenetworks have been thoroughly studied most of the existentmodels focus on aspects of this part of the supply chain otherthan the NGSC-POC interaction in the system balancingsuch as network operation optimization [51 52] or deploy-ment of facilities [53] There are also papers considering thenatural gas supply chain in amultilevel scheme inwhich boththeNGSC and the POCare present and accounted for such asthe related [54 55] These works are remarkable in the sensethat they span thewhole supply chainwithmuch emphasis onthe traders (financial front-ends of the natural gas producers)so that there is little to nomention of imbalances in the systemresulting from the dealings of the NGSCs and the POCs eventhough both actors are present in the models
Many authors do acknowledge [56 57] the existence ofa problematic situation in the NGSC-POC system followingthe paradigm shift yet we have found very few sources thatexplain plausible ways in which this problem is nowadayssolved For example [58] shows how storage is required by theNGSC when no flexibility exists in the network volumeman-agement either because it is not allowed or because it is nottechnically possible Nevertheless balancing is an importantpart of the modern natural gas supply chain managementand to date no policy has been accepted as optimal regardingthe way in which the imbalances produced by the NGSC
6 Mathematical Problems in Engineering
are physically and economically handled Among the mostimportant tools that aid the POC in its task of restoring thebalance of the system is the arbitrage penalization policies inwhich the POC performs a maintenance redistribution of theimbalances in the system and charges the NGSC(s) for thecost of this operation
In [59] one finds a modeling framework (which we aregoing to follow) of the penalization part of this problemThispenalization refers only to the cash-out that occurs betweenthe NGSC and the POC it leaves outside any reference toactual market conditions which are obviously important tothe NGSC The paper presents a solution method through amodification of the original problem as well as the analysisof how this modification affects the objective function andthe obtained solutions In [47] the authors compare twoalgorithms that solve the problem making use of certainnumerical procedures In the present paper we adapt thesealgorithms to our extended model We also make use of theidea proposed in [43] to divide our problem into several gen-eralized transportation problems when finding its numericalsolution
In [60 61] we study a modified version of the above-described problem in which the upper level objective func-tion includes certain new terms based upon the net profitof the leadermdashthe natural gas shipping company This for-mulation assumes however the complete knowledge (perfectinformation) about the changes in the prices of natural gasduring the process which is somewhat nonrealistic and notquite useful as the resulting function does not clearly reflectthe reasons behind the actions of the NGSC
In [62] a stochastic reformulation of the problem ispresented so that the NGSC is now able to forecast thenext several values of the natural gas demand and then toplan the extraction of natural gas from the pipeline Theresultingmodel is a stochastic variation of the originalmixed-integer bilevel optimization problem for which two differentsolution methods are proposed and compared
To the best of our knowledge there is no literaturebeyond the works listed in the paragraphs above and theirderivatives that explicitly deals with the NGSC-POC sub-system in the same way we propose formulating a bileveloptimization problem out of the balancing operations Weattribute this to the relatively recent nature of the problemwe are dealing with as well as the difficulty of its accurateformulation for specific instances
41 Problem Specification Following the scheme constructedin [47 59] wewill consider a leader-follower system inwhichthe first agent (the leader) namely the Natural Gas ShippingCompany (NGSC) buys the gas at the wells arranges for itsinjection into an (interstate) pipeline at its starting point andextracts some amount of gasmdashideally equal to the depositedamountmdashfrom pipeline meters in several pool zones acrossthe country The follower here is the administrator of thepipeline which we call the Pipeline Operating Company(POC) who permits the NGSC to extract amounts of naturalgas that may differ from the originally injected volumes thuscreating positive or negative imbalances The latter is a kind
of usual market practice that allows for a dynamic flow of thefuel within the natural gas supply chain
However since disrupting the system in this way impliesextra costs for the NGSC the company attempts to do itonly when its predictions of future market conditions showthat the total revenues overcome the losses incurred by thepenalization policy applied to the NGSC It is clear thatthe NGSC needs tools that provide it with the best possibleinformation and help it make advantage of the latter
The NGSC-POC system operates in the following way
(1) The NGSC makes a forecast of the demand it is likelyto have during the next period (month year etc) andconsiders different scenarios in which this can occur
(2) The NGSC books certain capacity 119863119888 for every poolzone and stage (day week month etc)
(3) For each subsequent stage the NGSC determines theamount of gas to extract and sell which possibly cre-ates positive and negative imbalances in the processthis continues until the period is over
(4) ThePOC studies the resulting last day imbalances andrearranges them according to certain business rules
(5) The POC charges the NGSC with certain penaltyfor the final (rearranged) imbalances The latter mayoccur to be negative that is the POC may pay to theNGSC
(6) The NGSC calculates the net profits as its salesrevenue minus the penalty
The resulting model is a bilevel multistage stochasticoptimization problem [63] in which the upper level decisionmaker (the leader) is the NGSC who has the objective ofmaximizing its net profit as the revenue from the sales of itsgas in the pipeline minus the penalty imposed by the POCThe lower level decisionmaker (the follower) is the POCwhoaims at minimizing the absolute value of the penalizationcash-out flow either from the POC to the NGSC or viceversaThe first stage of the stochastic problem corresponds tothe capacity booking made by the NGSC and these capacityvalues remain unchanged throughout the whole process Atthe next stages the decision variables are the daily extractionamounts (and hence the imbalances) unsatisfied demandand the penalty cash-outs imposed by the POC
Note that while the POC may appear as the party withmore influence in the system the NGSC is the leader of thebilevel problem The only reason why the NGSC is the upperlevel (leader) is because of the timing of the decision processIndeed it would seem logical that the POC enjoying strongercontrol over its own facilities has to abide to the decisions(regarding final day imbalances) that the NGSC has alreadymade This is because of the relative freedom that has beenawarded (in the current businessrsquo practice) to the NGSC increating imbalances to maintain healthy business in favor ofits customers
42 Stochastic Model In [62] the authors present a bilevelmultistage stochastic optimizationmodel which is developedto deal with a certain subsystem of the natural gas supply
Mathematical Problems in Engineering 7
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
solved only as mixed-integer bilevel programs Among themare the natural gas cash-out problem the deregulated electric-ity market equilibrium problem biofuel problems a problemof designing coupled energy carrier networks and so forth ifwe mention only part of such applications Bilevel models todescribe migration processes are also in the list of the mostpopular new themes of bilevel programming
This special volume of the Hindawi journalMathematicalProblems in Engineering comprises papers dealingwith threemain themes bilevel programming equilibriummodels andcombinatorial (integer programming) problems and theirapplications to engineering Because of that it opens withthis survey paper ldquoBilevel Programming and Applicationsrdquosumming up some recent and new directions and resultsof the development of the mathematical methods aimed atthe solution of bilevel programs of different types and theirapplications to real-life problems
The paper is organized as follows the survey of theliterature dealing with the formulation and history of bilevelprogramming problems is given in Section 2 Section 3describes the ways the linear bilevel programs are treatedwhile Section 4 surveys the recent results in an importantapplication of BLP to the well-known imbalance cash-outproblem arising in the natural gas industry Section 5 reviewsthe new methods of reducing the number of upper levelvariables which helps a lot in applying stochastic program-ming algorithms to solve the optimal cash-out problemsSection 6 describes various promising bilevel approachesto the mixed-integer allocation model Finally Section 7presents the latest bilevelmechanisms to solve very importantinformation protection and cybersecurity problems Theconclusion acknowledgements and the list of referencesfinish the survey
2 Bilevel Programs Statement and History
A bilevel program is an optimization problem where thefeasible set is partly determined through a solution setmapping of a second parametric optimization problem [1]The latter problem is given as
where 119891 119877119899 times 119877119898 rarr 119877 119892 119877119899 times 119877119898 rarr 119877119901 119879 sube 119877119898 is a(closed) set
Let 119884 119877119899 rarr 119877119898 denote the feasible set mapping let
119884 (119909) = 119910 119892 (119909 119910) le 0
where 119865 119877119899 times 119877119898 rarr 119877119866 119877119899 rarr 119877119902 119883 sube 119877119899 is a closedset
Problems (1) and (5) can be interpreted as an hierarchicalgame of two decision makers (or players) which make theirdecisions according to the hierarchical order The first player(called the leader)makes his selection first and communicatesit to the second player (the so-called follower)Then knowingthe choice of the leader the follower selects his responseas an optimal solution of problem (1) and gives this backto the leader Thus the leaderrsquos task is to determine a bestdecision that is a point 119909 which is feasible for problem (5)119866(119909) le 0 119909 isin 119883 minimizing together with the response119910 isin Ψ(119909) the function 119865(119909 119910) Therefore problem (1) iscalled the followerrsquos problem and (5) the leaderrsquos problemProblem (5) is the bilevel programming problem
21 Optimistic and Pessimistic Approaches Strictly speakingproblem (5) is ill-posed in the case when the set Ψ(119909) is nota singleton for some 119909 which means that the mapping 119909 997891rarr119865(119909 119910(119909)) is not a function but a multivalued mapping Thisis implied by an ambiguity in the computation of the upperlevel objective function value which is rather an element inthe set 119865(119909 119910) 119910 isin Ψ(119909) The quotation marks in (5) areused purely to indicate this ambiguity To cope with such anobstacle there are several ways out
(1) The leader can assume that the follower is willing(and able) to cooperate In this case the leader simplyselects the solution within the setΨ(119909) that is the bestone with respect to the upper level objective functionThis leads then to the function
to be minimized over the set 119909 119866(119909) le 0 119909 isin 119883This is the optimistic approach leading to the opti-mistic bilevel programming problem Roughly speak-ing this problem is closely related to the problem
If 119909 is a local minimum point of the function 120593119900(sdot) onthe set
119909 119866 (119909) le 0 119909 isin 119883 (8)
and 119910 isin Ψ(119909) then the point (119909 119910) is also a localminimum point of problem (7) The converse is ingeneral not true For more information about therelation between both problems the interested readeris referred to Dempe [1]
Mathematical Problems in Engineering 3
(2) The leader has no possibility to influence the fol-lowerrsquos selection neither has heshe any guess aboutthe followerrsquos choice In this case the leader has totake into account the followerrsquos ability to select theworst solution with respect to the leaderrsquos objectivefunction hence the leader has to diminish the damageresulting from such an unlucky selection This bringsup the function
to be minimized on the set 119909 119866(119909) le 0 119909 isin 119883
min 120593119901 (119909) 119866 (119909) le 0 119909 isin 119883 (10)
This is the pessimistic approach resulting in the pes-simistic bilevel programming problem This problemis often much more complicated than the optimisticbilevel programming problem see Dempe [1]There is also another pessimistic bilevel optimizationproblem in the literature To describe this problemconsider the bilevel optimization problem with con-necting upper level constraints and an upper levelobjective function depending only on the upper levelvariable 119909
This problem has been investigated in Wiesemann etal [7] The relations between (13) and (10) should bestudied in the future
(3) The leader is able to predict a selection of the follower119910(119909) isin Ψ(119909) for all 119909 If this function is inserted intothe upper level objective function this leads to theproblem
Such a function 119910(sdot) is called a selection function ofthe point-to-set mapping Ψ(sdot) Hence we call thisapproach the selection function approach One specialcase of this approach arises if the optimal solution ofthe lower level problem is unique for all values of 119909It is obvious that the optimistic and the pessimisticproblems are special cases of the selection functionapproach
Even under restrictive assumptions (as in the case oflinear bilevel optimization or if the followerrsquos problem hasa unique optimal solution for all 119909) the function 119910(sdot) is in
general nondifferentiable Hence the bilevel programmingproblem is a nonsmooth optimization problem
Various results and examplescounterexamples concern-ing the existence of solutions to different formulations ofbilevel programming problems can be found in [1 8ndash10] tomention only few
22 A Short History of Bilevel Programming The history ofbilevel programming dates back to von Stackelberg who (in1934 in monograph [11]) formulated a hierarchical game oftwo players now called Stackelberg game The formulationof the bilevel programming problem goes back to Brackenand McGill [12] the notion ldquobilevel programmingrdquo has beencoined probably by Candler and Norton [13] see also Vicenteet al [14] With the beginning of the 80rsquos of the last centurya very intensive investigation of bilevel programming startedA number of monographs for example Bard [15] Shimizuet al [16] and Dempe [1] edited volumes see Dempe andKalashnikov [17] and Migdalas et al [18] and (annotated)bibliographies for example Vicente and Calamai [6] andDempe [19] have been published in that field
One possibility to investigate bilevel programs is totransform them into single-level (or ordinary) optimizationproblems In the first years linear bilevel programmingproblems (where all the involved functions are affine (linear)and the sets 119883 and 119879 are whole spaces) were usuallytransformed making use of linear programming duality orequivalently the Karush-Kuhn-Tucker conditions for linearprogramming Applying this approach solution algorithmshave been developed compare for example Candler andTownsley [20] The transformed problem is a special caseof a mathematical program with equilibrium constraintsMPEC (now sometimes called mathematical program withcomplementarity constraints MPCC) We can call this theKKT transformation of the bilevel programming problemThis approach is also possible for convex parametric lowerlevel problems satisfying some regularity assumption
General MPCCs have been the topic of some mono-graphs see Luo et al [21] and Outrata et al [22] Solutionalgorithms for MPCCs (see for instance Outrata et al [22]Demiguel et al [23] Leyffer et al [24] and many others)have also been suggested for solving bilevel programmingproblems
Since MPCCs are nonconvex optimization problemssolution algorithms will hopefully compute local optimalsolutions of the MPCCs Thus it is interesting if a localoptimal solution of the KKT transformation of a bilevelprogramming problem is related to a local optimal solutionof the latter problem This has been the topic of the paper byDempe and Dutta [3]
Later on the selection function approach to bilevelprogramming has been investigated in the case when theoptimal solution of the lower level problem is uniquelydetermined and strongly stable in the sense of Kojima [25]Then under some assumptions the optimal solution of thelower level problem is a 1198751198621-function see Ralph and Dempe[26] and Scholtes [27] for the definition and properties of1198751198621-functionsThis can then be used to determine necessary
4 Mathematical Problems in Engineering
and sufficient optimality conditions for bilevel programmingcompare Dempe [28]
Using the optimal value function 120593(119909) of the lower levelproblem (1) the bilevel programming problem (7) can bereplaced with
min119909119910
119865 (119909 119910) 119866 (119909) le 0 119892 (119909 119910) le 0
119891 (119909 119910) le 120593 (119909) 119909 isin 119883 (15)
This is the so-called optimal value transformationSince the optimal value function is nonsmooth even
under very restrictive assumptions this is a nonsmooth non-convex optimization problem Using nonsmooth analysissee for example Mordukhovich [29 30] and Rockafellarand Wets [31] optimality conditions for the optimal valuetransformation can be obtained compareOutrata [32] Ye andZhu [33] and Dempe et al [34]
Nowadays a large number of PhD theses have beenwritten on bilevel programming problems very differenttypes of (necessary and sufficient) optimality conditions canbe found in the literature the number of applications is hugeand both exact and heuristic solution algorithms have beensuggested
3 Linear Bilevel Programming Problems
The linear bilevel program is the problem of the followingstructure
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (17)
Here 119860 119861 and 119862 are matrices of sizes 119901 times 119899 119901 times 119898 and119899 times 119898 respectively and all variables and vectors used areof appropriate dimensions Note that we have used here theso-called optimistic bilevel optimization problem which isrelated to problem (7)
The so-called connecting constraints 119860119909 + 119861119910 le 119888are included in the upper level problem Validity of suchconstraints is beyond the selection of the leader and can beverified only after the follower has selected hisher (possiblynot unique) optimal solution In the case especially whenΨ(119909) does not reduce to a singleton certain difficulties mayarise In order to examine the bilevel programming problemin the case that Ψ(119909) does not reduce to a singleton Ishizukaand Aiyoshi [35] introduced their double penalty method Ingeneral connecting constraints may imply that the feasibleset of the bilevel programming problem is disconnectedThissituation is illustrated by the following example
Example 1 (Mersha and Dempe [36]) Consider the problem
119911 = minus119909 minus 2119910 997888rarr min119909119910 (18)
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 1The problem with upper level connecting constraintsThefeasible set is depicted with bold lines The point 119862 is global optimalsolution and point 119860 is a local optimal solution
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 2The problem when the upper level connecting constraintsare shifted into the lower level problem The feasible set is depictedwith bold lines The global optimal solution is point 119861
subject to
2119909 minus 3119910 ge minus12
119909 + 119910 le 14
119910 isin Argmin119910
minus119910 minus3119909 + 119910 le minus3 3119909 + 119910 le 30
(19)
The optimal solution for this problem is the point 119862 at(119909 119910) = (8 6) (see Figure 1) But if we shift the two upperlevel constraints to the lower level we get the point 119861 at(119909 119910) = (6 8) as an optimal solution (see Figure 2) Fromthis example one can easily notice that if we shift constraintsfrom the upper level to the lower one the optimal solutionobtained prior to shifting is not optimal any more in generalHence ideas based on shifting constraints from one levelto another will lead to a solution which may not solve theoriginal problem
In Example 1 the optimal solution of the lower levelproblem was unique for all 119909 If this is not the case feasibilityof a selection of the upper level decision maker possibly
Mathematical Problems in Engineering 5
depends on the selection of the follower In the optimisticcase this means that the leader selects within the set ofoptimal solutions of the followerrsquos problem one point whichis at the same time feasible for the upper level connectingconstraints and gives the best objective function value for theupper level objective function
As we can see in Example 1 the existence of connectingupper level constraints might lead in general to a dis-connected feasible set in the bilevel programming problemTherefore solution algorithms will live in one of the con-nected components of the feasible set (ie a sequence offeasible points which all belong to one of the connected partsis computed) or they need to jump from one of the connectedparts of the feasible set to another one Some ideas of discreteoptimization are needed in such cases
In order to avoid the above-mentioned difficulties someresearchers restrict themselves to the cases when the upperlevel constraints depend on the upper level variables only (iematrix 119861 is zero-matrix 119861 = 0) Thus the bilevel problem(16)-(17) reduces to a simpler one
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (21)
In this problem parametric linear optimization (see forexample Nozicka et al [37]) can be used to show that thegraph of the mapping Ψ(sdot) equals the connected union offaces of the set (119909 119910)⊤ 119862119910 le 119909
4 Application of Bilevel Programming toImbalance Cash-Out Problem
In the early 1990s several regulations were passed in the USAand the European Union [38 39] changing the way naturalgas was marketed and traded Particularly this liberalization[40] effectively ended a period in which natural gas was astate-driven industry The liberalization has also created theemergent natural gas markets as well as a strong demandfor models to better tackle the new problems and profit fromthis new setting [41 42] It is possible to say that the above-mentioned processes formed the natural gas supply chainThe resulting market configuration demanded the indepen-dence of the transportation and commercialization processesAs a result of this paradigm shiftmdashand the accompanyingrestructurization of the marketmdasha systematic analysis ofseveral new features becomes indispensable
Of particular interest is a problem that arises in thenatural gas supply chain namely that of balancing the fuelvolumes over a distribution network Such a balancing pro-cedure directly concerns the Pipeline Operating Company(POC) since the correct operation of the pipeline means thewell controlled volumes of the transported gasMoreover anynatural gas shipping company (NGSC) is also concernedwiththe balancing of the volumes because it is often impossible
to avoid an imbalance justified by certain economic reasonsA natural gas shipping companyrsquos business is to sell the gasby moving it through the pipeline to its clients it has tofulfill signed contracts first and then market excesses of thegas to achieve the maximum profits In order to do that theNGSC has to manage the volumes at each selling point (so-called pipeline meters) taking into account the balance theselling prices and the total revenue The basic mathematicalframework of this problemrsquos modeling is found in [43]
Owing not only to this liberalization but also to the newlocal conditions that aremore open to competition new smallplayers entered the natural gas industry especially at the localscale Indeed the USA has over 80 interstate long-distancepipelines [44] serving different regions with various climaticdemographic economic andpolitical circumstancesNaturalgas usage in Alabama for example intuitively is not the sameas in Oregon thus the market dynamics of the fuel are alsodifferent and this we presume should be reflected in someway in the econometric data of the states
Not only macroeconomic trends however are affectedby this setting When doing cross-regions studies of variousaspects of the supply chain such as the forecasting of demand[45 46] the balancing of the pipelines after imbalances havebeen created by the natural gas shippers [43 47 48] or thedynamics of interstate-intrastate systems [49] one has to takeinto account the existence of different markets
The existence of a common relationship between priceand consumption of natural gas across several zones allowsfor strong claims of uniformity which are useful when forexample we are building scenarios for a stochastic problemIndeed if we manage to group the regions in clusters withsimilar price and consumption functions we can reduce thenumber of variables needed in a scenario tree formulation[42 50]
It must be emphasized that while natural gas pipelinenetworks have been thoroughly studied most of the existentmodels focus on aspects of this part of the supply chain otherthan the NGSC-POC interaction in the system balancingsuch as network operation optimization [51 52] or deploy-ment of facilities [53] There are also papers considering thenatural gas supply chain in amultilevel scheme inwhich boththeNGSC and the POCare present and accounted for such asthe related [54 55] These works are remarkable in the sensethat they span thewhole supply chainwithmuch emphasis onthe traders (financial front-ends of the natural gas producers)so that there is little to nomention of imbalances in the systemresulting from the dealings of the NGSCs and the POCs eventhough both actors are present in the models
Many authors do acknowledge [56 57] the existence ofa problematic situation in the NGSC-POC system followingthe paradigm shift yet we have found very few sources thatexplain plausible ways in which this problem is nowadayssolved For example [58] shows how storage is required by theNGSC when no flexibility exists in the network volumeman-agement either because it is not allowed or because it is nottechnically possible Nevertheless balancing is an importantpart of the modern natural gas supply chain managementand to date no policy has been accepted as optimal regardingthe way in which the imbalances produced by the NGSC
6 Mathematical Problems in Engineering
are physically and economically handled Among the mostimportant tools that aid the POC in its task of restoring thebalance of the system is the arbitrage penalization policies inwhich the POC performs a maintenance redistribution of theimbalances in the system and charges the NGSC(s) for thecost of this operation
In [59] one finds a modeling framework (which we aregoing to follow) of the penalization part of this problemThispenalization refers only to the cash-out that occurs betweenthe NGSC and the POC it leaves outside any reference toactual market conditions which are obviously important tothe NGSC The paper presents a solution method through amodification of the original problem as well as the analysisof how this modification affects the objective function andthe obtained solutions In [47] the authors compare twoalgorithms that solve the problem making use of certainnumerical procedures In the present paper we adapt thesealgorithms to our extended model We also make use of theidea proposed in [43] to divide our problem into several gen-eralized transportation problems when finding its numericalsolution
In [60 61] we study a modified version of the above-described problem in which the upper level objective func-tion includes certain new terms based upon the net profitof the leadermdashthe natural gas shipping company This for-mulation assumes however the complete knowledge (perfectinformation) about the changes in the prices of natural gasduring the process which is somewhat nonrealistic and notquite useful as the resulting function does not clearly reflectthe reasons behind the actions of the NGSC
In [62] a stochastic reformulation of the problem ispresented so that the NGSC is now able to forecast thenext several values of the natural gas demand and then toplan the extraction of natural gas from the pipeline Theresultingmodel is a stochastic variation of the originalmixed-integer bilevel optimization problem for which two differentsolution methods are proposed and compared
To the best of our knowledge there is no literaturebeyond the works listed in the paragraphs above and theirderivatives that explicitly deals with the NGSC-POC sub-system in the same way we propose formulating a bileveloptimization problem out of the balancing operations Weattribute this to the relatively recent nature of the problemwe are dealing with as well as the difficulty of its accurateformulation for specific instances
41 Problem Specification Following the scheme constructedin [47 59] wewill consider a leader-follower system inwhichthe first agent (the leader) namely the Natural Gas ShippingCompany (NGSC) buys the gas at the wells arranges for itsinjection into an (interstate) pipeline at its starting point andextracts some amount of gasmdashideally equal to the depositedamountmdashfrom pipeline meters in several pool zones acrossthe country The follower here is the administrator of thepipeline which we call the Pipeline Operating Company(POC) who permits the NGSC to extract amounts of naturalgas that may differ from the originally injected volumes thuscreating positive or negative imbalances The latter is a kind
of usual market practice that allows for a dynamic flow of thefuel within the natural gas supply chain
However since disrupting the system in this way impliesextra costs for the NGSC the company attempts to do itonly when its predictions of future market conditions showthat the total revenues overcome the losses incurred by thepenalization policy applied to the NGSC It is clear thatthe NGSC needs tools that provide it with the best possibleinformation and help it make advantage of the latter
The NGSC-POC system operates in the following way
(1) The NGSC makes a forecast of the demand it is likelyto have during the next period (month year etc) andconsiders different scenarios in which this can occur
(2) The NGSC books certain capacity 119863119888 for every poolzone and stage (day week month etc)
(3) For each subsequent stage the NGSC determines theamount of gas to extract and sell which possibly cre-ates positive and negative imbalances in the processthis continues until the period is over
(4) ThePOC studies the resulting last day imbalances andrearranges them according to certain business rules
(5) The POC charges the NGSC with certain penaltyfor the final (rearranged) imbalances The latter mayoccur to be negative that is the POC may pay to theNGSC
(6) The NGSC calculates the net profits as its salesrevenue minus the penalty
The resulting model is a bilevel multistage stochasticoptimization problem [63] in which the upper level decisionmaker (the leader) is the NGSC who has the objective ofmaximizing its net profit as the revenue from the sales of itsgas in the pipeline minus the penalty imposed by the POCThe lower level decisionmaker (the follower) is the POCwhoaims at minimizing the absolute value of the penalizationcash-out flow either from the POC to the NGSC or viceversaThe first stage of the stochastic problem corresponds tothe capacity booking made by the NGSC and these capacityvalues remain unchanged throughout the whole process Atthe next stages the decision variables are the daily extractionamounts (and hence the imbalances) unsatisfied demandand the penalty cash-outs imposed by the POC
Note that while the POC may appear as the party withmore influence in the system the NGSC is the leader of thebilevel problem The only reason why the NGSC is the upperlevel (leader) is because of the timing of the decision processIndeed it would seem logical that the POC enjoying strongercontrol over its own facilities has to abide to the decisions(regarding final day imbalances) that the NGSC has alreadymade This is because of the relative freedom that has beenawarded (in the current businessrsquo practice) to the NGSC increating imbalances to maintain healthy business in favor ofits customers
42 Stochastic Model In [62] the authors present a bilevelmultistage stochastic optimizationmodel which is developedto deal with a certain subsystem of the natural gas supply
Mathematical Problems in Engineering 7
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
(2) The leader has no possibility to influence the fol-lowerrsquos selection neither has heshe any guess aboutthe followerrsquos choice In this case the leader has totake into account the followerrsquos ability to select theworst solution with respect to the leaderrsquos objectivefunction hence the leader has to diminish the damageresulting from such an unlucky selection This bringsup the function
to be minimized on the set 119909 119866(119909) le 0 119909 isin 119883
min 120593119901 (119909) 119866 (119909) le 0 119909 isin 119883 (10)
This is the pessimistic approach resulting in the pes-simistic bilevel programming problem This problemis often much more complicated than the optimisticbilevel programming problem see Dempe [1]There is also another pessimistic bilevel optimizationproblem in the literature To describe this problemconsider the bilevel optimization problem with con-necting upper level constraints and an upper levelobjective function depending only on the upper levelvariable 119909
This problem has been investigated in Wiesemann etal [7] The relations between (13) and (10) should bestudied in the future
(3) The leader is able to predict a selection of the follower119910(119909) isin Ψ(119909) for all 119909 If this function is inserted intothe upper level objective function this leads to theproblem
Such a function 119910(sdot) is called a selection function ofthe point-to-set mapping Ψ(sdot) Hence we call thisapproach the selection function approach One specialcase of this approach arises if the optimal solution ofthe lower level problem is unique for all values of 119909It is obvious that the optimistic and the pessimisticproblems are special cases of the selection functionapproach
Even under restrictive assumptions (as in the case oflinear bilevel optimization or if the followerrsquos problem hasa unique optimal solution for all 119909) the function 119910(sdot) is in
general nondifferentiable Hence the bilevel programmingproblem is a nonsmooth optimization problem
Various results and examplescounterexamples concern-ing the existence of solutions to different formulations ofbilevel programming problems can be found in [1 8ndash10] tomention only few
22 A Short History of Bilevel Programming The history ofbilevel programming dates back to von Stackelberg who (in1934 in monograph [11]) formulated a hierarchical game oftwo players now called Stackelberg game The formulationof the bilevel programming problem goes back to Brackenand McGill [12] the notion ldquobilevel programmingrdquo has beencoined probably by Candler and Norton [13] see also Vicenteet al [14] With the beginning of the 80rsquos of the last centurya very intensive investigation of bilevel programming startedA number of monographs for example Bard [15] Shimizuet al [16] and Dempe [1] edited volumes see Dempe andKalashnikov [17] and Migdalas et al [18] and (annotated)bibliographies for example Vicente and Calamai [6] andDempe [19] have been published in that field
One possibility to investigate bilevel programs is totransform them into single-level (or ordinary) optimizationproblems In the first years linear bilevel programmingproblems (where all the involved functions are affine (linear)and the sets 119883 and 119879 are whole spaces) were usuallytransformed making use of linear programming duality orequivalently the Karush-Kuhn-Tucker conditions for linearprogramming Applying this approach solution algorithmshave been developed compare for example Candler andTownsley [20] The transformed problem is a special caseof a mathematical program with equilibrium constraintsMPEC (now sometimes called mathematical program withcomplementarity constraints MPCC) We can call this theKKT transformation of the bilevel programming problemThis approach is also possible for convex parametric lowerlevel problems satisfying some regularity assumption
General MPCCs have been the topic of some mono-graphs see Luo et al [21] and Outrata et al [22] Solutionalgorithms for MPCCs (see for instance Outrata et al [22]Demiguel et al [23] Leyffer et al [24] and many others)have also been suggested for solving bilevel programmingproblems
Since MPCCs are nonconvex optimization problemssolution algorithms will hopefully compute local optimalsolutions of the MPCCs Thus it is interesting if a localoptimal solution of the KKT transformation of a bilevelprogramming problem is related to a local optimal solutionof the latter problem This has been the topic of the paper byDempe and Dutta [3]
Later on the selection function approach to bilevelprogramming has been investigated in the case when theoptimal solution of the lower level problem is uniquelydetermined and strongly stable in the sense of Kojima [25]Then under some assumptions the optimal solution of thelower level problem is a 1198751198621-function see Ralph and Dempe[26] and Scholtes [27] for the definition and properties of1198751198621-functionsThis can then be used to determine necessary
4 Mathematical Problems in Engineering
and sufficient optimality conditions for bilevel programmingcompare Dempe [28]
Using the optimal value function 120593(119909) of the lower levelproblem (1) the bilevel programming problem (7) can bereplaced with
min119909119910
119865 (119909 119910) 119866 (119909) le 0 119892 (119909 119910) le 0
119891 (119909 119910) le 120593 (119909) 119909 isin 119883 (15)
This is the so-called optimal value transformationSince the optimal value function is nonsmooth even
under very restrictive assumptions this is a nonsmooth non-convex optimization problem Using nonsmooth analysissee for example Mordukhovich [29 30] and Rockafellarand Wets [31] optimality conditions for the optimal valuetransformation can be obtained compareOutrata [32] Ye andZhu [33] and Dempe et al [34]
Nowadays a large number of PhD theses have beenwritten on bilevel programming problems very differenttypes of (necessary and sufficient) optimality conditions canbe found in the literature the number of applications is hugeand both exact and heuristic solution algorithms have beensuggested
3 Linear Bilevel Programming Problems
The linear bilevel program is the problem of the followingstructure
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (17)
Here 119860 119861 and 119862 are matrices of sizes 119901 times 119899 119901 times 119898 and119899 times 119898 respectively and all variables and vectors used areof appropriate dimensions Note that we have used here theso-called optimistic bilevel optimization problem which isrelated to problem (7)
The so-called connecting constraints 119860119909 + 119861119910 le 119888are included in the upper level problem Validity of suchconstraints is beyond the selection of the leader and can beverified only after the follower has selected hisher (possiblynot unique) optimal solution In the case especially whenΨ(119909) does not reduce to a singleton certain difficulties mayarise In order to examine the bilevel programming problemin the case that Ψ(119909) does not reduce to a singleton Ishizukaand Aiyoshi [35] introduced their double penalty method Ingeneral connecting constraints may imply that the feasibleset of the bilevel programming problem is disconnectedThissituation is illustrated by the following example
Example 1 (Mersha and Dempe [36]) Consider the problem
119911 = minus119909 minus 2119910 997888rarr min119909119910 (18)
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 1The problem with upper level connecting constraintsThefeasible set is depicted with bold lines The point 119862 is global optimalsolution and point 119860 is a local optimal solution
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 2The problem when the upper level connecting constraintsare shifted into the lower level problem The feasible set is depictedwith bold lines The global optimal solution is point 119861
subject to
2119909 minus 3119910 ge minus12
119909 + 119910 le 14
119910 isin Argmin119910
minus119910 minus3119909 + 119910 le minus3 3119909 + 119910 le 30
(19)
The optimal solution for this problem is the point 119862 at(119909 119910) = (8 6) (see Figure 1) But if we shift the two upperlevel constraints to the lower level we get the point 119861 at(119909 119910) = (6 8) as an optimal solution (see Figure 2) Fromthis example one can easily notice that if we shift constraintsfrom the upper level to the lower one the optimal solutionobtained prior to shifting is not optimal any more in generalHence ideas based on shifting constraints from one levelto another will lead to a solution which may not solve theoriginal problem
In Example 1 the optimal solution of the lower levelproblem was unique for all 119909 If this is not the case feasibilityof a selection of the upper level decision maker possibly
Mathematical Problems in Engineering 5
depends on the selection of the follower In the optimisticcase this means that the leader selects within the set ofoptimal solutions of the followerrsquos problem one point whichis at the same time feasible for the upper level connectingconstraints and gives the best objective function value for theupper level objective function
As we can see in Example 1 the existence of connectingupper level constraints might lead in general to a dis-connected feasible set in the bilevel programming problemTherefore solution algorithms will live in one of the con-nected components of the feasible set (ie a sequence offeasible points which all belong to one of the connected partsis computed) or they need to jump from one of the connectedparts of the feasible set to another one Some ideas of discreteoptimization are needed in such cases
In order to avoid the above-mentioned difficulties someresearchers restrict themselves to the cases when the upperlevel constraints depend on the upper level variables only (iematrix 119861 is zero-matrix 119861 = 0) Thus the bilevel problem(16)-(17) reduces to a simpler one
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (21)
In this problem parametric linear optimization (see forexample Nozicka et al [37]) can be used to show that thegraph of the mapping Ψ(sdot) equals the connected union offaces of the set (119909 119910)⊤ 119862119910 le 119909
4 Application of Bilevel Programming toImbalance Cash-Out Problem
In the early 1990s several regulations were passed in the USAand the European Union [38 39] changing the way naturalgas was marketed and traded Particularly this liberalization[40] effectively ended a period in which natural gas was astate-driven industry The liberalization has also created theemergent natural gas markets as well as a strong demandfor models to better tackle the new problems and profit fromthis new setting [41 42] It is possible to say that the above-mentioned processes formed the natural gas supply chainThe resulting market configuration demanded the indepen-dence of the transportation and commercialization processesAs a result of this paradigm shiftmdashand the accompanyingrestructurization of the marketmdasha systematic analysis ofseveral new features becomes indispensable
Of particular interest is a problem that arises in thenatural gas supply chain namely that of balancing the fuelvolumes over a distribution network Such a balancing pro-cedure directly concerns the Pipeline Operating Company(POC) since the correct operation of the pipeline means thewell controlled volumes of the transported gasMoreover anynatural gas shipping company (NGSC) is also concernedwiththe balancing of the volumes because it is often impossible
to avoid an imbalance justified by certain economic reasonsA natural gas shipping companyrsquos business is to sell the gasby moving it through the pipeline to its clients it has tofulfill signed contracts first and then market excesses of thegas to achieve the maximum profits In order to do that theNGSC has to manage the volumes at each selling point (so-called pipeline meters) taking into account the balance theselling prices and the total revenue The basic mathematicalframework of this problemrsquos modeling is found in [43]
Owing not only to this liberalization but also to the newlocal conditions that aremore open to competition new smallplayers entered the natural gas industry especially at the localscale Indeed the USA has over 80 interstate long-distancepipelines [44] serving different regions with various climaticdemographic economic andpolitical circumstancesNaturalgas usage in Alabama for example intuitively is not the sameas in Oregon thus the market dynamics of the fuel are alsodifferent and this we presume should be reflected in someway in the econometric data of the states
Not only macroeconomic trends however are affectedby this setting When doing cross-regions studies of variousaspects of the supply chain such as the forecasting of demand[45 46] the balancing of the pipelines after imbalances havebeen created by the natural gas shippers [43 47 48] or thedynamics of interstate-intrastate systems [49] one has to takeinto account the existence of different markets
The existence of a common relationship between priceand consumption of natural gas across several zones allowsfor strong claims of uniformity which are useful when forexample we are building scenarios for a stochastic problemIndeed if we manage to group the regions in clusters withsimilar price and consumption functions we can reduce thenumber of variables needed in a scenario tree formulation[42 50]
It must be emphasized that while natural gas pipelinenetworks have been thoroughly studied most of the existentmodels focus on aspects of this part of the supply chain otherthan the NGSC-POC interaction in the system balancingsuch as network operation optimization [51 52] or deploy-ment of facilities [53] There are also papers considering thenatural gas supply chain in amultilevel scheme inwhich boththeNGSC and the POCare present and accounted for such asthe related [54 55] These works are remarkable in the sensethat they span thewhole supply chainwithmuch emphasis onthe traders (financial front-ends of the natural gas producers)so that there is little to nomention of imbalances in the systemresulting from the dealings of the NGSCs and the POCs eventhough both actors are present in the models
Many authors do acknowledge [56 57] the existence ofa problematic situation in the NGSC-POC system followingthe paradigm shift yet we have found very few sources thatexplain plausible ways in which this problem is nowadayssolved For example [58] shows how storage is required by theNGSC when no flexibility exists in the network volumeman-agement either because it is not allowed or because it is nottechnically possible Nevertheless balancing is an importantpart of the modern natural gas supply chain managementand to date no policy has been accepted as optimal regardingthe way in which the imbalances produced by the NGSC
6 Mathematical Problems in Engineering
are physically and economically handled Among the mostimportant tools that aid the POC in its task of restoring thebalance of the system is the arbitrage penalization policies inwhich the POC performs a maintenance redistribution of theimbalances in the system and charges the NGSC(s) for thecost of this operation
In [59] one finds a modeling framework (which we aregoing to follow) of the penalization part of this problemThispenalization refers only to the cash-out that occurs betweenthe NGSC and the POC it leaves outside any reference toactual market conditions which are obviously important tothe NGSC The paper presents a solution method through amodification of the original problem as well as the analysisof how this modification affects the objective function andthe obtained solutions In [47] the authors compare twoalgorithms that solve the problem making use of certainnumerical procedures In the present paper we adapt thesealgorithms to our extended model We also make use of theidea proposed in [43] to divide our problem into several gen-eralized transportation problems when finding its numericalsolution
In [60 61] we study a modified version of the above-described problem in which the upper level objective func-tion includes certain new terms based upon the net profitof the leadermdashthe natural gas shipping company This for-mulation assumes however the complete knowledge (perfectinformation) about the changes in the prices of natural gasduring the process which is somewhat nonrealistic and notquite useful as the resulting function does not clearly reflectthe reasons behind the actions of the NGSC
In [62] a stochastic reformulation of the problem ispresented so that the NGSC is now able to forecast thenext several values of the natural gas demand and then toplan the extraction of natural gas from the pipeline Theresultingmodel is a stochastic variation of the originalmixed-integer bilevel optimization problem for which two differentsolution methods are proposed and compared
To the best of our knowledge there is no literaturebeyond the works listed in the paragraphs above and theirderivatives that explicitly deals with the NGSC-POC sub-system in the same way we propose formulating a bileveloptimization problem out of the balancing operations Weattribute this to the relatively recent nature of the problemwe are dealing with as well as the difficulty of its accurateformulation for specific instances
41 Problem Specification Following the scheme constructedin [47 59] wewill consider a leader-follower system inwhichthe first agent (the leader) namely the Natural Gas ShippingCompany (NGSC) buys the gas at the wells arranges for itsinjection into an (interstate) pipeline at its starting point andextracts some amount of gasmdashideally equal to the depositedamountmdashfrom pipeline meters in several pool zones acrossthe country The follower here is the administrator of thepipeline which we call the Pipeline Operating Company(POC) who permits the NGSC to extract amounts of naturalgas that may differ from the originally injected volumes thuscreating positive or negative imbalances The latter is a kind
of usual market practice that allows for a dynamic flow of thefuel within the natural gas supply chain
However since disrupting the system in this way impliesextra costs for the NGSC the company attempts to do itonly when its predictions of future market conditions showthat the total revenues overcome the losses incurred by thepenalization policy applied to the NGSC It is clear thatthe NGSC needs tools that provide it with the best possibleinformation and help it make advantage of the latter
The NGSC-POC system operates in the following way
(1) The NGSC makes a forecast of the demand it is likelyto have during the next period (month year etc) andconsiders different scenarios in which this can occur
(2) The NGSC books certain capacity 119863119888 for every poolzone and stage (day week month etc)
(3) For each subsequent stage the NGSC determines theamount of gas to extract and sell which possibly cre-ates positive and negative imbalances in the processthis continues until the period is over
(4) ThePOC studies the resulting last day imbalances andrearranges them according to certain business rules
(5) The POC charges the NGSC with certain penaltyfor the final (rearranged) imbalances The latter mayoccur to be negative that is the POC may pay to theNGSC
(6) The NGSC calculates the net profits as its salesrevenue minus the penalty
The resulting model is a bilevel multistage stochasticoptimization problem [63] in which the upper level decisionmaker (the leader) is the NGSC who has the objective ofmaximizing its net profit as the revenue from the sales of itsgas in the pipeline minus the penalty imposed by the POCThe lower level decisionmaker (the follower) is the POCwhoaims at minimizing the absolute value of the penalizationcash-out flow either from the POC to the NGSC or viceversaThe first stage of the stochastic problem corresponds tothe capacity booking made by the NGSC and these capacityvalues remain unchanged throughout the whole process Atthe next stages the decision variables are the daily extractionamounts (and hence the imbalances) unsatisfied demandand the penalty cash-outs imposed by the POC
Note that while the POC may appear as the party withmore influence in the system the NGSC is the leader of thebilevel problem The only reason why the NGSC is the upperlevel (leader) is because of the timing of the decision processIndeed it would seem logical that the POC enjoying strongercontrol over its own facilities has to abide to the decisions(regarding final day imbalances) that the NGSC has alreadymade This is because of the relative freedom that has beenawarded (in the current businessrsquo practice) to the NGSC increating imbalances to maintain healthy business in favor ofits customers
42 Stochastic Model In [62] the authors present a bilevelmultistage stochastic optimizationmodel which is developedto deal with a certain subsystem of the natural gas supply
Mathematical Problems in Engineering 7
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
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[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
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[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
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[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
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[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
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[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
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[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
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[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
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[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
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[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
and sufficient optimality conditions for bilevel programmingcompare Dempe [28]
Using the optimal value function 120593(119909) of the lower levelproblem (1) the bilevel programming problem (7) can bereplaced with
min119909119910
119865 (119909 119910) 119866 (119909) le 0 119892 (119909 119910) le 0
119891 (119909 119910) le 120593 (119909) 119909 isin 119883 (15)
This is the so-called optimal value transformationSince the optimal value function is nonsmooth even
under very restrictive assumptions this is a nonsmooth non-convex optimization problem Using nonsmooth analysissee for example Mordukhovich [29 30] and Rockafellarand Wets [31] optimality conditions for the optimal valuetransformation can be obtained compareOutrata [32] Ye andZhu [33] and Dempe et al [34]
Nowadays a large number of PhD theses have beenwritten on bilevel programming problems very differenttypes of (necessary and sufficient) optimality conditions canbe found in the literature the number of applications is hugeand both exact and heuristic solution algorithms have beensuggested
3 Linear Bilevel Programming Problems
The linear bilevel program is the problem of the followingstructure
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (17)
Here 119860 119861 and 119862 are matrices of sizes 119901 times 119899 119901 times 119898 and119899 times 119898 respectively and all variables and vectors used areof appropriate dimensions Note that we have used here theso-called optimistic bilevel optimization problem which isrelated to problem (7)
The so-called connecting constraints 119860119909 + 119861119910 le 119888are included in the upper level problem Validity of suchconstraints is beyond the selection of the leader and can beverified only after the follower has selected hisher (possiblynot unique) optimal solution In the case especially whenΨ(119909) does not reduce to a singleton certain difficulties mayarise In order to examine the bilevel programming problemin the case that Ψ(119909) does not reduce to a singleton Ishizukaand Aiyoshi [35] introduced their double penalty method Ingeneral connecting constraints may imply that the feasibleset of the bilevel programming problem is disconnectedThissituation is illustrated by the following example
Example 1 (Mersha and Dempe [36]) Consider the problem
119911 = minus119909 minus 2119910 997888rarr min119909119910 (18)
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 1The problem with upper level connecting constraintsThefeasible set is depicted with bold lines The point 119862 is global optimalsolution and point 119860 is a local optimal solution
0
10
x
A C
B
10
y minusx minus 2y rarr min minusy rarr min
Figure 2The problem when the upper level connecting constraintsare shifted into the lower level problem The feasible set is depictedwith bold lines The global optimal solution is point 119861
subject to
2119909 minus 3119910 ge minus12
119909 + 119910 le 14
119910 isin Argmin119910
minus119910 minus3119909 + 119910 le minus3 3119909 + 119910 le 30
(19)
The optimal solution for this problem is the point 119862 at(119909 119910) = (8 6) (see Figure 1) But if we shift the two upperlevel constraints to the lower level we get the point 119861 at(119909 119910) = (6 8) as an optimal solution (see Figure 2) Fromthis example one can easily notice that if we shift constraintsfrom the upper level to the lower one the optimal solutionobtained prior to shifting is not optimal any more in generalHence ideas based on shifting constraints from one levelto another will lead to a solution which may not solve theoriginal problem
In Example 1 the optimal solution of the lower levelproblem was unique for all 119909 If this is not the case feasibilityof a selection of the upper level decision maker possibly
Mathematical Problems in Engineering 5
depends on the selection of the follower In the optimisticcase this means that the leader selects within the set ofoptimal solutions of the followerrsquos problem one point whichis at the same time feasible for the upper level connectingconstraints and gives the best objective function value for theupper level objective function
As we can see in Example 1 the existence of connectingupper level constraints might lead in general to a dis-connected feasible set in the bilevel programming problemTherefore solution algorithms will live in one of the con-nected components of the feasible set (ie a sequence offeasible points which all belong to one of the connected partsis computed) or they need to jump from one of the connectedparts of the feasible set to another one Some ideas of discreteoptimization are needed in such cases
In order to avoid the above-mentioned difficulties someresearchers restrict themselves to the cases when the upperlevel constraints depend on the upper level variables only (iematrix 119861 is zero-matrix 119861 = 0) Thus the bilevel problem(16)-(17) reduces to a simpler one
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (21)
In this problem parametric linear optimization (see forexample Nozicka et al [37]) can be used to show that thegraph of the mapping Ψ(sdot) equals the connected union offaces of the set (119909 119910)⊤ 119862119910 le 119909
4 Application of Bilevel Programming toImbalance Cash-Out Problem
In the early 1990s several regulations were passed in the USAand the European Union [38 39] changing the way naturalgas was marketed and traded Particularly this liberalization[40] effectively ended a period in which natural gas was astate-driven industry The liberalization has also created theemergent natural gas markets as well as a strong demandfor models to better tackle the new problems and profit fromthis new setting [41 42] It is possible to say that the above-mentioned processes formed the natural gas supply chainThe resulting market configuration demanded the indepen-dence of the transportation and commercialization processesAs a result of this paradigm shiftmdashand the accompanyingrestructurization of the marketmdasha systematic analysis ofseveral new features becomes indispensable
Of particular interest is a problem that arises in thenatural gas supply chain namely that of balancing the fuelvolumes over a distribution network Such a balancing pro-cedure directly concerns the Pipeline Operating Company(POC) since the correct operation of the pipeline means thewell controlled volumes of the transported gasMoreover anynatural gas shipping company (NGSC) is also concernedwiththe balancing of the volumes because it is often impossible
to avoid an imbalance justified by certain economic reasonsA natural gas shipping companyrsquos business is to sell the gasby moving it through the pipeline to its clients it has tofulfill signed contracts first and then market excesses of thegas to achieve the maximum profits In order to do that theNGSC has to manage the volumes at each selling point (so-called pipeline meters) taking into account the balance theselling prices and the total revenue The basic mathematicalframework of this problemrsquos modeling is found in [43]
Owing not only to this liberalization but also to the newlocal conditions that aremore open to competition new smallplayers entered the natural gas industry especially at the localscale Indeed the USA has over 80 interstate long-distancepipelines [44] serving different regions with various climaticdemographic economic andpolitical circumstancesNaturalgas usage in Alabama for example intuitively is not the sameas in Oregon thus the market dynamics of the fuel are alsodifferent and this we presume should be reflected in someway in the econometric data of the states
Not only macroeconomic trends however are affectedby this setting When doing cross-regions studies of variousaspects of the supply chain such as the forecasting of demand[45 46] the balancing of the pipelines after imbalances havebeen created by the natural gas shippers [43 47 48] or thedynamics of interstate-intrastate systems [49] one has to takeinto account the existence of different markets
The existence of a common relationship between priceand consumption of natural gas across several zones allowsfor strong claims of uniformity which are useful when forexample we are building scenarios for a stochastic problemIndeed if we manage to group the regions in clusters withsimilar price and consumption functions we can reduce thenumber of variables needed in a scenario tree formulation[42 50]
It must be emphasized that while natural gas pipelinenetworks have been thoroughly studied most of the existentmodels focus on aspects of this part of the supply chain otherthan the NGSC-POC interaction in the system balancingsuch as network operation optimization [51 52] or deploy-ment of facilities [53] There are also papers considering thenatural gas supply chain in amultilevel scheme inwhich boththeNGSC and the POCare present and accounted for such asthe related [54 55] These works are remarkable in the sensethat they span thewhole supply chainwithmuch emphasis onthe traders (financial front-ends of the natural gas producers)so that there is little to nomention of imbalances in the systemresulting from the dealings of the NGSCs and the POCs eventhough both actors are present in the models
Many authors do acknowledge [56 57] the existence ofa problematic situation in the NGSC-POC system followingthe paradigm shift yet we have found very few sources thatexplain plausible ways in which this problem is nowadayssolved For example [58] shows how storage is required by theNGSC when no flexibility exists in the network volumeman-agement either because it is not allowed or because it is nottechnically possible Nevertheless balancing is an importantpart of the modern natural gas supply chain managementand to date no policy has been accepted as optimal regardingthe way in which the imbalances produced by the NGSC
6 Mathematical Problems in Engineering
are physically and economically handled Among the mostimportant tools that aid the POC in its task of restoring thebalance of the system is the arbitrage penalization policies inwhich the POC performs a maintenance redistribution of theimbalances in the system and charges the NGSC(s) for thecost of this operation
In [59] one finds a modeling framework (which we aregoing to follow) of the penalization part of this problemThispenalization refers only to the cash-out that occurs betweenthe NGSC and the POC it leaves outside any reference toactual market conditions which are obviously important tothe NGSC The paper presents a solution method through amodification of the original problem as well as the analysisof how this modification affects the objective function andthe obtained solutions In [47] the authors compare twoalgorithms that solve the problem making use of certainnumerical procedures In the present paper we adapt thesealgorithms to our extended model We also make use of theidea proposed in [43] to divide our problem into several gen-eralized transportation problems when finding its numericalsolution
In [60 61] we study a modified version of the above-described problem in which the upper level objective func-tion includes certain new terms based upon the net profitof the leadermdashthe natural gas shipping company This for-mulation assumes however the complete knowledge (perfectinformation) about the changes in the prices of natural gasduring the process which is somewhat nonrealistic and notquite useful as the resulting function does not clearly reflectthe reasons behind the actions of the NGSC
In [62] a stochastic reformulation of the problem ispresented so that the NGSC is now able to forecast thenext several values of the natural gas demand and then toplan the extraction of natural gas from the pipeline Theresultingmodel is a stochastic variation of the originalmixed-integer bilevel optimization problem for which two differentsolution methods are proposed and compared
To the best of our knowledge there is no literaturebeyond the works listed in the paragraphs above and theirderivatives that explicitly deals with the NGSC-POC sub-system in the same way we propose formulating a bileveloptimization problem out of the balancing operations Weattribute this to the relatively recent nature of the problemwe are dealing with as well as the difficulty of its accurateformulation for specific instances
41 Problem Specification Following the scheme constructedin [47 59] wewill consider a leader-follower system inwhichthe first agent (the leader) namely the Natural Gas ShippingCompany (NGSC) buys the gas at the wells arranges for itsinjection into an (interstate) pipeline at its starting point andextracts some amount of gasmdashideally equal to the depositedamountmdashfrom pipeline meters in several pool zones acrossthe country The follower here is the administrator of thepipeline which we call the Pipeline Operating Company(POC) who permits the NGSC to extract amounts of naturalgas that may differ from the originally injected volumes thuscreating positive or negative imbalances The latter is a kind
of usual market practice that allows for a dynamic flow of thefuel within the natural gas supply chain
However since disrupting the system in this way impliesextra costs for the NGSC the company attempts to do itonly when its predictions of future market conditions showthat the total revenues overcome the losses incurred by thepenalization policy applied to the NGSC It is clear thatthe NGSC needs tools that provide it with the best possibleinformation and help it make advantage of the latter
The NGSC-POC system operates in the following way
(1) The NGSC makes a forecast of the demand it is likelyto have during the next period (month year etc) andconsiders different scenarios in which this can occur
(2) The NGSC books certain capacity 119863119888 for every poolzone and stage (day week month etc)
(3) For each subsequent stage the NGSC determines theamount of gas to extract and sell which possibly cre-ates positive and negative imbalances in the processthis continues until the period is over
(4) ThePOC studies the resulting last day imbalances andrearranges them according to certain business rules
(5) The POC charges the NGSC with certain penaltyfor the final (rearranged) imbalances The latter mayoccur to be negative that is the POC may pay to theNGSC
(6) The NGSC calculates the net profits as its salesrevenue minus the penalty
The resulting model is a bilevel multistage stochasticoptimization problem [63] in which the upper level decisionmaker (the leader) is the NGSC who has the objective ofmaximizing its net profit as the revenue from the sales of itsgas in the pipeline minus the penalty imposed by the POCThe lower level decisionmaker (the follower) is the POCwhoaims at minimizing the absolute value of the penalizationcash-out flow either from the POC to the NGSC or viceversaThe first stage of the stochastic problem corresponds tothe capacity booking made by the NGSC and these capacityvalues remain unchanged throughout the whole process Atthe next stages the decision variables are the daily extractionamounts (and hence the imbalances) unsatisfied demandand the penalty cash-outs imposed by the POC
Note that while the POC may appear as the party withmore influence in the system the NGSC is the leader of thebilevel problem The only reason why the NGSC is the upperlevel (leader) is because of the timing of the decision processIndeed it would seem logical that the POC enjoying strongercontrol over its own facilities has to abide to the decisions(regarding final day imbalances) that the NGSC has alreadymade This is because of the relative freedom that has beenawarded (in the current businessrsquo practice) to the NGSC increating imbalances to maintain healthy business in favor ofits customers
42 Stochastic Model In [62] the authors present a bilevelmultistage stochastic optimizationmodel which is developedto deal with a certain subsystem of the natural gas supply
Mathematical Problems in Engineering 7
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
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[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
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[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
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[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
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[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
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[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
depends on the selection of the follower In the optimisticcase this means that the leader selects within the set ofoptimal solutions of the followerrsquos problem one point whichis at the same time feasible for the upper level connectingconstraints and gives the best objective function value for theupper level objective function
As we can see in Example 1 the existence of connectingupper level constraints might lead in general to a dis-connected feasible set in the bilevel programming problemTherefore solution algorithms will live in one of the con-nected components of the feasible set (ie a sequence offeasible points which all belong to one of the connected partsis computed) or they need to jump from one of the connectedparts of the feasible set to another one Some ideas of discreteoptimization are needed in such cases
In order to avoid the above-mentioned difficulties someresearchers restrict themselves to the cases when the upperlevel constraints depend on the upper level variables only (iematrix 119861 is zero-matrix 119861 = 0) Thus the bilevel problem(16)-(17) reduces to a simpler one
where Ψ(sdot) is the solution set mapping of the lower levelproblem
Ψ (119909) = Argmin119910
119889⊤119910 119862119910 le 119909 (21)
In this problem parametric linear optimization (see forexample Nozicka et al [37]) can be used to show that thegraph of the mapping Ψ(sdot) equals the connected union offaces of the set (119909 119910)⊤ 119862119910 le 119909
4 Application of Bilevel Programming toImbalance Cash-Out Problem
In the early 1990s several regulations were passed in the USAand the European Union [38 39] changing the way naturalgas was marketed and traded Particularly this liberalization[40] effectively ended a period in which natural gas was astate-driven industry The liberalization has also created theemergent natural gas markets as well as a strong demandfor models to better tackle the new problems and profit fromthis new setting [41 42] It is possible to say that the above-mentioned processes formed the natural gas supply chainThe resulting market configuration demanded the indepen-dence of the transportation and commercialization processesAs a result of this paradigm shiftmdashand the accompanyingrestructurization of the marketmdasha systematic analysis ofseveral new features becomes indispensable
Of particular interest is a problem that arises in thenatural gas supply chain namely that of balancing the fuelvolumes over a distribution network Such a balancing pro-cedure directly concerns the Pipeline Operating Company(POC) since the correct operation of the pipeline means thewell controlled volumes of the transported gasMoreover anynatural gas shipping company (NGSC) is also concernedwiththe balancing of the volumes because it is often impossible
to avoid an imbalance justified by certain economic reasonsA natural gas shipping companyrsquos business is to sell the gasby moving it through the pipeline to its clients it has tofulfill signed contracts first and then market excesses of thegas to achieve the maximum profits In order to do that theNGSC has to manage the volumes at each selling point (so-called pipeline meters) taking into account the balance theselling prices and the total revenue The basic mathematicalframework of this problemrsquos modeling is found in [43]
Owing not only to this liberalization but also to the newlocal conditions that aremore open to competition new smallplayers entered the natural gas industry especially at the localscale Indeed the USA has over 80 interstate long-distancepipelines [44] serving different regions with various climaticdemographic economic andpolitical circumstancesNaturalgas usage in Alabama for example intuitively is not the sameas in Oregon thus the market dynamics of the fuel are alsodifferent and this we presume should be reflected in someway in the econometric data of the states
Not only macroeconomic trends however are affectedby this setting When doing cross-regions studies of variousaspects of the supply chain such as the forecasting of demand[45 46] the balancing of the pipelines after imbalances havebeen created by the natural gas shippers [43 47 48] or thedynamics of interstate-intrastate systems [49] one has to takeinto account the existence of different markets
The existence of a common relationship between priceand consumption of natural gas across several zones allowsfor strong claims of uniformity which are useful when forexample we are building scenarios for a stochastic problemIndeed if we manage to group the regions in clusters withsimilar price and consumption functions we can reduce thenumber of variables needed in a scenario tree formulation[42 50]
It must be emphasized that while natural gas pipelinenetworks have been thoroughly studied most of the existentmodels focus on aspects of this part of the supply chain otherthan the NGSC-POC interaction in the system balancingsuch as network operation optimization [51 52] or deploy-ment of facilities [53] There are also papers considering thenatural gas supply chain in amultilevel scheme inwhich boththeNGSC and the POCare present and accounted for such asthe related [54 55] These works are remarkable in the sensethat they span thewhole supply chainwithmuch emphasis onthe traders (financial front-ends of the natural gas producers)so that there is little to nomention of imbalances in the systemresulting from the dealings of the NGSCs and the POCs eventhough both actors are present in the models
Many authors do acknowledge [56 57] the existence ofa problematic situation in the NGSC-POC system followingthe paradigm shift yet we have found very few sources thatexplain plausible ways in which this problem is nowadayssolved For example [58] shows how storage is required by theNGSC when no flexibility exists in the network volumeman-agement either because it is not allowed or because it is nottechnically possible Nevertheless balancing is an importantpart of the modern natural gas supply chain managementand to date no policy has been accepted as optimal regardingthe way in which the imbalances produced by the NGSC
6 Mathematical Problems in Engineering
are physically and economically handled Among the mostimportant tools that aid the POC in its task of restoring thebalance of the system is the arbitrage penalization policies inwhich the POC performs a maintenance redistribution of theimbalances in the system and charges the NGSC(s) for thecost of this operation
In [59] one finds a modeling framework (which we aregoing to follow) of the penalization part of this problemThispenalization refers only to the cash-out that occurs betweenthe NGSC and the POC it leaves outside any reference toactual market conditions which are obviously important tothe NGSC The paper presents a solution method through amodification of the original problem as well as the analysisof how this modification affects the objective function andthe obtained solutions In [47] the authors compare twoalgorithms that solve the problem making use of certainnumerical procedures In the present paper we adapt thesealgorithms to our extended model We also make use of theidea proposed in [43] to divide our problem into several gen-eralized transportation problems when finding its numericalsolution
In [60 61] we study a modified version of the above-described problem in which the upper level objective func-tion includes certain new terms based upon the net profitof the leadermdashthe natural gas shipping company This for-mulation assumes however the complete knowledge (perfectinformation) about the changes in the prices of natural gasduring the process which is somewhat nonrealistic and notquite useful as the resulting function does not clearly reflectthe reasons behind the actions of the NGSC
In [62] a stochastic reformulation of the problem ispresented so that the NGSC is now able to forecast thenext several values of the natural gas demand and then toplan the extraction of natural gas from the pipeline Theresultingmodel is a stochastic variation of the originalmixed-integer bilevel optimization problem for which two differentsolution methods are proposed and compared
To the best of our knowledge there is no literaturebeyond the works listed in the paragraphs above and theirderivatives that explicitly deals with the NGSC-POC sub-system in the same way we propose formulating a bileveloptimization problem out of the balancing operations Weattribute this to the relatively recent nature of the problemwe are dealing with as well as the difficulty of its accurateformulation for specific instances
41 Problem Specification Following the scheme constructedin [47 59] wewill consider a leader-follower system inwhichthe first agent (the leader) namely the Natural Gas ShippingCompany (NGSC) buys the gas at the wells arranges for itsinjection into an (interstate) pipeline at its starting point andextracts some amount of gasmdashideally equal to the depositedamountmdashfrom pipeline meters in several pool zones acrossthe country The follower here is the administrator of thepipeline which we call the Pipeline Operating Company(POC) who permits the NGSC to extract amounts of naturalgas that may differ from the originally injected volumes thuscreating positive or negative imbalances The latter is a kind
of usual market practice that allows for a dynamic flow of thefuel within the natural gas supply chain
However since disrupting the system in this way impliesextra costs for the NGSC the company attempts to do itonly when its predictions of future market conditions showthat the total revenues overcome the losses incurred by thepenalization policy applied to the NGSC It is clear thatthe NGSC needs tools that provide it with the best possibleinformation and help it make advantage of the latter
The NGSC-POC system operates in the following way
(1) The NGSC makes a forecast of the demand it is likelyto have during the next period (month year etc) andconsiders different scenarios in which this can occur
(2) The NGSC books certain capacity 119863119888 for every poolzone and stage (day week month etc)
(3) For each subsequent stage the NGSC determines theamount of gas to extract and sell which possibly cre-ates positive and negative imbalances in the processthis continues until the period is over
(4) ThePOC studies the resulting last day imbalances andrearranges them according to certain business rules
(5) The POC charges the NGSC with certain penaltyfor the final (rearranged) imbalances The latter mayoccur to be negative that is the POC may pay to theNGSC
(6) The NGSC calculates the net profits as its salesrevenue minus the penalty
The resulting model is a bilevel multistage stochasticoptimization problem [63] in which the upper level decisionmaker (the leader) is the NGSC who has the objective ofmaximizing its net profit as the revenue from the sales of itsgas in the pipeline minus the penalty imposed by the POCThe lower level decisionmaker (the follower) is the POCwhoaims at minimizing the absolute value of the penalizationcash-out flow either from the POC to the NGSC or viceversaThe first stage of the stochastic problem corresponds tothe capacity booking made by the NGSC and these capacityvalues remain unchanged throughout the whole process Atthe next stages the decision variables are the daily extractionamounts (and hence the imbalances) unsatisfied demandand the penalty cash-outs imposed by the POC
Note that while the POC may appear as the party withmore influence in the system the NGSC is the leader of thebilevel problem The only reason why the NGSC is the upperlevel (leader) is because of the timing of the decision processIndeed it would seem logical that the POC enjoying strongercontrol over its own facilities has to abide to the decisions(regarding final day imbalances) that the NGSC has alreadymade This is because of the relative freedom that has beenawarded (in the current businessrsquo practice) to the NGSC increating imbalances to maintain healthy business in favor ofits customers
42 Stochastic Model In [62] the authors present a bilevelmultistage stochastic optimizationmodel which is developedto deal with a certain subsystem of the natural gas supply
Mathematical Problems in Engineering 7
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
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[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
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[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
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[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
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[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
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[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
are physically and economically handled Among the mostimportant tools that aid the POC in its task of restoring thebalance of the system is the arbitrage penalization policies inwhich the POC performs a maintenance redistribution of theimbalances in the system and charges the NGSC(s) for thecost of this operation
In [59] one finds a modeling framework (which we aregoing to follow) of the penalization part of this problemThispenalization refers only to the cash-out that occurs betweenthe NGSC and the POC it leaves outside any reference toactual market conditions which are obviously important tothe NGSC The paper presents a solution method through amodification of the original problem as well as the analysisof how this modification affects the objective function andthe obtained solutions In [47] the authors compare twoalgorithms that solve the problem making use of certainnumerical procedures In the present paper we adapt thesealgorithms to our extended model We also make use of theidea proposed in [43] to divide our problem into several gen-eralized transportation problems when finding its numericalsolution
In [60 61] we study a modified version of the above-described problem in which the upper level objective func-tion includes certain new terms based upon the net profitof the leadermdashthe natural gas shipping company This for-mulation assumes however the complete knowledge (perfectinformation) about the changes in the prices of natural gasduring the process which is somewhat nonrealistic and notquite useful as the resulting function does not clearly reflectthe reasons behind the actions of the NGSC
In [62] a stochastic reformulation of the problem ispresented so that the NGSC is now able to forecast thenext several values of the natural gas demand and then toplan the extraction of natural gas from the pipeline Theresultingmodel is a stochastic variation of the originalmixed-integer bilevel optimization problem for which two differentsolution methods are proposed and compared
To the best of our knowledge there is no literaturebeyond the works listed in the paragraphs above and theirderivatives that explicitly deals with the NGSC-POC sub-system in the same way we propose formulating a bileveloptimization problem out of the balancing operations Weattribute this to the relatively recent nature of the problemwe are dealing with as well as the difficulty of its accurateformulation for specific instances
41 Problem Specification Following the scheme constructedin [47 59] wewill consider a leader-follower system inwhichthe first agent (the leader) namely the Natural Gas ShippingCompany (NGSC) buys the gas at the wells arranges for itsinjection into an (interstate) pipeline at its starting point andextracts some amount of gasmdashideally equal to the depositedamountmdashfrom pipeline meters in several pool zones acrossthe country The follower here is the administrator of thepipeline which we call the Pipeline Operating Company(POC) who permits the NGSC to extract amounts of naturalgas that may differ from the originally injected volumes thuscreating positive or negative imbalances The latter is a kind
of usual market practice that allows for a dynamic flow of thefuel within the natural gas supply chain
However since disrupting the system in this way impliesextra costs for the NGSC the company attempts to do itonly when its predictions of future market conditions showthat the total revenues overcome the losses incurred by thepenalization policy applied to the NGSC It is clear thatthe NGSC needs tools that provide it with the best possibleinformation and help it make advantage of the latter
The NGSC-POC system operates in the following way
(1) The NGSC makes a forecast of the demand it is likelyto have during the next period (month year etc) andconsiders different scenarios in which this can occur
(2) The NGSC books certain capacity 119863119888 for every poolzone and stage (day week month etc)
(3) For each subsequent stage the NGSC determines theamount of gas to extract and sell which possibly cre-ates positive and negative imbalances in the processthis continues until the period is over
(4) ThePOC studies the resulting last day imbalances andrearranges them according to certain business rules
(5) The POC charges the NGSC with certain penaltyfor the final (rearranged) imbalances The latter mayoccur to be negative that is the POC may pay to theNGSC
(6) The NGSC calculates the net profits as its salesrevenue minus the penalty
The resulting model is a bilevel multistage stochasticoptimization problem [63] in which the upper level decisionmaker (the leader) is the NGSC who has the objective ofmaximizing its net profit as the revenue from the sales of itsgas in the pipeline minus the penalty imposed by the POCThe lower level decisionmaker (the follower) is the POCwhoaims at minimizing the absolute value of the penalizationcash-out flow either from the POC to the NGSC or viceversaThe first stage of the stochastic problem corresponds tothe capacity booking made by the NGSC and these capacityvalues remain unchanged throughout the whole process Atthe next stages the decision variables are the daily extractionamounts (and hence the imbalances) unsatisfied demandand the penalty cash-outs imposed by the POC
Note that while the POC may appear as the party withmore influence in the system the NGSC is the leader of thebilevel problem The only reason why the NGSC is the upperlevel (leader) is because of the timing of the decision processIndeed it would seem logical that the POC enjoying strongercontrol over its own facilities has to abide to the decisions(regarding final day imbalances) that the NGSC has alreadymade This is because of the relative freedom that has beenawarded (in the current businessrsquo practice) to the NGSC increating imbalances to maintain healthy business in favor ofits customers
42 Stochastic Model In [62] the authors present a bilevelmultistage stochastic optimizationmodel which is developedto deal with a certain subsystem of the natural gas supply
Mathematical Problems in Engineering 7
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
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[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
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[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
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[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
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[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
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[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
chain While former models were focused on the arbitragepolicies in a deterministic setting here we have expandedthe problem to include such elements as gas sales andbooking costs and added a stochastic framework to modelthe uncertainty in demand and prices faced by the upper leveldecision maker (the leader)
The developed model was implemented numerically andcompared to the perfect information solution (PIS) and theexpected value solutions (EVS) Experimental findings showthat 19 of the 21 instances deliver implementation values ofover half of the PIS whereas only one of the EVS implemen-tation values has a relative error below 075 The stochasticsolution implementation values are better than those of theEVS values in all but one casemdashwhich corresponds to thesimplest instance testedmdashwhich testifies in favour of ourapproach The performed linear reformulation also provedadvantageous as solving the original model with nonlinearlevels takes considerably longer time and does not providebetter solutions after up to 10 hours of running time in 20of the 21 experiments
Future work includes assessing the convenience of usingheuristic approaches for solving the lower level (as opposed tousing a specialized linear solver) and reformulating the linearlower level in the form of its duality conditions adding theseto the upper level to solve a single-level problem instead of abilevel one It is also worthwhile to study these models underdifferent time series not showing seasonality is also plannedas it is the implementation of a rolling horizon approach toremedy the lack of accuracy over long-period problems (suchas problem B011 involving 28 periods)
43 Penalty Function Method Paper [64] studies a specialbilevel programming problem that arises from the dealings ofa Natural Gas Shipping Company (NGSC) and the PipelineOperator Company (POC) with facilities of the latter usedby the former Because of the business relationships betweenthese two actors the timing and objectives of their decision-making process are different and sometimes even opposed
In order to model that bilevel programming was tra-ditionally used in the above-mentioned works Later theproblem was expanded and theoretically studied to facilitateits solution this included extension of the upper level objec-tive function linear reformulation heuristic approaches andbranch-and-bound techniques
In this paper the authors presented a linear programmingreformulation of the latest version of the model which is sig-nificantly faster to solve when implemented computationallyMore importantly this new formulation makes it easier totheoretically analyze the problem allowing one to draw someconclusions about the nature of the solution of the modifiedproblem
When aNGSC and a POC engage in a contract the result-ing dynamics may be subject to multilevel programminganalysis In this work an inexact penalization approach (IPA)was developed to solve the related bilevel linear programmingproblem in which the NGSC is the upper level decisionmaker and tries to maximize its earnings In the meantimethe POC is the lower level decisionmaker trying to minimize
the cash-out between both parties while balancing thepipeline network to guarantee an adequate operation of thelatter
The IPA algorithm is adapted to the linearized versions ofthe problems found in [65] and theoretical work is thenmadeto demonstrate the convergence of this solution method
Combining the inexact penalization approach and amodified Nelder-Mead simplex algorithm has resulted in afast and efficient enough optimization scheme in which newiterations are generated corrected and then evaluated foroptimality To summarize the numerical experiments theIPMNMapproachworks considerably better than both directimplementations and IPA versionswithout linearizationThismakes a support for our linearization attempts as well as forthe advantageous usage of the IPA algorithms developed in[47] Altogether numerical results concerning the runningtime convergence and optimal values are presented andcompared to previous reports showing a significant improve-ment in speedwithout actual sacrifice of the solutionrsquos quality
In conclusion it is possible to believe that the newsolution speed achieved allows one to reach a quick andmorefrequent balancing Indeed the more accurate the solution isthe more dynamic and successful the industryrsquos response tomarket necessities will be
5 Reduction of Upper Level Dimension inBilevel Programming Problem
As we have already seen from the previous sections bilevelprogrammingmodeling is a new and dynamically developingarea of mathematical programming and game theory Forinstance when we study value chains the general rule usuallyis that decisions are made by different parties along thechain and these parties have often different even opposedgoals This raises the difficulty of supply chain analysisbecause regular optimization techniques (eg like linearprogramming) cannot be readily applied so that tweaks andreformulations are often needed (cf [59])
The latter is exactly the case with the Natural GasValue Chain From extraction at the wellheads to the finalconsumption points (households power plants etc) naturalgas goes through several processes and changes ownershipmany a time
Bilevel programming is especially relevant in the caseof the interaction between a Natural Gas Shipping Com-pany (NGSC) and a Pipeline Operating Company (POC)The first one owns the gas since the moment it becomesa consumption-grade fuel (usually at wellheadrefinementcomplexes from now onward called the extraction points)and sells it to Local Distributing Companies (LCD) whoown small city-size pipelines that serve final costumersTypically NGSCs neither engage in business with end-usersnor actually handle the natural gas physically
Whenever the volumes extracted by the NGSCs differfrom those stipulated in the contracts we say an imbalanceoccurs Since imbalances are inevitable and necessary in ahealthy industry the POC is allowed to apply control mech-anisms in order to avoid and discourage abusive practices
8 Mathematical Problems in Engineering
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
(the so-called arbitrage) on part of the NGSCs One ofsuch tools is cash-out penalization techniques after a givenoperative period Namely if a NGSC has created imbalancesin one or more pool zones then the POC may proceedto ldquomoverdquo gas from positive-imbalanced zones to negative-imbalanced ones up to the point where every pool zone hasthe imbalance of the same sign that is either all nonnegativeor all nonpositive thus rebalancing the networkAt this pointthe POCwill either charge the NGSC a higher (than the spot)price for each volume unit of natural gas withdrawn in excessfrom its facilities or pay back a lower (than the sale) price ifthe gas was not extracted
Prices as a relevant factor induce us into the areaof stochastic programming instead of the deterministicapproach The formulated bilevel problem is reduced to thealso bilevel one but with linear constraints at both levels (cf[62]) However this reduction involves introduction of manyartificial variables on the one hand and generation of a lotof scenarios to apply the essentially stochastic tools on theother hand The latter makes the dimension of the upperlevel problem simply unbearable burden even for the mostmodern and powerful PC systems First attempts to diminishthe number of decision variables were made by the authors in[66 67]
The aim of chapters [68 69] is a mathematical formal-ization of the task of reduction of the upper level problemrsquosdimension without affecting (if possible) the optimal solu-tion of the original nonlinear bilevel programming prob-lem Under a couple of quite reasonable assumptions aboutthe data of the original bilevel programming problem theauthors of [68 69] established that the modified problemobtained by translating part of upper level variables to thelower level and replacing the original lower level programwith an appropriate equilibrium problem will have the samesolution set as the original bilevel program
Abitmore specialized and profound results were deducedin [68] for the linear bilevel program bymaking use of certaintools from the previous works [70ndash74] As paradoxicallyit could sound in the linear case the problem is muchmore complicated Indeed the uniqueness of a generalizedNash equilibrium (GNE) at the lower level of is much toorestrictive a demand As was shown by Rosen [72] theuniqueness of a so-called normalized GNE is rather morerealistic assumptionThis idea was further developed later bymany authors including the authors of [69 73]
Following the line proposed in [72] the authors of [69]introduce and study the concept of normalized generalizedNash equilibrium (NGNE) defined similarly to the conceptfrom [72] Based upon the revealed properties of such a entitythey establish the existence and uniqueness results for thelower level problem Hence the coincidence of the solutionsets of the original bilevel (linear or nonlinear) program andthe modified model obtained by the translation of part ofvariables from the upper to the lower level is demonstrated
To conlcude chapters [68 69] deal with an interestingproblem of reducing the number of variables at the upperlevel of bilevel programming problems The latter problemsare widely used to model various applications in particularthe natural gas cash-out problems described in [59 62] To
solve these problems with stochastic programming tools it isimportant that part of the upper level variables be governedat the lower level to reduce the number of (upper level)variables which are involved in generating the scenario trees
The chapters present certain preliminary results recentlyobtained in this direction In particular it has been demon-strated that the desired reduction is possible when the lowerlevel optimal response is determined uniquely for each vectorof upper level variables In [69] the necessary base forsimilar results is prepared for the general case of bilevelprograms with linear constraints when the uniqueness of thelower level optimal response is quite a rare case However ifthe optimal response is defined for a fixed set of Lagrangemultipliers then it is possible to demonstrate (following theideas and techniques from [72]) that the so-called normalizedNash equilibrium is uniqueThe latter gives one a hope to getthe positive results about reducing the dimension of the upperlevel problem without affecting the solution of the originalbilevel programming problem
6 Allocation Models as BilevelProgramming Problems
Bilevel programming has also served as a suitable optionfor modeling allocation problems where two-hierarchizedlevels with different objectives are involved At each levelthe decision maker aims to optimize his own interest Thepredefined existing hierarchy allows that the upper level hascomplete information about the lower levelrsquos decision on theallocation but not on the vice versa manner In particularbilevel programming offers a convenient framework fordealing with the allocation problems
An important and very common problem that appears inthese kinds of situations is the allocation of resources or theallocation of parties in the whole process considered Hencewe are going to divide this literature review in two directionsfirst the previous works done where the optimal allocationof resources are described and then the papers related tooptimally allocate customers distribution centers plants orother parties involved in a specific supply chain are refereed
61 Bilevel Allocation of Resources When considering a com-panyrsquos personnel and workers as limited resources we couldmention paper [75] where the main department boastingseveral branching divisions needs to allocate the personnel(workers technicians and management personnel) for thecompanyrsquos tasks The leader intends to maximize its benefitby allocating the specific workers to the divisions whilethe followers aim to maximize their own benefits using theassigned personnel The authors of [75] solved the proposedmodel by applying a simulation bionic algorithm The mainissue is that they did not make any conclusions about thequality of the thus obtained solution due to the complexityinherent to the bilevel model
In [76] theminimum total time for finishing jobs in a sys-tem is sought In that problem the leader is the job schedulerwho tries to optimize the system performance by allocatingthe workers to the machines On the other hand the follower
Mathematical Problems in Engineering 9
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
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[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
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[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
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[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
is represented by many noncooperative workers seeking touse a set of commonmachinesminimizing the latency of theirwork scheduleThree polynomial-time algorithms for solvingthe problem are proposed and complexity results are givendemonstrating that this problem is NP-hard
Next one can find a plenty of papers devoted to theanalyzis of the allocation of water to different regions of theworld For example in [77] a nonlinear bilevel programmingmodel with fuzzy random variables for distributing (inan equitable way) the water in a region is studied Thewhole community (society) is seen as the leader and thefollowers are seen as the subareas contained in the regionBoth decision levels strive to maximize their economic gainThe authors of [77] proposed a hybrid heuristic based onan interactive fuzzy programming technique and a geneticalgorithm Also an application to a real case study wasmade showing the reasonable performance of the developedsolution method
Paper [78] examines a similar situation a bilevel mul-tiobjective linear programming model is considered It isimportant to note that the lower level problem containsmultiple objective functions The leader has to allocate theamount ofwater destined for irrigation industry domesticityand ecology in order to maximize the benefit for the regionThen the follower optimizes its gain using the water resourcedoomed for each purpose The problem is solved by usingfuzzy goal programming in the upper level and a tolerance-based approach in the lower level Their model and method-ology was validated in a case study from China In [79] morereferences concerning this particular topic can be found (inJapanese)
Another interesting application is about housing alloca-tion In [80] this problem for a continuum transportationsystem is analyzed The leader selects the optimal housingdevelopment pattern while the follower decides about theallocation of the houses based on their renting and travelcostsThe lower level problem is defined by a set of differentialequations and it is solved by the finite element method Theresults obtained from numerical experimentation show thatthe algorithm seems to be efficient enough An extension ofthe previous work was done in [81] The main difference isin that the leader optimizes the housing allocation in order toachieve theminimumCO2 emissions while the followers aimis to find the equilibrium among the users in a transportationsystem The authors of [81] also adapted the finite elementmethod and proposed two alternative solution algorithmsbased on the Newton-Raphson procedure and the convexcombination approach The computational tests showed thattraffic intensity CO2 emissions and transport demand arebalanced along with the best housing allocation
Bilevel programs related to the optimal allocation of aspecific product can also be found in the literature Forexample [82] presents a problem where a company marketsproducts and allocate resources to two producer factoriesthat consume the resources Hence the model can be viewedand treated as a Stackelberg equilibrium problem becausein the lower level both followers compete for the commonallocated resources trying to optimize their own criteria Ahybrid intelligent algorithm based on fuzzy simulation as
well as neural network and genetic algorithms are proposedfor solving this bilevel problem
Wang and Lootsma [83] introduced a bilevel model forthe case when the general manager tries to allocate resourcesamong the different departments of the company In theupper level the correct allocation of the resources to thedepartments is made in order to maximize the companyrsquostotal revenue On the other hand in the lower level eachdepartment estimates its own benefit generated with theallocated resources A numerical example is given to illustratethe proposed exact method
As we mentioned before bilevel programming allows arealistic mathematical modeling for a very wide applicationareas We are going to confirm this fact with the work doneby Burgard et al [84] where a genomic problem is addressedIn that problem the leader maximizes the bioengineeringobjective that is the chemical production and the followeroptimizes the flux allocation based on the biomass generatedthrough the gene deletions
62 Bilevel Allocation for the Supply Chain Models It iswell known that supply chains involve many components inthe whole process At some point of the supply chain anallocation is required for example to allocate customers toplants demanded zones to distribution centers vehicles toproducers and so forthUnder this scheme Calvete et al[85] introduced a production-distribution bilevel problemin which a company (the leader) is dedicated exclusively tothe allocation of customers to distribution centers satisfyingtheir demand of products Another company (the follower) isdoomed to produce these productsThe leader will distributethe products and purchase them from some plants andthen the distribution centers will transport them to theircustomers meeting their requirements in order to minimizethe distribution costs On the other hand the follower decidesits own production plan based on the production capacity ofthe plants and by considering the requirements of the demandgrouped in the distribution centers seeking to minimizethe operation costs The authors of [85] considered a realcase from a company in Spain and also some benchmarkinstances Furthermore they solved this problem by usinga heuristic algorithm based on an ant colony optimizationmethod delivering pretty good quality solutions in a reason-able time
Also Legillon et al [86] considered the same problemproposing a coevolutionary algorithm without improvingthe solution quality given in the seminal paper Camacho-Vallejo et al [87] developed a method based on scattersearch obtaining the best known results for the benchmarkinstances In [88] a single-commodity multiplant networkwith multiple depots is studiedThe leader seeks to minimizethe total cost (ie the cost associated with the distributionfrom the plants to depots and then to the customers plusthe warehousing costs and the operation costs of the depots)of locating depots and allocating customers to them Thefollower intends to balance the workload of the systemimproving the customer service and finding a trade-offbetween cost and efficiency A standard genetic algorithmwasproposed [88] in order to solve some randomly generated test
10 Mathematical Problems in Engineering
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
problems with demonstrating certain opportunity areas forimproving its performance
Humanitarian logistics have given rise to application ofbilevel programming frameworks for dealing with situationsthat appear in that area Feng and Wen [89] consideredthe bilevel program where an earthquake affected the localtransportation network Here the leader tries to maximizethe flow of vehicles entering the affected area to provideassistance whereas the followers attempt to travel throughan unaffected route to minimize their total travel time Sincethis situation generates traffic jams andnegatively impacts therecovery and relief efforts a government agency regulates theuse of existing roads In order to solve the proposed modela genetic algorithm was implemented and validated in a casestudy showing that this algorithm is an effective tool to solvethe problem in question
In their turn Wang et al [90] proposed a model forlocating storage centers and allocating the sent aid Theleader minimizes the cost of locating the storage centersallocation of sent aid distribution and penalties while thefollower (an affected community) optimizes its own costbased on the resources allocated to each community Asmall test instance was created for testing the developedparticle swarm optimization algorithm showing the ease ofits implementation
Similar to the models discussed above Sun et al [91]seek an optimal decision about locating distribution centersby the search of an equilibrium among the customersrsquo costsThe leader will locate new distribution centers to minimizefixed and variable costs while meeting the demand by aset of customers In its turn the follower will allocate thecustomers to the distribution centers so as to minimize thecost of meeting their demand An algorithm that exploits thespecial structure of the lower level problem and a branch-and-bound (BampB) scheme in the upper level is proposed to dealwith this bilevel program In a different context (but with asimilar structure) Xu andWei [92]modeled a problem relatedto the waste management of constructions and demolitionsThe government is the leader that has to make the decisionabout locating the waste collection depots and processingcenters The administrators of different construction wastemanagement systems control the allocation of thewaste to thelocated facilities Both objectives functions minimize theirown costs in a fuzzy random environment An improvedparticle swarm optimization algorithm was designed to treatand solve the latter problem
It seems that facility location and customersrsquo allocationrequirements can be effectively modeled with bilevel pro-gramming when taking into account the customersrsquo demandat the facilities that will serve them Various papers in whichthe customers are allocated to the facilities according toa predetermined list of preferences can be found in theliterature see [93ndash96] In all those papers the facility locationproblem under customersrsquo preferences is studied In thebilevel program induced the leader has to locate some facili-ties while the follower will allocate the customers optimizingtheir preestablished preferences towards the facilities Thefirst three papers (ie [93ndash95]) developed valid two-sidedbounds for the objective functions invloved in this problem
and the last two works (ie [95 96]) implemented heuristicalgorithms to process the bilevel model
Moreover competitive facility location models have beenapproached with bilevel programming too In that problemtwo competing firms have to locate some facilities in orderto capture the maximum demand of the existing customersWith an aim to classify the problem as a bilevel programa hierarchy among the firms must exist in the model Alot of variations of these models have been published Thedifferentiation relies on the customersrsquo behavior for examplethe customers may be allocated to the facilities based ona predefined criterion such as the shortest distance a listof preferences preestablished contracts or in a randomway Another important factor is the characteristics of thecompeting firms for instance (i) if they have an exactnumber of facilities to be located that is (119903 | 119901)-centroidproblem (ii) whether one firm already has located facilitiesand the other firm has to locate new ones that is (119903 119883119901)-medianoid problemThe existence of facilities the possibilityof closing some ormake themmore attractive and so forthmdashall them are the issues that are addressed in these modelsIt is important to note that in competitive facility locationproblems neither the leader nor the follower will makethe decision of the customersrsquo allocation but this allocationimplicitly appears in the process and clearly affects bothlevelsrsquo decisions The reader is referred to [97ndash106] in orderto have a closer look to particular models in this area
Further the design of telecommunication networks hasalso been analyzed as a bilevel programming scheme Aproblemwithin this area is the one studied by Kim et al [107]in which the topological design of a local area network isproposedThe problem consists of allocating users to clustersand the union of clusters by bridges in order to obtaina minimum response time network boasting at the sametime the minimum connection costs Therefore the decisionconcerning the optimal allocation of users to clusters will bemade by the leader while the follower will make the decisionabout connecting all the clusters by forming a spanning treeThe authors [107] applied a coevolutionary genetic algorithmbased on Nash equilibrium to solve the problem
Finally optimization in ports has also attracted theattention of researches and found applications of bilevelprogramming compare Lee et al [108] where a problemfor scheduling berth and quay cranes is studied In thatproblem the leader deals with the berth allocation problemminimizing the sum of waiting and handling times of eachvessel On the other hand the follower solves the quay cranescheduling problem in order to minimize the total time untilall the vessels and the quay cranes have finished up theiractivities Owing to the difficulty of the exact solution of thisbilevel problem a genetic algorithm that finds reasonablequality solutions is proposed in [108]
7 Information Protection and CybersecurityProblems as Bilevel Programs
The methods and approaches solving bilevel programmingproblems also are actual in the areas of information
Mathematical Problems in Engineering 11
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
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[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
protection and cybersecurity However the solutions inthese similar areas have some special features especiallyconcerning certain cryptographic applications One of thesefeatures is the followingThere exists a standard (single-level)mathematical formalization of the cryptographic problembut it has been shown to bear some flawsThus the proposedmethods and approaches based on the bilevel programmingtechniques helps eliminate those deficiences and enhance theprocessing of problems of the information protection andcybersecurity on the new quality level
At the same time the problems of information protec-tion and cybersecurity clealy lack good interpretations withthe help of the bilevel programming apparatus Thereforethis section of the survey is presenting the first-time andoriginal review of the possible treatment of these importantinformation protection and cybersecurity problems as bilevelprograms
71 Some Cryptographic Applications One of the urgentproblems of public key cryptosystem improvements is theincrease of the qulity of software performance and hardwareimplementations One of the approaches helping improve thefunctioning of cryptosystems is marking up the performanceof finite field arithmetic concerning operations of multipli-cation A possible way to do is to widely apply the bilevelprogramming techniques
As the well-known publications show (cf [109ndash112] tomention only few) the most effective multiplication algo-rithms have been provided by Comba [109] and Karatsuba[112] However Combarsquos algorithm shows somewhat betterresults in numerous rendition (benchmark) tests of soft-ware implementations on modern platforms The combinedKaratsuba-Comba multiplication (KCM) algorithm for pro-cessors of the reduced instruction set computers (RISC-processors) is described in paper [113]
The KCM-algorithm is an example of a promising com-bination of those by Comba and Karatsuba while Karatsubarsquosalgorithm is especially often used for the machine wordmultiplication As a result the main goal of that paper [113]is to provide a suggestion for the effective increasing ofsoftware implementation of the finite field 119866119865(119901) multipli-cation (squaring) with the aid of Combarsquos algorithm Suchresearchwasmotivated by the necessity to obtain the effectiveconfirmation of software implementation of some knownalgorithms for continuous development of the modern 32-bit and 64-bit platforms It is important to mention that thelast ten years have seen a rapid development of multicoreprocessors and multiprocessor systems [113]
72 Software Implementation With the recent boost of infor-mation technology in modern society the problem of infor-mation security became of special urgencyThemost difficulttask is to provide secure handling and storage of criticaland confidential data for government and private companiesbanks and other systems A solution to this problem is toimplement systems that provide for information confiden-tiality integrity authenticity and accessibility by means of
cryptographic software and cryptographic hardware based onsome approaches making use of bilevel programming
At the same time cryptoanalyticalmethods taking advan-tage of the progress in capabilities of modern computersdemand high requirements on the security parameters ofmodern cryptosystems with the use of the well-knowntechniques and devices of bilevel programming Moreoverthe increased amount of data processed in modern infor-mation systems needs a quite high-level performance ofthe modern cryptosystems Hence the timing requirementsto cryptographic applications have increased dramaticallythat is prospective cryptoalgorithms must provide efficientprocessing of bulk data when applying bilevel programmingand at the same time a high level of security
So far most research activity has been carried outabout some theoretical aspects of hyperelliptic curve cryp-tosystems (HECC) including many improvements of theunderlying arithmetic of the hyperelliptic curves On theimplementation side improvements for specific processorsand hardware platforms have been analyzed The presentapproach provides a very important contribution towardspractical implementation of HECC by showing how to buildan efficient hyperelliptic curve of digital signature algorithm(HECDSA) implementation and provides cryptographicallysuitable curves Unfortunately the published results on prac-tical implementations of HECC are rare (see for example[114 115]) This solution is intended to provide very practicalfacts for the implementation of an HECDSA system with allits necessary details at the interpretation with the help of thebilevel programming techniques There are numerous mod-ern publications dealing with HECC but they describe novalidated system parameters for the efficient implementationof a workable cryptosystem
The lack of publications dedicated to this topic was themotivation behind the thorough summary of all results forefficient HECC implementation presented in this reviewand the comparison of HECC (HECDSA) with the existingelliptic curve cryptosystems (ECC) andor elliptic curve ofdigital signature algorithms (ECDSA) based on the use ofsome bilevel programming methods
73 Cybersecurity Applications The bilevel formulation isinvestigated through a problem in which the goal of thedestructive agent is to minimize the number of power systemcomponents that must be destroyed in order to cause aloss of load greater than or equal to a specied level Thisgoal is tempered by the logical assumption that followinga deliberate outage the system operator will implement allfeasible corrective actions to minimize the level of systemload shed
The resulting nonlinear mixed-integer bilevel program-ming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the lowerlevel optimization problem with its Karush-Kuhn-Tucker(KKT) optimality conditions and also converting a numberof nonlinearities to linear equivalents using somewell-knowninteger algebra results The equivalent formulation has beentested in [116] on two case studies including the 24-bus IEEE
12 Mathematical Problems in Engineering
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
reliability test system (RTS) through the use of commerciallyavailable software
The bilevel model specically allows one to dene differentobjective functions for the terrorist and the system opera-tor and permits to impose constraints on the upper leveloptimization problem The latter are functions of both theupper and lower level variables This degree of exibility is notpossible to implement through the existingmax-minmodels
As present researchers have begun to look into somenewways of addressing the security assessment problem herecalled the Terrorist Threat Problem (TTP) For example in[117] a multiagent system was proposed capable of assessingpower system vulnerability monitoring hidden failures ofprotection devices and providing adaptive control actionsto prevent catastrophic failures and cascading sequences ofevents
Attack tree (AT) is another widely used combinatorialmodel in the cybersecurity analysis The basic formalism ofAT does not take into account defense mechanisms Defensetrees (DT) have been developed to investigate the effect ofdefense mechanisms using measures such as attackerrsquos costand security cost return on investment (ROI) and returnon attack (ROA) DT however places defense mechanismsonly at the leaf node level while the corresponding ROIROAanalysis does not incorporate the probability of attack In anattack response tree (ART) an attacker-defender game wasused to find an optimal policy from the countermeasuresrsquopool The latter suffers from the problem of state-spaceexplosion since a solution in ART is sought by means ofa partially observable stochastic game model In [118] theauthors have presented a novel attack tree named the attackcountermeasure tree (ACT) in which (i) defense mechanismscan be applied at any node of the tree not just at the leaf nodelevel (ii) some qualitative analysis (usingmin-cuts structuraland Birnbaum importance measures) and probabilistic anal-ysis (using attackerrsquos and security costs the system risktheimpact of an attack ROI and ROA) can be performed (iii)the optimal countermeasure set can be selected from thepool of defense mechanisms without constructing a state-space model They have used single- and multiobjectiveoptimization tools to find suitable countermeasures underdifferent constraints In addition they have illustrated thefeatures of ACT using a practical case study namely asupervisory control and data acquisition (SCADA) attack
Finally some authors [119] have proposed a trilevelmodelCybersecurity is becoming an area of growing concern inthe electric power industry with the development of smartgrid A false data injection attack which is against thestate estimation through a SCADA network has recentlyattracted the ever wider interest of researchers This review[119] further develops the concept of a Load redistribution(LR) attack a special type of the false data injection attackThe damage from LR attacks to power system operationscan manifest in an immediate or a delayed fashion For theimmediate attacking goal they have shown in [119] that themost damaging attack can be identified through a max-minattacker-defender model Benders decomposition within arestart framework is used to solve the bilevel immediate LRattack problem with a moderate computational effort Its
efficiency has been validated by the Karush-Kuhn-Tucker-(KKT-) basedmethod solution in their previouswork For thedelayed attacking goal the authors of [119] have proposed atrilevelmodel to identify themost damaging attack and trans-form the model into an equivalent single-level mixed-integerproblem for its final solution In order to summarize thetechniques developed in [119] enable a quantitative analysis ofthe damage from LR attacks to the power system operationsand security and hence provide an in-depth insight intoan effective attack prevention when resources (budgets) arelimited A 14-bus system is used to test the correctness of theproposed model and algorithm
8 Concluding Remarks
In this paper we present a survey of Bilevel ProgrammingandApplication area closely related to applied problems suchas natural gas imbalance cash-out problem toll optimizationproblem and others Recent results and trends in the mixed-integer bilevel programming models with linear objectivefunction and constraints are also described
Many open questions still exist in Bilevel Program-ming theory especially in relation to applications Newtopicsquestions arise as for example application of non-smoothvariational analysis Many new applications arefound much is yet open with respect to solution algorithmsimportant are also mixed-discrete bilevel optimization prob-lems All these items have not been included in this surveyonly due to the space limitations but we hope to enlight themin the nearest future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research activity of the second author was financiallysupported by the RampD Department (Catedra de Investi-gacion) CAT-174 of the Tecnologico de Monterrey (ITESM)CampusMonterrey and by the SEP-CONACYT Projects CB-2008-01-106664 and CB-2013-01-221676 Mexico Also thework of the fourth author was supported by the NationalCouncil of Science and Technology (CONACyT) of Mex-ico as part of the Projects CB-2011-01-169765 PROMEP1035114330 and PAICYT 464-10 The research activity ofthe fifth author was financially supported by the ProjectPROMEP1035103889 the Academic Groups ResearchProject PROMEP1035124953 and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) with theProject CE960-11 The coauthors would also like to expresstheir profound gratitude to the two anonymous refereeswhose comments and suggestions have helped improve thepaper essentially
Mathematical Problems in Engineering 13
References
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
[1] S Dempe Foundations of Bilevel Programming Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] S Dempe B S Mordukhovich and A B Zemkoho ldquoNecessaryoptimality conditions in pessimistic bilevel programmingrdquoOptimization vol 63 no 4 pp 505ndash533 2014
[3] S Dempe and J Dutta ldquoIs bilevel programming a special caseof amathematical programwith complementarity constraintsrdquoMathematical Programming A vol 131 no 1-2 pp 37ndash48 2012
[4] S Dewez M Labb P Marcotte and G Savard ldquoNew formu-lations and valid inequalities for a bilevel pricing problemrdquoOperations Research Letters vol 36 no 2 pp 141ndash149 2008
[5] L Thi T Duc and P Dinh ldquoA DC programming approach fora class of bilevel programming problems and its application inportfolio selectionrdquo Numerical Algebra Control and Optimiza-tion vol 2 no 1 pp 167ndash185 2012
[6] L N Vicente and P H Calamai ldquoBilevel and multilevelprogramming a bibliography reviewrdquo Journal of Global Opti-mization vol 5 no 3 pp 291ndash306 1994
[7] W Wiesemann A Tsoukalas P-M Kleniati and B RustemldquoPessimistic bi-level optimisationrdquo Tech Rep Imperial CollegeLondon and Massachusetts Institute of Technology 2012
[8] B Bank J Guddat D Klatte B Kummer and K TammerNon-Linear Parametric Optimization Birkhaauser Basel Switzer-land 1983
[9] R Lucchetti F Mignanego and G Pieri ldquoExistence theoremsof equilibrium points in Stackelberg games with constraintsrdquoOptimization vol 18 no 6 pp 857ndash866 1987
[10] D Fanghanel Zwei-Ebenen-Optimierung mit Diskreter UntererEbene und Stetiger Oberer Ebene TU Bergakademie FreibergFreiberg Germany 2006
[11] H von Stackelberg Marktform und Gleichgewicht JuliusSpringer Vienna Austria 1934
[12] J Bracken and J T McGill ldquoMathematical programs withoptimization problems in the constraintsrdquoOperations Researchvol 21 pp 37ndash44 1973
[13] WCandler andRNorton ldquoMultilevel programming and devel-opment policyrdquo Tech Rep 258 World Bank Staff WashingtonWash USA 1977
[14] L N Vicente C A Floudas and P M Pardalos ldquoBilevel pro-gramming introduction history and overviewrdquo inEncyclopediaof Optimization C A Floudas and P M Pardalos Eds KluwerAcademic Dordrecht The Netherlands 2001
[15] J F Bard Practical Bilevel Optimization Algorithms and Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 1998
[16] K Shimizu Y Ishizuka and J F Bard Nondifferentiable andtwo-level mathematical programming Kluwer Academic Pub-lishers Boston Mass USA 1997
[17] S Dempe and V V Kalashnikov Eds Optimization withMultivalued Mappings Theory Applications and AlgorithmsSpringer Berlin Germany 2006
[18] A Migdalas P M Pardalos and P Varbrand Eds MultilevelOptimization Algorithms and Applications Kluwer AcademicDordrecht The Netherlands 1998
[19] S Dempe ldquoAnnotated bibliography on bilevel programmingand mathematical programs with equilibrium constraintsrdquoOptimization vol 52 no 3 pp 333ndash359 2003
[20] W Candler and R Townsley ldquoA linear two-level programmingproblemrdquo Computers amp Operations Research vol 9 no 1 pp59ndash76 1982
[21] Z Luo J Pang andDRalphMathematical Programswith Equi-librium Constraints Cambridge University Press CambridgeUK 1996
[22] J Outrata M Kocvara and J Zowe Nonsmooth Approach toOptimization Problems with Equilibrium Constraints KluwerAcademic Publishers Dordrecht The Netherlands 1998
[23] V Demiguel M P Friedlander F J Nogales and S Scholtes ldquoAtwo-sided relaxation scheme for mathematical programs withequilibrium constraintsrdquo SIAM Journal onOptimization vol 16no 2 pp 587ndash609 2005
[24] S Leyffer G Lopez-Calva and J Nocedal ldquoInterior methodsfor mathematical programs with complementarity constraintsrdquoSIAM Journal on Optimization vol 17 no 1 pp 52ndash77 2006
[25] M Kojima ldquoStrongly stable stationary solutions in nonlinearprogramsrdquo in Analysis and Computation of Fixed Points S MRobinson Ed vol 43 pp 93ndash138 Academic Press New YorkNY USA 1980
[26] D Ralph and S Dempe ldquoDirectional derivatives of the solutionof a parametric nonlinear programrdquo Mathematical Program-ming vol 70 no 2 pp 159ndash172 1995
[27] S Scholtes ldquoIntroduction to piecewise differentiable equa-tionsrdquo Tech Rep 531994 Universitat Karlsruhe Institutfur Statistik und Mathematische Wirtschaftstheorie 1994httpwwwengcamacuksimss248publicationsindexhtml
[28] S Dempe ldquoA necessary and a sufficient optimality condition forbilevel programming problemsrdquoOptimization vol 25 no 4 pp341ndash354 1992
[29] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Vol 1 Basic Theory Springer Berlin Germany2006
[30] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation vol 2 Springer Berlin Germany 2006
[31] R T Rockafellar and R J Wets Variational Analysis SpringerBerlin Germany 1998
[32] J V Outrata ldquoNecessary optimality conditions for Stackelbergproblemsrdquo Journal of OptimizationTheory andApplications vol76 no 2 pp 305ndash320 1993
[33] J J Ye and D Zhu ldquoNew necessary optimality conditions forbilevel programs by combining the MPEC and value functionapproachesrdquo SIAM Journal on Optimization vol 20 no 4 pp1885ndash1905 2010
[34] S Dempe J Dutta and B S Mordukhovich ldquoNew necessaryoptimality conditions in optimistic bilevel programmingrdquoOpti-mization vol 56 no 5-6 pp 577ndash604 2007
[35] Y Ishizuka and E Aiyoshi ldquoDouble penalty method for bileveloptimization problemsrdquo Annals of Operations Research vol 34no 1ndash4 pp 73ndash88 1992
[36] A G Mersha and S Dempe ldquoLinear bilevel programming withupper level constraints depending on the lower level solutionrdquoApplied Mathematics and Computation vol 180 no 1 pp 247ndash254 2006
[37] F Nozicka J Guddat H Hollatz and B Bank Theorie derLinearen Parametrischen Optimierung Akademie Berlin Ger-many 1974
[38] Energy Information Administration FERC Order 636 TheRestructuring Rule 2005 httpwwweiagovforecastssteo
[39] Energy Information Administration ldquoFERC Policy on SystemOwnership Since 1992rdquo httpwwweiadoegovoil gasnaturalgasanalysis publicationsngmajorlegfercpolicyhtml
[40] IHS Engineering ldquoEC Proposes New Legislat ion for Euro-pean Energy Policyrdquo 2008 httpengineersihscomnewseu-en-energy-policy-9-07html
14 Mathematical Problems in Engineering
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
[41] Environmental Protection Agency ldquoThe Impacts of FERCOrder 636 on coal mine gas project developmentrdquo 2008httpwwwepagovcmopdocspol004pdf
[42] K T Midthun Optimization Models for Liberalized NaturalGas Markets Norwegian University of Science and Technology(NTNU) Faculty of Social Science and Technology Manage-ment Department of Industrial Economics and TechnologyManagement Trondheim Norway 2008
[43] S Dempe V V Kalashnikov and R Z Rıos-Mercado ldquoDiscretebilevel programming application to a natural gas cash-outproblemrdquo European Journal of Operational Research vol 166no 2 pp 469ndash488 2005
[44] M J Doane and D F Spulber ldquoOpen access and the evolutionof the US spot market for natural gasrdquo Journal of Law andEconomics vol 34 no 2 pp 447ndash517 1994
[45] R Gutierrez A Nafidi and R G Sanchez ldquoForecasting totalnatural-gas consumption in Spain by using the stochasticGompertz innovation diffusion modelrdquo Applied Energy vol 80no 2 pp 115ndash124 2005
[46] F K Lyness ldquoGas demand forecastingrdquo Journal of the RoyalStatistical Society D vol 33 no 1 pp 9ndash12 1984
[47] V V Kalashnikov and R Z Rıos-Mercado ldquoA natural gas cash-out problem a bilevel programming framework and a penaltyfunction methodrdquo Optimization and Engineering vol 7 no 4pp 403ndash420 2006
[48] N Keyaerts L Meeus and W DrsquoHaeseleer ldquoAnalysis ofbalancing-system design and contracting behavior in the natu-ral gas marketsrdquo in European Doctoral Seminar on Natural GasResearch vol 24 Delft The Netherlands 2009
[49] H G Huntington ldquoFederal price regulation and the supply ofnatural gas in a segmented field marketrdquo Land Economics vol54 no 3 pp 337ndash347 1978
[50] A Tomasgard F Romo M Fodstad and K T MidthunldquoOptimization models for the natural gas value chainrdquo inGeometric Modeling Numerical Simulation and OptimizationAppliedMathematics at SINTEF ChapterOptimizationModelsfor the Natural Gas Value Chain Springer 2007
[51] C Borraz-Sanchez and R Z Rıos-Mercado ldquoA hybrid meta-heuristic approach for natural gas pipeline network optimiza-tionrdquo in Hybrid Metaheuristics vol 3636 of Lecture Notes inComputer Science pp 54ndash65 Springer New York NY USA2005
[52] A Chebouba F Yalaoui A Smati L Amodeo K Younsi andATairi ldquoOptimization of natural gas pipeline transportation usingant colony optimizationrdquo Computers and Operations Researchvol 36 no 6 pp 1916ndash1923 2009
[53] A Kabirian and M R Hemmati ldquoA strategic planning modelfor natural gas transmission networksrdquo Energy Policy vol 35no 11 pp 5656ndash5670 2007
[54] S A Gabriel J Zhuang and S Kiet ldquoA large-scale linearcomplementarity model of the North American natural gasmarketrdquo Energy Economics vol 27 no 4 pp 639ndash665 2005
[55] R Egging S A Gabriel F Holz and J Zhuang ldquoA comple-mentarity model for the European natural gas marketrdquo EnergyPolicy vol 36 no 7 pp 2385ndash2414 2008
[56] D Hawdon ldquoEfficiency performance and regulation of theinternational gas industry a bootstrap DEA approachrdquo EnergyPolicy vol 31 no 11 pp 1167ndash1178 2003
[57] K G Arano and B F Blair ldquoAn ex-post welfare analysis of nat-ural gas regulation in the industrial sectorrdquo Energy Economicsvol 30 no 3 pp 789ndash806 2008
[58] B Esnault ldquoThe need for regulation of gas storage the case ofFrancerdquo Energy Policy vol 31 no 2 pp 167ndash174 2003
[59] V V Kalashnikov and R Z Rıos-Mercado ldquoA penalty-functionapproach to a mixed-integer bilevel programming problemrdquoin Proceedings of the 3rd International Meeting on ComputerScience C R Zozaya Ed vol 2 pp 1045ndash1054 AguascalientesMexico September 2001
[60] S Dempe V V Kalashnikov and G A Perez-Valdes ldquoMixed-integer bilevel programming application to an extended gascash-out problemrdquo in Proceedings of the International Businessand Economics Research Conference (IBERC amp TLC rsquo06) p 14Las Vegas Nev USA 2006
[61] V V Kalashnikov N I Kalashnykova andG A Perez ldquoNaturalgas cash-out problem with price predictionsrdquo in Proceedings ofthe International Applied Business Research Conference (ABRCrsquo07) R D Clute Ed p 13 Mazatlan Mexico 2007
[62] V V Kalashnikov G A Perez-Valdes A Tomasgard and N IKalashnykova ldquoNatural gas cash-out problem bilevel stochas-tic optimization approachrdquo European Journal of OperationalResearch vol 206 no 1 pp 18ndash33 2010
[63] P Kall and S W Wallace Stochastic Programming John Wileyamp Sons Chichester UK 1994
[64] S Dempe V V Kalashnikov G A Perez-Valdes andN Kalash-nykova ldquoNatural gas bilevel cash-out problem convergence ofa penalty function methodrdquo European Journal of OperationalResearch vol 215 no 3 pp 532ndash538 2011
[65] V V Kalashnikov G A Perez and N I Kalashnykova ldquoAlinearization approach to solve the natural gas cash-out bilevelproblemrdquoAnnals of Operations Research vol 181 no 1 pp 423ndash442 2010
[66] V V Kalashnikov T I Matis and G A Perez-Valdes ldquoTimeseries analysis applied to construct US natural gas price func-tions for groups of statesrdquo Energy Economics vol 32 no 4 pp887ndash900 2010
[67] V V Kalashnikov G A Perez-Valdes T I Matis and NI Kalashnykova ldquoUS natural gas market classification usingpooled regressionrdquo Mathematical Problems in Engineering vol2014 Article ID 695084 9 pages 2014
[68] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programmingmodel Part 1rdquo inAdvances inIntelligent Decision Technologies J Watada J Phillips-Wren GJain and R J Howlett Eds vol 10 of Smart Innovation Systemsand Technologies pp 255ndash264 Springer Heidelberg Germany2011
[69] V V Kalashnikov S Dempe G A Perez-Valdes and N IKalashnykova ldquoReduction of dimension of the upper levelproblem in a bilevel programming model Part 2rdquo in Advancesin Intelligent Decision Technologies J Watada G Phillips-WrenL C Jain and R J Howlett Eds vol 10 of Smart InnovationSystems and Technologies pp 265ndash272 Springer HeidelbergGermany 2011
[70] D Kinderlehrer and G Stampacchia An Introduction to Varia-tional Inequalities andTheir Applications Academic Press NewYork NY USA 1980
[71] O L Mangasarian ldquoUniqueness of solution in linear program-mingrdquo Linear Algebra and Its Applications vol 25 pp 151ndash1621979
[72] J B Rosen ldquoExistence and uniqueness of equilibrium points forconcave N-person gamesrdquo Econometrica vol 33 no 3 pp 520ndash534 1965
Mathematical Problems in Engineering 15
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
[73] R Nishimura S Hayashi and M Fukushima ldquoRobust Nashequilibria inN-person non-cooperative games uniqueness andreformulationrdquo Pacific Journal of Optimization vol 5 no 2 pp237ndash259 2009
[74] G K Saharidis and M G Ierapetritou ldquoResolution method formixed integer bi-level linear problems based on decompositiontechniquerdquo Journal of Global Optimization vol 44 no 1 pp 29ndash51 2009
[75] L Lei CGuang-Nian andLChen-Xin ldquoResearch onproblemsbilevel programming for personnel allocation in enterpriserdquo inProceedings of the 17th International Conference onManagementScience amp Engineering (ICMSE rsquo10) pp 293ndash298 MelbourneAustralia November 2010
[76] T Roughgarden ldquoStackelberg scheduling strategiesrdquo SIAMJournal on Computing vol 33 no 2 pp 332ndash350 2004
[77] J Xu Y Tu and Z Zeng ldquoBilevel optimization of regional waterresources allocation problem under fuzzy random environ-mentrdquo Journal of Water Resources Planning and Managementvol 139 no 3 pp 246ndash264 2013
[78] S Fang PGuoM Li and L Zhang ldquoBilevelmultiobjective pro-gramming applied to water resources allocationrdquoMathematicalProblems in Engineering vol 2013 Article ID 837919 9 pages2013
[79] G Hongli L Juntao and G Hong ldquoA survey of bilevelprogramming model and algorithmrdquo in Proceedings of the 4thInternational Symposium on Computational Intelligence andDesign (ISCID rsquo11) pp 199ndash203 Hangzhou China October2011
[80] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007
[81] J Yin S CWongNN Sze andHWHo ldquoA continuummodelfor housing allocation and transportation emission problems ina polycentric cityrdquo International Journal of Sustainable Trans-portation vol 7 no 4 pp 275ndash298 2013
[82] R Liang J Gao and K Iwamura ldquoFuzzy random dependent-chance bilevel programming with applicationsrdquo in Advancesin Neural Networks-ISNN 2007 vol 4492 of Lecture Notesin Computer Science pp 257ndash266 Springer Berlin Germany2007
[83] S Y Wang and F A Lootsma ldquoA hierarchical optimizationmodel of resource allocationrdquoOptimization vol 28 no 3-4 pp351ndash365 1994
[84] A P Burgard P Pharkya and C D Maranas ldquoOptKnock abilevel programming framework for identifying gene knockoutstrategies for microbial strain optimizationrdquo Biotechnology andBioengineering vol 84 no 6 pp 647ndash657 2003
[85] H I Calvete C Gale and M Oliveros ldquoBilevel model forproductiondistribution planning solved by using ant colonyoptimizationrdquo Computers and Operations Research vol 38 no1 pp 320ndash327 2011
[86] F Legillon A Liefooghe and E G Talbi ldquoA coevolutionarymeta-heuristic for bi-level optimizationrdquo Rapport de RechercheINRIA vol 7741 pp 1ndash22 2011
[87] J F Camacho-Vallejo A E Cordero-Franco and R GGonzalez-Ramırez ldquoSolving the bilevel facility location prob-lem under preferences by a Stackelberg-evolutionary algo-rithmrdquoMathematical Problems in Engineering vol 2014 ArticleID 430243 14 pages 2014
[88] B Huang and N Liu ldquoBilevel programming approach tooptimizing a logistic distribution network with balancing
requirementsrdquo Transportation Research Record no 1894 pp188ndash197 2004
[89] C M Feng and C C Wen ldquoA bi-level programming modelfor allocating private and emergency vehicle flows in seismicdisaster areasrdquo in Proceedings of the Eastern Asia Society forTransportation Studies vol 5 pp 1408ndash1423 2005
[90] J Wang J Zhu J Huang and M Zhang ldquoMulti-level emer-gency resources location and allocationrdquo in Proceedings of theIEEE International Conference on Emergency Management andManagement Sciences (ICEMMS rsquo10) pp 202ndash205 August 2010
[91] H Sun Z Gao and J Wu ldquoA bi-level programming modeland solution algorithm for the location of logistics distributioncentersrdquoAppliedMathematicalModelling vol 32 no 4 pp 610ndash616 2008
[92] J Xu and P Wei ldquoA bi-level model for location-allocationproblem of construction and demolition waste managementunder fuzzy random environmentrdquo International Journal ofCivil Engineering vol 10 no 1 pp 1ndash12 2012
[93] P Hansen Y A Kochetov and N Mladenovic ldquoLower boundsfor the uncapacitated facility location problem with user pref-erencesrdquo G-2004-24 GERAD-HEC Montreal Canada 2004
[94] I L Vasilrsquoev K B Klimentova and Y A Kochetov ldquoNew lowerbounds for the facility location problemwith client preferencesrdquoComputational Mathematics and Mathematical Physics vol 49no 6 pp 1055ndash1066 2009
[95] I L Vasilrsquoev and K B Klimentova ldquoThe branch and cut methodfor the facility location problem with clients preferencesrdquoJournal of Applied and Industrial Mathematics vol 4 no 3 pp441ndash454 2010
[96] M Maric Z Stanimirovic and N Milenkovic ldquoMetaheuristicmethods for solving the bilevel uncapacitated facility locationproblem with clientsrsquo preferencesrdquo in EURO Mini Conferencevol 39 of Electronic Notes in Discrete Mathematics pp 43ndash50Elsevier 2012
[97] J Bhadury J H Jaramillo and R Batta ldquoOn the use of geneticalgorithms to solve location problemsrdquoComputersampOperationsResearch vol 29 no 6 pp 761ndash779 2002
[98] T Uno H Katagiri and H Kato ldquoAn application of particleswarm optimization to bilevel facility location problems withquality of facilitiesrdquo Asia Pacific Management Review vol 12no 4 pp 183ndash189 2007
[99] VMarianovM Rıos andM Icaza ldquoFacility location formarketcapture when users rank facilities by shorter travel and waitingtimesrdquo European Journal of Operational Research vol 191 no 1pp 32ndash44 2008
[100] C M C Rodrıguez D R S Penate and J A M PerezldquoCompetencia espacial por cuotas de mercado el problema dellıder-seguidormediante programacion linealrdquoRecta vol 12 no1 pp 69ndash84 2011 (Spanish)
[101] H Kucukaydin N Aras and I K Altinel ldquoCompetitive facilitylocation problemwith attractiveness adjustment of the followera bilevel programming model and its solutionrdquo EuropeanJournal of Operational Research vol 208 no 3 pp 206ndash2202011
[102] V L Beresnev ldquoLocal search algorithms for the problem ofcompetitive location of enterprisesrdquo Automation and RemoteControl vol 73 no 3 pp 425ndash439 2012
[103] D Kress and E Pesch ldquoSequential competitive location onnetworksrdquo European Journal of Operational Research vol 217no 3 pp 483ndash499 2012
16 Mathematical Problems in Engineering
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
[104] D Kress and E Pesch ldquo(119903 | 119901)-centroid problems onthe networks with vertex and edge demandrdquo Computers ampOperations Research vol 39 no 12 pp 2954ndash2967 2012
[105] H Kucukaydın N Aras and K Altınel ldquoA leader-followergame in competitive facility locationrdquo Computers amp OperationsResearch vol 39 no 2 pp 437ndash448 2012
[106] M G Ashtiani A Makui and R Ramezanian ldquoA robust modelfor a leader-follower competitive facility location problem in adiscrete spacerdquo Applied Mathematical Modelling vol 37 no 1-2pp 62ndash71 2013
[107] J R Kim J U Lee and J Jo ldquoHierarchical spanning treenetwork design with Nash genetic algorithmrdquo Computers andIndustrial Engineering vol 56 no 3 pp 1040ndash1052 2009
[108] D H Lee L Song and H Wang ldquoBilevel programming modeland solutions of berth allocation and quay crane schedulingrdquoin Proceedings of the 85th Annual Meeting of TransportationResearch Board Washington DC USA 2006
[109] P G Comba ldquoExponentiation cryptosystems on the IBM PCrdquoIBM Systems Journal vol 29 no 4 pp 526ndash538 1990
[110] M Brown D Hankerson J Lopez and A Menezes ldquoSoftwareimpl ementation of the NIST elliptic curves over prime fieldsrdquoResearch Report CORR 2000-55 Department of Combina-torics and Optimization University of Waterloo WaterlooCanada 2000
[111] S-M Hong S-Y Oh and H Yoon ldquoNew modular multiplica-tion algorithms for fast modular exponentiationrdquo in Advancesin CryptologymdashEUROCRYPT rsquo96 vol 1070 of Lecture Notes inComputer Science pp 166ndash177 1996
[112] A Weimerskirch and C Paar ldquoGeneralizations of the Karat-suba algorithm for efficient implementationsrdquo CryptologyePrint Archive Report 2006224 2006 httpeprintiacrorg2006224
[113] J Groszligschadl R M Avanzi E Sava and S Tillich ldquoEnergy-efficient software implementation of long integer modu-lar arithmeticrdquo in Cryptographic Hardware and EmbeddedSystemsmdashCHES 2005 vol 3659 of Lecture Notes in ComputerScience pp 75ndash90 Springer Berlin Germany 2005
[114] R Moreno J M Miret and F Sebe ldquoA hyperelliptic cryp-tosystem based on the P1363 IEEE standardrdquo in Proceedings ofthe International Meeting on Coding Theory And Cryptography(IMCTC rsquo99) Medina del Campo Spain 1999
[115] N P Smart ldquoOn the performance of hyperelliptic cryptosys-temsrdquo in Advances in CryptologymdashEurocryptrsquo99 vol 1592 ofLecture Notes in Computer Science pp 165ndash175 Springer BerlinGermany 1999
[116] J M Arroyo and F D Galiana ldquoOn the solution of the bilevelprogramming formulation of the terrorist threat problemrdquo IEEETransactions on Power Systems vol 20 no 2 pp 789ndash797 2005
[117] C-C Liu J Jung G T Heydt V Vittal and A G PhadkeldquoThe strategic power infrastructure defense (SPID) system Aconceptual designrdquo IEEE Control Systems Magazine vol 20 no4 pp 40ndash52 2000
[118] A Roy D S Kim and K S Trivedi ldquoCyber security analysisusing attack countermeasure treesrdquo in Proceedings of the 6thAnnualWorkshop onCyber Security and Information IntelligenceResearch 2010
[119] Y-L Yuan Z-I Li and K-I Ren ldquoQuantitative analysis of loadredistribution attacks in power systemsrdquo IEEE Transactions onParallel and Distributed Systems vol 23 no 9 pp 1731ndash17382012
![arXiv:1408.4805v2 [astro-ph.HE] 10 Oct 2014 · ... O. Martinez14, J. Mart´ınez ... Universidad Nacional Aut´onoma de M´exico, Mexico D ... 25 Universidad Aut´onoma del Estado](https://static.documents.pub/doc/80x56/5ac080937f8b9ac6688c3eff/arxiv14084805v2-astro-phhe-10-oct-2014-o-martinez14-j-martinez-.jpg)
