A Quadratic Programming Bibliography
NicholasI. M. Gould
ComputationalScienceandEngineeringDepartment
RutherfordAppletonLaboratory, Chilton
Oxfordshire,OX11 0QX, England,EU
Email : [email protected]
and
PhilippeL. Toint
Departmentof Mathematics,Universityof Namur
61, ruedeBruxelles,B-5000Namur, Belgium,EU.
Email : [email protected]
RAL NumericalAnalysisGroupInternalReport2000–1
Thisversion:February27,2001
Thefollowing is alist of all of thepublishedandunpublishedworksonquadraticprogramming
thatweareawareof. Somearegeneralreferencesto backgroundmaterial,while othersarecen-
tral to thedevelopmentof thequadraticprogrammingmethodsandto theapplicationswe in-
tendto coverin ourevolvingbookonthesubject.Wehavedeliberatelynotincludedany but the
mostrelevantof thehundreds,if not thousands,of citationsto sequential/successive/recursive
quadraticprogrammingmethodsfor nonlinearprogramming,nor to thoseon linearprogram-
ming or quadraticprogrammingwith quadraticconstraints.ThecompleteLATEX bibliography,
togetherwith up-to-dateadditions,is availableonlineat
ftp://ftp.numerical.rl.ac.uk/pub/qpbook/qpbook.bib
and
ftp://thales.math.fundp.ac.be/pub/qpbook/qpbook.bib .
We would bedelightedto receiveany correctionsor updatesto this list.
A Quadratic Programming Bibliography
N. N. Abdelmalek. Restorationof imageswith missinghigh-frequency componentsusingquadraticprogramming.AppliedOptics, 22(14),2182–2188,1983.
Abstract. A methodfor restoringan optical imagewhich is subjectedto low-passfrequency filtering is
presented.It is assumedthat the objectwhoseimageis restoredis of finite spatialextent.The problemis
treatedasanalgebraicimage-restorationproblemwhich is thensolvedasaquadraticprogrammingproblem
with boundedvariables.The regularizationtechniquefor the ill-posedsystemis to replacethe consistent
systemof thequadraticprogrammingproblemby anapproximatesystemof smallerrank.Therank which
givesa bestor near-bestsolutionis estimated.This methodis a novel one,andit comparesfavorablywith
otherknown methods.Computer-simulatedexamplesarepresented.Commentsandconclusionsaregiven.
N. N. AbdelmalekandT. Kasvand. Digital imagerestorationusingquadraticprogramming.AppliedOptics, 19(19),3407–3415,1980.
Abstract. The problemof digital imagerestorationis consideredby obtainingan approximatesolutionto
the Fredholmintegral equationof the first kind in two variables.The systemof linear equationsresulting
from thediscretizationof the integral equationis convertedto a consistentsystemof linearequations.The
problemis thensolved asa quadraticprogrammingproblemwith boundedvariableswherethe unknown
solutionis minimizedin theL2 norm.In thismethodminimumcomputerstorageis needed,andtherepeated
solutionsareobtainedin anefficient way. Also therankof theconsistentsystemwhich givesa bestor near
bestsolutionis estimated.Computersimulatedexamplesusingspatiallyseparablepointspreadfunctionsare
presented.Commentsandconclusionsaregiven.
R. A. Abramsand A. Ben Israel. A duality theoremfor complex quadraticprogramming.Journalof OptimizationTheoryandApplications, 4(4), 245–252,1969.
Abstract. A duality theoryfor complex quadraticprogrammingover polyhedralconesis developed,follow-
ing Dorn,by usinglinearduality theory.
J. W. Adams. Quadraticprogrammingapproachesto new optimal windows andantennaar-rays. ConferenceRecord. TwentyFourth AsilomarConferenceon Signals,SystemsandComputersMaplePress,SanJose, CA,USA, 1, 69–72,1990a.
Abstract. New window designproblemsareformulatedin termsof quadraticprogramming.Thenew win-
dowspermitthedesignerto controlthetradeoff betweenthepeaksidelobelevel andthetotalsidelobeenergy.
In addition,linearconstraintscanbeimposedon thedesignproblem.Theproposedmethodsareapplicable
to applicationsin thefieldsof signalprocessingandantennaarrays.
J.W. Adams.Quadraticprogrammingapproachesto new problemsin digital filter design.Con-ferenceRecord. TwentyFourthAsilomarConferenceonSignals,SystemsandComputersMaplePress,SanJose, CA,USA, 1, 307–310,1990b.
Abstract. New digital filter designproblemsareformulatedin termsof quadraticprogramming.The new
filters permit the designerto control the tradeoff betweenthe peakerrorsand the total squarederrors.In
2 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
particular, themean-squarederrorcanbeminimizedsubjectto peakerrorsconstraints,asrequiredin many
practicaldesignproblems.In addition,equalityconstraintscanbeimposedfor specialapplications.
J. W. Adams,P. Kruethong,R. Hashemi,J. L. Sullivan, andD. R. Gleeson. New quadraticprogrammingalgorithmsfor designingFIR digital filters. ConferenceRecord of TheTwentySeventhAsilomarConferenceonSignals,SystemsandComputers.IEEEComput.SocPress,LosAlamitos,CA,USA, 2, 1206–1210,1993.
Abstract. TheParks-McClellanalgorithm(1973)is verypopularfor designingFIR digital filters. It is based
on a linear programmingalgorithmcalledthe Remezexchange.Our new algorithmis basedon quadratic
programming,which includeslinear programmingas a specialcase.The filters in this paperpermit the
designerto controlthetradeoff betweenthepeakerrorandthetotal squarederror. Thesefilters aredesigned
accordingto thepeak-constrainedleast-squares(PCLS)optimalitycriterion.
J. W. Adams,J. L. Sullivan, D. R. Gleeson,P. H. Chang,andR. Hashemi. Application ofquadraticprogrammingto FIR digital filter designproblems.ConferenceRecord of theTwentyEighthAsilomarConferenceon Signals,SystemsandComputers. IEEE Comput.SocPress,LosAlamitos,CA,USA, 1, 314–318,1994.
Abstract. Quadraticprogrammingproblemshavelongbeenof interestin thebusinesscommunity. Quadratic
programmingis often usedasthe basisfor ”programtrading” wherestocksareautomaticallyboughtand
soldby mutual fundsto optimizeprofits.Quadraticprogrammingalgorithmscanalsobe usedto optimize
digital filters, asdiscussedin this paper. We presentthe generalizedmultiple exchange(GME), simplified
generalizedmultipleexchange(SGME)andmodifiedgeneralizedmultipleexchange(MGME) algorithmsfor
designingconstrainedleast-squares(CLS)filters.TheCLSfiltersaregeneralizationsof thepopularminimax
andleast-squaresfilters.TheCLS filters areimportantnot only becauseof their generality, but alsobecause
they areneededfor many practicalapplications.
W. P. AdamsandP. M. Dearing. On the equivalencebetweenroof duality andLagrangianduality for unconstrained0–1quadraticprogrammingproblems.DiscreteAppliedMath-ematics, 48(1), 1–20,1994.
Abstract. Considerstechniquesfor computingupperboundson theoptimalobjective functionvalueto any
unconstrained0–1 quadraticprogrammingproblem(maximization).In particular, the authorsstudy three
methodsfor obtainingupperboundsaspresentedin arecentpaperby Hammer, Hansen,andSimeone(1984)
(1) generatingtwo classesof upper-boundinglinearfunctionsreferredto aspavedupperplanesandroofs,(2)
solving thecontinuousrelaxationof a mixed-integer linear problemby Rhys(1970),and(3) the quadratic
complementationof variableswhich resultsin a boundcalled the height.The authorsshow that all three
methodsdirectly result from standardpropertiesof a reformulationof the quadraticproblemasa mixed-
integer linear program,with methods(1) and (3) resultingfrom a Lagrangiandual of this reformulation.
Basedon this reformulation,they expanduponthepublishedresults.
W. P. Adamsand H. D. Sherali. A tight linearizationand an algorithm for 0–1 quadraticprogrammingproblems.ManagementScience, 32(10),1274–1290,1986.
Abstract. Thepaperis concernedwith thesolutionof linearlyconstrained0–1quadraticprogrammingprob-
lems.Problemsof thiskind arisein numerouseconomic,locationdecision,andstrategic planningsituations,
includingcapitalbudgeting,facility location,quadraticassignment,mediaselection,anddynamicsetcov-
ering.A new linearizationtechniqueis presentedfor this problemwhich is demonstratedto yield a tighter
continuousor linearprogrammingrelaxationthanis availablethroughothermethods.An implicit enumera-
tion algorithmwhichusesLagrangianrelaxation,Benders’cuttingplanes,andlocalexplorationsis designed
to exploit thestrengthof this linearization.Computationalexperienceis providedto demonstratetheuseful-
nessof theproposedlinearizationandalgorithm.
S.N. Afriat. Thequadraticform definiteon a manifold. Proceedingsof theCambridge Philo-
sophicalSociety, 47, 1–6,1951.
N. I. M. GOULD & PH.L. TOINT 3
A. Aggarwal andC. A. Floudas.A decompositionapproachfor globaloptimumsearchin QP,NLP andMINLP problems.Annalsof OperationsResearch, 25(1–4),119–145,1990.
Abstract. A new approachfor global optimumsearchis presentedwhich involvesa decompositionof the
variableset into two sets-complicatingandnoncomplicatingvariables.This resultsin a decompositionof
the constraintset leadingto two subproblems.The decompositionof the original probleminducesspecial
structurein the resultingsubproblemsanda seriesof thesesubproblemsarethensolved, usingthe gener-
alisedBenders’decompositiontechnique,to determinetheoptimalsolution.Mathematicalpropertiesof the
proposedapproacharepresented.Even thoughthe proposedapproachcannotguaranteethe determination
of theglobaloptimum,computationalexperienceon anumberof nonconvex QP, NLP andMINLP example
problemsindicatesthataglobaloptimumsolutioncanbeobtainedfrom variousstartingpoints.
F. G. Akhmadov. Computationalmethodfor solving the quadraticprogrammingproblemin Ln
2
0 T space. IzvestiyaAkademiiNauk Azerbaidzhanskoi SSR,Seriya Fiziko–
Tekhnicheskikhi MatematicheskikhNauk, 4(4), 102–106,1983.
Abstract. A methodfor solving the quadraticprogrammingproblemin the spaceof all vector-functions,
eachcomponentsquareof which is integrable,is given.It is supposedthat thematrix of thequadraticform
is positively defined.
F. A. Al-Khayyal. Linear, quadraticandbilinearprogrammingapproachesto the linearcom-
plementarityproblems.EuropeanJournalof OperationsResearch, 24, 216–227,1987.
F. A. Al-Khayyal. Jointly constrainedbilinearprogramsandrelatedproblems:An overview.
Computers in MathematicalApplications, 19, 53–62,1990.
F. A. Al-Khayyal andJ. E. Falk. Jointly constrainedbiconvex programming.Mathematicsof
OperationsResearch, 8, 273–286,1983.
C. AlessandriandA. Tralli. Frictionlesscontactwith BEM usingquadraticprogramming—
discussion.Journalof EngineeringMechanics-ASCE, 119(12),2538–2540,1993.
B. Alidaee, G. A. Kochenberger, andA. Ahmadian. 0–1 quadraticprogrammingapproachfor optimum solutionsof two schedulingproblems. International Journal of SystemsScience, 25(2), 401–408,1994.
Abstract. Two schedulingproblemsareconsidered:(1) schedulingn jobsnon-preemptively onasinglema-
chineto minimizetotal weightedearlinessandtardiness(WET); (2) schedulingn jobsnon-preemptively on
two parallelidenticalprocessorstominimizeweightedmeanflow time.In thesecondproblem,apre-ordering
of the jobs is assumedthatmustbe satisfiedfor any setof jobs scheduledon eachspecificmachine.Both
problemsareknown to be NP-complete.A 0–1 quadraticassignmentformulationof the problemsis pre-
sented.An equivalent0–1mixedinteger linearprogrammingapproachfor theproblemsareconsideredand
a numericalexampleis given.Theformulationspresentedenableoneto useoptimalandheuristicavailable
algorithmsof 0–1quadraticassignmentfor theproblemsconsideredhere.
K. S. Alsultan and K. G. Murty. Exterior point algorithmsfor nearestpoints and convex
quadraticprograms.MathematicalProgramming, 57(2), 145–161,1992.
A. Altman. QHOPDM—ahigher-orderprimal-dualmethodfor large-scaleconvex quadratic
programming.EuropeanJournalof OperationalResearch, 87(1), 200–202,1995.
A. Altman. Higher order primal-dual interior point methodfor separableconvex quadratic
optimization.Control andCybernetics, 25(4), 761–772,1996.
4 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
A. AltmanandJ.Gondzio.Regularizedsymmetricindefinitesystemsin interiorpointmethodsfor linearandquadraticoptimization. Logilab TechnicalReport1998.6,DepartmentofManagementSciences,Universityof Geneva,Geneva,Switzerland,1998.
Abstract. This paperpresentslinear algebratechniquesusedin the implementationof an interior point
methodfor solving linearprogramsandconvex quadraticprogramswith linearconstraints.New regulariza-
tion techniquesfor Newton systemsapplicableto bothsymmetricpositive definiteandsymmetricindefinite
systemsaredescribed.They transformthe latter to quasidefinitesystemsknown to bestronglyfactorizable
to a form of Cholesky-like factorization.Two differentregularizationtechniques,primal anddual,arevery
well suitedto the(infeasible)primal-dualinterior point algorithm.This particularalgorithm,with anexten-
sionof multiplecentralitycorrectors,is implementedin oursolverHOPDM.Computationalresultsaregiven
to illustratethe potentialadvantagesof the approachwhenappliedto the solutionof very large linear and
convex quadraticprograms.
H. Amato and G. Mensch. Rank restrictionson the quadraticform in indefinite quadraticprogramming.Unternehmensforschung, 15(3), 214–216,1971.
Abstract. A quadraticprogrammingproblem,whereq x aTx xT Qx is anindefiniteobjective function,
canbesolvedwith Swarup’s approachto optimizing cTx α dT x β only if therankof Q is two; if Q is
definite,therankof Q mustbeone.
D. E. AmosandM. L. Slater. Polynomialandsplineapproximationby quadraticprogramming.Communicationsof theACM, 12(7),379–381,1969.Seealso,CollectedAlgorithmsfromACM, 1985.
Abstract. The problemof approximationto a given function, or of fitting a given setof data,wherethe
approximatingfunctionis requiredto havecertainof its derivativesof specifiedsignover thewholerangeof
approximation,is studied.Two approachesarepresented,in eachof which quadraticprogrammingis used
to provide both theconstraintson thederivativesandtheselectionof thefunctionwhich yields thebestfit.
Thefirst is a modifiedBernsteinpolynomialscheme,andthesecondis asplinefit.
P. Anand. Decompositionprinciplefor indefinitequadraticprograme.TrabajosdeEstadistica
y deInvestigacion, 23, 61–71,1972.
S. C. Anand,F. E. Weisgerber, andH. S. Hwei. Direct solutionvs quadraticprogrammingtechniquein elastic-plasticfinite elementanalysis.ComputersandStructures, 7(2), 221–228,1977.
Abstract. Elastic-plasticplanestressfinite elementanalysisof adiskrolling onarigid trackis performedby
thedirectmethodaswell asthequadraticprogrammingtechnique.Trescaandvon Mises’ yield conditions
areusedin the former whereasanapproximatepiecewise linear Trescayield conditionis usedin the later
case.It is concludedfrom acomparisonof thecomputertimesneededin thetwo casesthatthedirectmethod
is far superiorto thequadraticprogrammingtechnique.
A. Anckonie. A quadraticprogramfor determiningefficient frontier portfolio compositions
using the SAS language. In ‘SUGI 10—Proceedingsof the TenthAnnual SAS Users
GroupInternationalConference’,Vol. 15,pp.55–60,1985.
W. Anfilof f. Gravity interpretationwith the aid of quadraticprogramming. Geophysics,
46(3), 340–341,1981.
P. L. DeAngelis,P. M. Pardalos,andG. Toraldo.Quadraticprogrammingwith boxconstraints.
In I. M. Bomze,ed., ‘Developmentsin Global optimization’, pp. 73–93,Kluwer Aca-
demicPublishers,Dordrecht,TheNetherlands,1997.
N. I. M. GOULD & PH.L. TOINT 5
K. M. Anstreicher. On long steppathfollowing andSUMT for linearandquadraticprogram-ming. SIAMJournalon Optimization, 6(1), 33–46,1996.
Abstract. We considera long stepbarrieralgorithmfor the minimizationof a convex quadraticobjective
subjectto linear inequality constraints.The algorithm is a dual versionof a methoddevelopedby K. M.
Anstreicheret al. (1993),andrequiresO nL or O sqrtnL iterationsto solve a problemwith n constraints,
dependingon how the barrier parameteris reduced.As a corollary of our analysiswe demonstratethat
the classicalSUMT algorithm, exactly as implementedin 1968, solves linear andquadraticprogramsin
O nLlogL iterations,with properinitializationandchoiceof parameters.
K. M. Anstreicher. On theequivalenceof convex programmingboundsfor booleanquadraticprogramming.Working paper, Departmentof ManagementScience,Universityof Iowa,IowaCity, USA, 1997.
Abstract. Recentpapershave shown theequivalenceof several tractableboundsfor Booleanquadraticpro-
gramming.In this note we give simplified proofs for theseresults,andalso show that all of the bounds
consideredaresimultaneouslyattainedby onediagonalperturbationof thequadraticform.
K. M. AnstreicherandN. W. Brixius. Solving quadraticassignmentproblemsusingconvexquadraticprogrammingrelaxations. Working paper, Departmentof ManagementSci-ence,Universityof Iowa,IowaCity, USA, 2000.
Abstract. Wedescribeabranch-and-boundalgorithmfor thequadraticassignmentproblem(QAP)thatuses
a convex quadraticprogramming(QP) relaxationto obtain a boundat eachnode.The QP subproblems
areapproximatelysolved using the Frank-Wolfe algorithm, which in this caserequiresthe solution of a
linearassignmentproblemoneachiteration.Ourbranchingstrategy makesextensiveuseof dualinformation
associatedwith the QP subproblems.We obtainstate-of-the-artcomputationalresultson large benchmark
QAPs.
K. M. AnstreicherandN. W. Brixius. A new boundfor thequadraticassignmentproblembasedon convex quadraticprogramming.MathematicalProgramming, 89(3), 341–357,2001.
Abstract. We describea new convex quadraticprogrammingboundfor the quadraticassignmentproblem
(QAP).Theconstructionof theboundusesasemidefiniteprogrammingrepresentationof abasiceigenvalue
boundfor QAP. The new bounddominatesthe well-known projectedeigenvaluebound,andappearsto be
competitive with existing boundsin thetrade-off betweenboundqualityandcomputationaleffort.
K. M. Anstreicher, D. denHertog,C. Roos,andT. Terlaky. A long-stepbarriermethodforconvex quadraticprogramming.Algorithmica, 10(5), 365–382,1993.
Abstract. In this paperwe proposea long-steplogarithmicbarrier function methodfor convex quadratic
programmingwith linear equalityconstraints.After a reductionof the barrierparameter, a seriesof long
stepsalongprojectedNewton directionsaretakenuntil the iterateis in thevicinity of thecenterassociated
with the currentvalueof the barrierparameter. We prove that the total numberof iterationsis O nL or
O nL , dependingonhow thebarrierparameteris updated.
K. Aoki andT. Fujikawa. VAR planningandnonconvex quadraticprogramming.Transactionsof theInstituteof ElectricalEngineersof Japan, 100(3), 78–88,1980.
Abstract. So far, many methodsbasedon linear programming,nonlinearprogrammingand integer pro-
gramminghavebeenproposedfor varplanning.In thelinearprogrammingmethod,therelationshipbetween
voltage,active power and reactive power is representedby linear equations.The nonlinearprogramming
method,in which this relationshipis expressedexactly, is not suitedfor a large-scalepower system.Both
thesemethodsneglect discretevariablesrepresentingthe numberof the condenseror reactorunitsandthe
transformertap positions.The integer programmingmethod,which enableshandlingthesediscretevari-
ables,is of coursenot suitedfor a large-scalepower system.Theauthorsshow that thecondenserplanning
problemis formulatedinto aparametricconvex quadraticprogrammingandthereactorplanningproblemis
formulatedinto aparametricnonconvex quadraticprogramming.
6 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
K. Aoki and M. Kanezashi. A decompositionalgorithm for a dual angulartype quadraticprogramming.In A. Lew, ed.,‘Proceedingsof the6th Hawaii InternationalConferenceon SystemsSciences.WesternPeriodicals,North Hollywood,CA, USA’, pp. 358–360,1973.
Abstract. This paperdealswith a decompositionprocedurefor the problemwhoseobjective function is
linear for a couplingvariable.Moreover it is describedby takingadvantageof suchcharacteristicsthatone
caneasilyobtainthevariationof a couplingvariableby computinglinearforms.
K. Aoki andT. Satoh.Economicdispatchwith network securityconstraintsusingparametricquadraticprogramming. IEEE Transactionson Power Apparatusand Systems, PAS-101(12),4548–4556,1982.
Abstract. Thispaperpresentsanefficient methodto solveaneconomicloaddispatchproblemwith DC load
flow type network securityconstraints.The conventionallinear programmingandquadraticprogramming
methodscannotdealwith transmissionlossesasaquadraticform of generatoroutputs.In orderto overcome
this defect, the extensionof the quadraticprogrammingmethodis proposed,which is designatedas the
parametricquadraticprogrammingmethod.The upperboundingtechniqueandthe relaxationmethodare
coupledwith theproposedmethodfor thepurposeof computationalefficiency. Thetestresultsshow thatthe
proposedmethodis practicalfor real-timeapplications.
M. Arioli. The useof QR factorizationin sparsequadraticprogrammingandbackwarderrorissues.SIAMJournalon Matrix AnalysisandApplications, 21(3), 825–839,2000.
Abstract. We presenta roundoff erroranalysisof a null spacemethodfor solving quadraticprogramming
minimizationproblems.Thismethodcombinestheuseof adirectQRfactorizationof theconstraintswith an
iterative solver on thecorrespondingnull space.Numericalexperimentsarepresentedwhich give evidence
of thegoodperformancesof thealgorithmonsparsematrices.
B. ArmstrongandB. A. Wade. Nonlinearpid control with partial stateknowledge: designby quadraticprogramming.In ‘Proceedingsof the2000AmericanControlConference,Danvers,MA, USA’, Vol. 2, pp.774–778,2000.
Abstract. NonlinearPID (NPID)controlis implementedbyallowing thecontrollergainsto varyasafunction
of systemstate.NPID controllerswill in generaldependon knowledgeof thefull statevector. In this work,
NPID controllerswhichoperatewithout knowledgeof somestatevariablesaredemonstrated.A generalbut
conservative designmethodis presentedwith anexperimentaldemonstration.For a specialcase,complete
necessaryandsufficient conditionsareestablished.
D. A. Arsamastsev, P. I. Bartolomey, and S. K. Okulovski. New improved quadratic-
programmingmethodsfor super-large power-systemsanalysis. In ‘Proceedingsof the
EighthPowerSystemsComputationConference’,Vol. 178,pp.710–716,1984.
L. ArseneauandM. J. Best. Resolutionof degeneratecritical parametervaluesin parametric
quadraticprogramming.TechnicalReportCORR99-47,Departmentof Combinatorics
andOptimization,Universityof Waterloo,Ontario,Canada,1999.
J. Atkociunas.Quadraticprogrammingfor degenerateshakedown problemsof barstructures.
MechanicsResearch Communications, 23(2), 195–203,1996.
A. M. Aurela and J. J. Torsti. A quadraticprogrammingmethodfor stabilizedsolution ofunstablelinearsystems.AnnalesUniversitatisTurkuensis,SerAI (AstronomicaChemicaPhysicaMathematica), 123, 1968.
N. I. M. GOULD & PH.L. TOINT 7
Abstract. An improved versionis presentedof thequadraticprogrammingmethodintroducedin 1967for
stabilizedsolutionof unstablesystemsof linear equations.The mostprobablesolutionandits confidence
limits arediscussed.In thepresentwork, themethodproperwasexaminedmoresystematicallyby usingthe
secondartificial exampleof Phillips (1962),in which thecorrectresultwasknown. Thedependenceof the
computingtimet onthenumberof variablesadjustedsimultaneously, l, wasstudied,takingatotalof m 15
variables,subjectto thenonnegativity constraint.Theoptimumwasachievedwith l 3 to 6 (final precision
ε 1 105, t 3 min in theIBM 1130).Thedifferentsolutionsexhibitedmarkedconsistency with eachother,
indicatingtheaccuracy andreliability of themethod
G. Auxenfants,L. Barthe,andP. Gibert. Architecturefor scientificsoftware.II. Analysisof aquadraticprogrammingalgorithm.RechercheAerospatiale, 4, 247–255,1982.
Abstract. Presentsa quadraticprogrammingalgorithmwith linearconstraints,working in thecaseof large-
scaleoptimizationproblems.The numberof variablesis reducedby a partial dualizationof constraintre-
lations.It enablesoneto determinewhetheror not theadmissiblesetis empty. Theprogramminghasbeen
implementedon a CYBER 170/750usinga methodof architecturebasedon (1) datacentralizationand(2)
managementof informationexchangebetweenprocessorsby databasemanagementsystem.This algorithm
representsoneof theelementsof amoreoptimizationcode.
T. Aykin. Onaquadraticinteger-programfor thelocationof interactinghubfacilities.European
Journalof OperationalResearch, 46(3), 409–411,1990.
A. BachemandB. Korte.An algorithmfor quadraticoptimizationovertransportationpolytopes.
Zeitschrift fur AngewandteMathematikundMechanik, 58, 459–461,1978.
R. BacherandH. P. vanMeeteren.Securitydispatchbasedon couplingof linearandquadraticprogrammingtechniques. Power Systems,Modelling and Control Applications.IFACSymposiumPergamon,Oxford, England, pp.211–217,1989.
Abstract. Securitydispatchcan be definedas the real-timeclosedloop cost-optimalallocationof active
generatoroutputwhile consideringbranchflow limits of theintactnetwork andlower anduppergeneration
limits. MostOPFalgorithmsfail to guaranteeaccuracy. reliability andspeedatthesametimeandcanthusnot
beusedin real-timeclosedloop application.Accuracy, reliability andspeedcanbe obtainedby executing
a LP basedOPF and a QP basedconstrainedeconomicdispatchat different executionfrequencies.The
QP basedalgorithmusesthe critical constraintsetasdeterminedby the LP basedalgorithm.Constrained
economicdispatchcansubstitutetheclassicaleconomicdispatchandwill provide asecuredispatch.
W. E. Baethgen,D. B. Taylor, andM. M. Alley. Quadratic-programmingmethodfor deter-
mining optimum nitrogenratesfor winter-wheatduring tillering. AgronomyJournal,
81(4), 557–559,1989.
A. BagchiandB. Kalantari. A methodfor computingapproximatesolutionof thetrust region
problemwith applicationto projectivemethodsfor quadraticprogramming.Workingpa-
per, Departmentof ComputerScience,RutgersUniversity, New Brunswick,New Jersey,
USA, 1988.
J. R. Baker. Determinationof an optimal forecastmodel for ambulancedemandusinggoal
andquadraticprogramming.In ‘Proceedings—SoutheasternChapterof the Instituteof
ManagementSciencesTwentiethAnnualMeeting’,Vol. 6, pp.154–157,1984.
E. Balas. Duality in discreteprogramming: II. The quadraticcase. ManagementScience,
16, 14–32,1969.
8 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
E. Balas.Nonconvex quadraticprogrammingvia generalizedpolars.SIAMJournalonAppliedMathematics, 28(2), 335–349,1975.
Abstract. A new approachis proposedto linearly constrainednonconvex quadraticprogramming.Theap-
proachis basedon generalizedpolar sets,andis akin to the convex analysisapproachto integer program-
ming.Theauthorconstructsageneralizedpolarof theKuhn-Tucker polyhedronassociatedwith a quadratic
program.This generalizedpolaris aconvex polyhedralconewhoseinterior containsnocomplementaryfea-
sible solutionbetterthanthe bestknown one.An algorithmis thenproposed,which doesnot usecutting
planes,but constructsapolytopecontainingthefeasiblesetandcontainedin thepolarof thelatter. Thebest
complementarysolutionfoundin theprocessis optimal,or noneexists.
C. C. Baniotopoulos,K. M. Abdalla, and P. D. Panagiotopoulos.A variational inequalityandquadraticprogrammingapproachto theseparationproblemof steelboltedbrackets.ComputersandStructures, 53(4), 983–991,1994.
Abstract. A variationalinequalityandquadraticprogrammingapproachis proposedfor theinvestigationof
the separationproblemof steelboltedbrackets.By applyingthe classicunilateralcontactlaw to describe
theseparationprocessalongthecontactsurfacesbetweenthebracket andthecolumnflange,thecontinuous
problemis formulatedasa variationalinequalityor asa quadraticprogrammingproblem.By applyingan
appropriatefinite elementdiscretizationscheme,thediscreteproblemis formulatedasaquadraticoptimiza-
tion problemwith inequalityconstraintswhich, in turn, canbeeffectively treatednumericallyby meansof
an appropriatequadraticoptimizationalgorithm.The applicability and the effectivenessof the methodis
illustratedby meansof anumericalapplication.
C. C. Baniotopoulosand K. M. Abdalla. Steelcolumn-to-columnconnectionsundercom-binedload—aquadratic-programmingapproach.Computers andStructures, 46(1), 13–20,1993.
Abstract. The aim of this paperis the investigationof the mechanicalbehaviour of boltedsteelcolumn-
to-columnconnectionsundermomentand axial loadsby meansof a methodthat takes into accountthe
possibility of the appearanceof detachmentphenomenabetweenthe spliceplates.As is well known, re-
gionsof detachment(callednonactive contactregionsbelow), dueto the appearanceof the prying-action
phenomenon,do appearon the adjacentfronts of suchsteelspliceplates,greatlyaffecting the mechanical
responseof steelconnectionsof this type.The significanceof theproblemunderinvestigationarisesfrom
thefactthatcolumn-to-columnsplicesareextensively appliedin any possiblecombinationto thedesignand
constructionof steelstructures.It is thereforeobvious that,sincelocal failurephenomenaon suchconnec-
tionsdueto undesirable-andnotapriori defined-detachmentbetweenthespliceplates(asconsequenceof the
developmentof theprying- actionphenomenon)maycausea total destructionof thewholesteelstructure.
For this reason,it is importantfor suchabehaviour to beaccuratelypredictedandthepreviously mentioned
nonactive contactregionson thespliceplatesto bedefined.In this sense,suchan investigationleadsto an
ameliorationof thedesignprinciplesfor boltedsteelcolumn-to-columnsplicesandto a refinementof the
respective steelconstructionstandards.
B. BankandR. Hansel.Stability of mixed-integerquadratic-programmingproblems.Mathe-
maticalProgrammingStudies, 21, 1–17,1982.
F. Barahona,M. Junger, andG. Reinelt. Experimentsin quadratic0–1programming.Mathe-
maticalProgramming, 44(2), 127–137,1989.
E.W. BarankinandR.Dorfman.Towardquadraticprogramming.Reportto thelogisticsbranch,
Officeof Naval Research,1955.
E. W. BarankinandR. Dorfman. A methodfor quadraticprogramming.Econometrica, 24,
1956.
N. I. M. GOULD & PH.L. TOINT 9
E. W. BarankinandR. Dorfman. On quadraticprogramming.University of California Publi-
cationsin Statistics, 2(13),285–318,1958.
H. J.C. Barbosa,F. M. P. Raupp,andC. C. H. Borges.Numericalexperimentswith algorithmsfor boundconstrainedquadraticprogrammingin mechanics.ComputersandStructures,64(1–4),579–594,1997.
Abstract. In this work, the computationalperformanceof somealgorithmsfor solving boundconstrained
quadraticprogrammingproblemsis comparedby meansof numericalexperiments.The model problems
usedto testthebehaviour of thealgorithmsconsideredweretheobstacleproblemfor a membraneandthe
contactproblemin infinitesimalelasticity. Bothproblemsinvolveddifferentloadconditionsandparameters.
Thefinite elementmethodwasusedfor thespatialdiscretizationprocess.
J.L. Barlow andG. Toraldo. Theeffect of diagonalscalingon projectedgradientmethodsfor
boundconstrainedquadraticprogrammingproblems. OptimizationMethodsand Soft-
ware, 5(3), 235–245,1995.
R. O. Barr. An efficient computationalprocedurefor a generalizedquadraticprogramming
problem.SIAMJournalon Control, 7(3), 415–429,1999.
R. H. Bartels,G. H. Golub,andM. A. Saunders.Numericaltechniquesin mathematicalpro-
gramming. In J. B. Rosen,O. L. MangasarianandK. Ritter, eds,‘NonlinearProgram-
ming’, pp.123–176.AcademicPress,London,England,1970.
G. Basheinand M. Enns. Computationof optimal controlsby a methodcombiningquasi-linearizationandquadraticprogramming.InternationalJournal of Control, 16(1), 177–187,1972.
Abstract. Quadraticprogramming(QP)haspreviously beenappliedto thecomputationof theoptimalcon-
trols for linear systemswith quadraticcost criteria. This paperextendsthe applicationof QP to nonlin-
earproblemsthroughquasi-linearizationand the solutionof a sequenceof linear-quadraticsub-problems
whosesolutionsconverge to the solutionof the original non-linearproblem.The methodis calledquasi-
linearization-quadraticprogrammingor Q-QP.
E. M. L. Beale. On quadraticprogramming.NavalResearch LogisticsQuarterly, 6(3), 227–
243,1959.
E.M. L. Beale.Theuseof quadraticprogrammingin stochasticlinearprogramming.Technical
ReportP-2404-1,TheRAND Corporation,SantaMonica,CA, USA, 1961.
E. M. L. Beale.Noteon ’a comparisonof two methodsin quadraticprogrammng.Operations
Research, 14, 442–443,1966.
E.M. L. Beale.An introductionto Beale’smethodof quadraticprogramming.In J.Abadie,ed.,
‘Nonlinear programming’,pp. 143–153,North Holland, Amsterdam,the Netherlands,
1967.
E. M. L. BealeandR. Benveniste. Quadraticprogramming. In ‘Design andImplementationof OptimizationSoftware’, pp. 249–258.Sijthoff andNoordhoff, AlphenaandenRijn,Netherlands,1978.
10 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. Following a generalintroductionto the theory of quadraticprogramming,the paperdescribes
computationalaspectsof anew algorithmfor convex quadraticprogramming.An essentialfeatureis thatthe
only informationneededaboutthe objective function is the gradientdirectionat successive trial solutions
(andthevalueof theobjective functionat thefinal solution).Theconstraintsarehandledasin theReduced
GradientMethod.The methodis essentiallya generalizationof the methodof ConjugateGradients.But
pureConjugateGradients,althoughfinite, requirea completerestartwhenever thesetof active constraints
changes.If storagespaceis available,thealgorithmstoresadditionaldirectionsin awaythatavoidstheneed
for acompleterestart.
J.E. Beasley. Heuristicalgorithmsfor theunconstrainedbinaryquadraticprogrammingprob-lem. Technicalreport, Departmentof Mathematics,Imperial College of ScienceandTechnology, London,England,1998.
Abstract. In this paperwe considerthe unconstrainedbinary quadraticprogrammingproblem.This is the
problemof maximisinga quadraticobjective by suitablechoiceof binary (zero-one)variables.We present
two heuristicalgorithmsbasedupontabu searchandsimulatedannealingfor this problem.Computational
resultsarepresentedfor a numberof publically availabledatasetsinvolving up to 2500variables.An inter-
estingfeatureof our resultsis thatwhilst for mostproblemstabu searchdominatessimulatedannealingfor
theverylargestproblemsweconsidertheconverseis true.Thispapertypifiesa”multiple solutiontechnique,
singlepaper”approach,i.e.anapproachthatwithin thesamepaperpresentsresultsfor anumberof different
heuristicsappliedto thesameproblem.Issuesrelatingto algorithmicdesignfor suchpapersarediscussed.
C. R. Bector. Indefinitequadraticprogrammingwith standarderrorsin objective. Cahiers du
Centre d’EtudesdeRechercheOperationalle, 10, 247–253,1968.
C. R. BectorandM. Dahl. Simplex type finite iterative techniqueandduality for a special
type of pseudo-concave quadraticprogram,. Cahiers du Centre d’Etudesde Recherche
Operationalle, 16, 207–222,1974.
L. BehjatandA. Vannelli.VLSI concentricpartitioningusinginteriorpointquadraticprogram-ming. In ‘ISCAS’99.Proceedingsof the1999IEEEInternationalSymposiumonCircuitsandSystemsVLSI. IEEE,Piscataway, NJ,USA’, Vol. 6, pp.93–96,1999.
Abstract. This paperpresentsa novel approachfor solvingthestandardcell placementproblem.A relaxed
quadraticformulationof theproblemis solvediteratively incorporatingtechniquesto increasethespreading
of cells,includingintroducingattractorsanddynamicfirst momentconstraints.At eachiteration,a percent-
ageof thecells thatarecloseto theboundaryof thechip arefixed.This procedureis donerecursively until
at leasteightypercentof thecellsarefixed.Numericalsimulationof theproposedapproachis presentedfor
testsystems.
L. Y. Belousov. Quadraticprogrammingin problemsof optimal planningof trajectorymea-surements.CosmicResearch, 9(6), 750–759,1971.
Abstract. Theproblemof optimalplanningof trajectorymeasurementsof two differentmeasuredparameters
is investigatedfor thecaseof alimiteddispersion,anarbitrarycorrelationcouplingof themeasurementerrors
of eachparameterseparately, andabsenceof correlationbetweenmeasurementsof differentparameters.It
is shown that the problemposedcanbe reducedto solving a problemin quadraticprogrammingbasedon
thelinear-programmingmethodgeneralizedfor thecontinuouscase.In conclusion,theproblemis statedby
inductionfor anarbitrarynumberof independentmeasuredparameters.
T. Belytschko. Discussionof elastic-plasticanalysisby quadraticprogramming. American
Societyof Civil Engineering, Journal of theEngineeringMechanicsDivision, 100, 130–
131,1974.
N. I. M. GOULD & PH.L. TOINT 11
A. Bemporad,M. Morari, V. Dua, andE. N. Pistikopoulos. The explicit solutionof modelpredictive control via multiparametricquadraticprogramming. In ‘Proceedingsof the2000AmericanControlConference,Danvers,MA, USA’, Vol. 2, pp.872–876,2000.
Abstract. The control basedon online optimization,popularlyknown asmodelpredictive control (MPC),
haslong beenrecognizedasthe winning alternative for constrainedsystems.The main limitation of MPC
is, however, its onlinecomputationalcomplexity. For discrete-timelinear time-invariantsystemswith con-
straintson inputsandstates,we developanalgorithmto determineexplicitly thestatefeedbackcontrol law
associatedwith MPC,andshow thatit is piecewise linearandcontinuous.Thecontrollerinheritsall thesta-
bility andperformancepropertiesof MPC,but theonlinecomputationis reducedto a simplelinearfunction
evaluationinsteadof theexpensive quadraticprogram.Thenew techniqueis expectedto enlarge thescope
of applicabilityof MPC to small-size/fast-samplingapplicationswhichcannotbecoveredsatisfactorily with
anti-windupschemes.
M. BenDaya.Line searchtechniquesfor thelogarithmicbarrierfunctionin quadraticprogram-ming. Journalof theOperationalResearch Society, 46(3), 332–338,1995.
Abstract. In thispaper, weproposealine-searchprocedurefor thelogarithmicbarrierfunctionin thecontext
of aninteriorpointalgorithmfor convex quadraticprogramming.Preliminarytestingshowsthattheproposed
procedureis superiorto someother line-searchmethodsdevelopedspecificallyfor the logarithmicbarrier
functionin theliterature.
M. Ben DayaandK. S. Al Sultan. A new penaltyfunction algorithmfor convex quadraticprogramming.EuropeanJournalof OperationalResearch, 101(1), 155–163,1997.
Abstract. Wedevelopanexteriorpointalgorithmfor convex quadraticprogrammingusingapenaltyfunction
approach.Eachiteration in the algorithmconsistsof a singleNewton stepfollowed by a reductionin the
valueof thepenaltyparameter. Thepointsgeneratedby thealgorithmfollow anexterior paththatwedefine.
Convergenceof thealgorithmis established.Theproposedalgorithmwasmotivatedby thework of Al-Sultan
andMurty (1991)on nearestpoint problems,a specialquadraticprogram.A preliminaryimplementationof
thealgorithmproducedencouragingresults.In particular, thealgorithmrequiresasmallandalmostconstant
numberof iterationsto solve thesmallto mediumsizeproblemstested.
M. BenDayaandC. M. Shetty. Polynomialbarrierfunctionalgorithmsfor convex quadratic
programming.ArabianJournal for ScienceandEngineering, 15(4B), 656–670,1990.
J.M. Bennett.Quadraticprogrammingandpiecewiselinearnetworkswith structuralengineer-
ing applications,. In A. Prekopa,ed., ‘Survey of MathematicalProgramming,Vol. 3’,
pp.95–105,North Holland,Amsterdam,theNetherlands,1979.
P. Benson,R. L. Smith, I. E. Schochetman,andJ. C. Bean. Optimal solutionapproximationfor infinite positive-definitequadraticprogramming.Journalof OptimizationTheoryandApplications, 85(2), 235–248,1995.
Abstract. We considera generaldoubly-infinite, positive-definite,quadraticprogrammingproblem.We
show that the sequenceof uniqueoptimal solutionsto the naturalfinite-dimensionalsubproblemsstrongly
converges to the uniqueoptimal solution.This offers the opportunityto arbitrarily well approximatethe
infinite-dimensionaloptimalsolutionby numericallysolving a sufficiently large finite-dimensionalversion
of theproblem.We thenapplyour resultsto a generaltime-varying, infinite- horizon,positive-definite,LQ
controlproblem.
R. Benveniste.A quadraticprogrammingalgorithmusingconjugatesearchdirections.Mathe-maticalProgramming, 16(1), 63–80,1979.
12 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. A quadraticprogrammingalgorithmis presented,resemblingBeale’s (1955)quadraticprogram-
ming algorithmandWolfe’s reducedgradientmethod.It usesconjugatesearchdirections.Thealgorithmis
conceivedasbeingparticularlyappropriatefor problemswith a largeHessianmatrix.An outlineof thesolu-
tion to thequadraticcapacity-constrained transportationproblemusingtheabove methodis alsopresented.
R.Benveniste.Onewayto solvetheparametricquadraticprogrammingproblem.MathematicalProgramming, 21(2), 224–228,1981.
Abstract. A methodis presentedfor thesolutionof theparametricquadraticprogrammingproblemby the
useof conjugatedirections.
C. Bergthaller. A quadraticequivalentof the minimum risk problem. RevueRoumainedo
MathematiquesPureet Appliquees, 15, 17–23,1970.
C. Bergthaller. Parametricquadraticprogramming.In ‘4th conferenceon probability theory.Abstracts.AcadSocialistRepublicof Rumania,Bucharest,Romania’,pp.23–24,1971a.
Abstract. Thispaperdealswith theparametricprogrammingproblemmincTx 1 2xT Dx A λ x b, x 0,
where: c x Rn, b Rm, D is a symmetricnxn positive definitematrix, A λ Ao λAl , Ao andAl are
fixedm n matrices,suchthattherankof Al is l andλ 0 is realparameter. Someparticularcasesare:1)
Oneelementof thematrix A is a linear functionof lambdaandall othersareconstant.2) A columnof mod
A is a linear (vectorial)function of λ aj aoj λal
j andtheothersareconstant.3) A row of A is a linear
(vectorial)functionof λ α j αoj λαl
i andtheothersareconstant.
C. Bergthaller. Quasi-convex quadraticprogramming. ComptesRendusHebdomadairesdesSeancesde l’AcademiedesSciences,SerieA (ScienceMathematiques), 273(15), 685–686,1971b.
Abstract. A simplicial algorithm is given for the programminqTx 1 2xT Qx Ax b x 0 whereq is
an n-dimensionalvector andQ is a symmetricalmatrix suchthat the objective function f x identical to
qTx 1 2xT Qx is quasi-convex for x 0 without beingconvex.
A. B. Berkelaar. Sensitivity analysisin (degenerate)quadraticprogramming. EconometricInstituteReport30, EconometricInstitute,ErasmusUniversity, Rotterdam,TheNether-lands,1997.
Abstract. In this paperwe dealwith sensitivity analysisin convex quadraticprogramming,without making
assumptionson nondegeneracy, strict convexity of the objective function, and the existenceof a strictly
complementarysolution.We show that theoptimalvalueasa functionof a right–handsideelement(or an
elementof the linearpartof theobjective) is piecewisequadratic,wherethepiecescanbecharacterizedby
maximalcomplementarysolutionsandtripartitions.Further, we investigatedifferentiabilityof this function.
A new algorithmto computetheoptimalvaluefunction is proposed.Finally, we discusstheadvantagesof
thisapproachwhenappliedto mean–varianceportfolio models.
A. B. Berkelaar, B. Jansen,C. Roos,andT. Terlaky. Sensitivity analysisin quadraticpro-gramming. Report96-11,EconometricInstitute,ErasmusUniversity, Rotterdam,TheNetherlands,1996.
Abstract. In this paperwe dealwith sensitivity analysisin convex quadraticprogramming,without making
assumptionson nondegeneracy, strict convexity of the objective function, and the existenceof a strictly
complementarysolution.We show that theoptimalvalueasa functionof a right–handsideelement(or an
elementof the linearpartof theobjective) is piecewisequadratic,wherethepiecescanbecharacterizedby
maximalcomplementarysolutionsandtripartitions.Further, we investigatedifferentiabilityof this function.
A new algorithmto computetheoptimalvaluefunction is proposed.Finally, we discusstheadvantagesof
thisapproachwhenappliedto mean–varianceportfolio models.
N. I. M. GOULD & PH.L. TOINT 13
A. B. Berkelaar, B. Jansen,K. Roos,andT. Terlaky. Basis-andpartition identificationforquadraticprogrammingandlinearcomplementarityproblems.MathematicalProgram-ming, 86(2), 261–282,1999.
Abstract. Optimal solutionsof interior point algorithmsfor linear andquadraticprogrammingand linear
complementarityproblemsprovide maximally complementarysolutions.Maximally complementarysolu-
tionscanbecharacterizedby optimalpartitions.Ontheotherhand,thesolutionsprovidedby simplex-based
pivot algorithmsaregivenin termsof complementarybases.A basisidentificationalgorithmis analgorithm
whichgeneratesacomplementarybasis,startingfrom any complementarysolution.A partitionidentification
algorithmis analgorithmwhichgeneratesa maximallycomplementarysolution(andits correspondingpar-
tition), startingfrom any complementarysolution.In linearprogrammingsuchalgorithmswererespectively
proposedby Megiddo in 1991andBalinski andTucker in 1969.In this paperwe will presentidentifica-
tion algorithmsfor quadraticprogrammingandlinear complementarityproblemswith sufficient matrices.
Thepresentedalgorithmsarebasedon theprincipalpivot transformandtheorthogonalitypropertyof basis
tableaus.
A. B. Berkelaar, K. Roos,and T. Terlaky. The optimal partition and optimal set approachto linear and quadraticprogramming. EconometricInstitute Report51, EconometricInstitute,ErasmusUniversity, Rotterdam,TheNetherlands,1997.
Abstract. In this chapterwe describetheoptimalsetapproachfor sensitivity analysisfor LP. We show that
optimalpartitionsandoptimalsetsremainconstantbetweentwo consecutive transition-pointsof theoptimal
valuefunction.Theadvantageof usingthisapproachinsteadof theclassicalapproach(usingoptimalbases)
is shown. Moreover, we presentanalgorithmto computethepartitions,optimalsetsandtheoptimalvalue
function. This is a new algorithm and usesprimal and dual optimal solutions.We also extend someof
the resultsto parametricquadraticprogramming,anddiscussdifferencesandresemblanceswith the linear
programmingcase.
H. Bernau. Upper boundtechniquesfor quadraticprogramming. AlkalmazottMatematikaiLapok, 3(1–2),161–170,1977. Seealso,A. Prekopa,ed.Survey of MathematicalPro-gramming, Vol.1, North-Holland,Amsterdam,pp.347–356,1979.
Abstract. An extensionof themethodsof Wolfe, JagannathanandBealeis presentedfor quadraticprogram-
ming problemswith upperboundsfor thevariables.It is shown that the upperboundstechniquefor linear
programmingproblemscanbeveryeasilyincorporatedin thesemethods.
H. Bernau. Quadraticprogrammingproblemsandrelatedlinear complementarityproblems.Journalof OptimizationTheoryandApplications, 65(2), 209–222,1990.
Abstract. Investigatesthegeneralquadraticprogrammingproblem,i.e. theproblemof finding theminimum
of aquadraticfunctionsubjectto linearconstraints.In thecasewhere,over thesetof feasiblepoints,theob-
jective functionis boundedfrom below, thisproblemcanbesolvedby theminimizationof a linearfunction,
subjectto thesolutionsetof a linearcomplementarityproblem,representingtheKuhn-Tucker conditionsof
the quadraticproblem.To detectin the quadraticproblemthe unboundednessfrom below of the objective
function,necessaryandsufficientconditionsarederived.It is shown that,whentheseconditionsareapplied,
thegeneralquadraticprogrammingproblembecomesequivalentto theinvestigationof anappropriatelyfor-
mulatedlinearcomplementaryproblem.
O.Bertoldi,M. V. Cazzol,A. Garzillo,andM. Innorta.A dualquadraticprogrammingalgorithmorientedto the probabilisticanalysisof large interconnectednetworks. In ‘PSCC.Pro-ceedingsof theTwelfth Power SystemsComputationConference.Power Syst.Comput.Conference,Zurich,Switzerland’,Vol. 2, pp.1249–1255,1996.
Abstract. A fastandrobust innovative computingprocedurehasbeendevelopedaimedat allowing theuse
of optimalpower flow techniquesin theframework of theprobabilisticadequacy assessmentof large inter-
connectedpower systems.Thepaperdescribesthemethodologicalapproachandthe relevant implemented
14 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
algorithm.Several numericalresultsaresuppliedwhich demonstratethe high computingefficiency of the
proceduresothat it is suitablein theprobabilisticsimulationdomain.
M. J.Best. Equivalenceof somequadraticprogrammingalgorithms.MathematicalProgram-ming, 30(1), 71–87,1984.
Abstract. The authorformulatesa generalalgorithmfor the solutionof a convex (but not strictly convex)
quadraticprogrammingproblem.Conditionsaregivenunderwhich theiteratesof thealgorithmareuniquely
determined.The quadraticprogrammingalgorithmsof Fletcher(1971),Gill andMurray (1978),Bestand
Ritter(1976),andvandePanneandWhinston/Dantzig(1969)areshown to bespecialcasesandconsequently
areequivalentin thesensethatthey constructidenticalsequencesof points.Thevariousmethodsareshown
to differ only in the mannerin which they solve the linear equationsexpressingthe Kuhn-Tucker system
for theassociatedequalityconstrainedsubproblems.Equivalenceresultshave beenestablishedby Goldfarb
(1972)andDjang (1979) for the positive definite Hessiancase.The analysisextendstheseresultsto the
positive semi-definitecase.
M. J.Best.An algorithmfor parametricquadraticprogramming.In H. Fischer, B. Rieduller and
S. Schaffler, eds,‘Applied MathematicsandParallel Computing—Festschriftfor Klaus
Ritter’, pp.57–76.Physica-Verlag,Heidelburg, 1996.
M. J. Best and R. J. Caron. A methodto increasethe computationalefficiency of certainquadraticprogrammingalgorithms.MathematicalProgramming, 25(3), 354–358,1983.
Abstract. Presentsamethodfor computingtheKuhn-Tuckermultipliersassociatedwith equalityconstraints
in quadraticprogrammingproblems.Whenappliedto acertainclassof algorithmsasignificantreductionin
computationtime andin storageis achieved.
M. J.BestandR. J.Caron.A parameterizedHessianquadraticprogrammingproblem.Annalsof OperationsResearch, 5(1–4),373–394,1986.
Abstract. Presentsa generalactive setalgorithmfor thesolutionof a convex quadraticprogrammingprob-
lemhaving aparametrizedHessianmatrix.TheparametricHessianmatrix is apositive semidefiniteHessian
matrix plusa real parametermultiplying a symmetricmatrix of rank oneor two. The algorithmsolvesthe
problemfor all parametervaluesin theopeninterval uponwhich theparametricHessianis positive semidef-
inite. The algorithm is generalin that any of several existing quadraticprogrammingalgorithmscan be
extendedin astraightforwardmannerfor thesolutionof theparametricHessianproblem.
M. J. BestandN. Chakravarti. Stability of linearly constrainedconvex quadraticprograms.
Journalof OptimizationTheoryandApplications, 64(1), 43–53,1990.
M. J. BestandN. Chakravarti. An On2 active setmethodfor solving a certainparametric
quadraticprogram. Journal of OptimizationTheoryand Applications, 72(2), 213–224,1992.
Abstract. The paper presents an O n2 method for solving the parametric quadratic program
min 1 2 xT Dx aT x λ 2 ∑nj 1 γ j xj c 2, having lower andupperboundson thevariables,for all non-
negative valuesof theparameterlambda. Here,D is apositive diagonalmatrix,aanarbitraryn-vector, each
γ j , j 1 n, andc arearbitraryscalars.An applicationto economicsis alsopresented.
M. J.BestandK. Ritter. An effectivealgorithmfor quadraticminimizationproblems.Technical
report1691,Universityof Wisconsin,Madison,Wisconsin,USA, 1976.
M. J. BestandK. Ritter. A quadraticprogrammingalgorithm. ZOR,MethodsandModelsofOperationsResearch, 32(5), 271–297,1988.
N. I. M. GOULD & PH.L. TOINT 15
Abstract. By usingconjugatedirectionsa methodfor solving convex quadraticprogrammingproblemsis
developed.Thealgorithmgeneratesa sequenceof feasiblesolutionsandterminatesaftera finite numberof
iterations.Extensionsof thealgorithmfor nonconvex andlargestructuredquadraticprogrammingproblems
arediscussed.
D. Bhatia. Duality for quadraticprogrammingin complex space.Zeitschrift fur AngewandteMathematikundMechanik, 54(1), 55–57,1974.
Abstract. The duality theoremsof Rami (1972) for symmetricdual quadraticprogramsare extendedin
complex spaceover arbitrarypolyhedralcones.
S. Bhowmik, S. K. Goswami, andP. K. Bhattacherjee.Distribution systemplanningthroughcombinedheuristicandquadraticprogrammingapproach.Electric MachinesandPowerSystems, 28(1), 87–103,2000.
Abstract. Thepresentpaperreportsanew techniquefor theplanningof a radialdistribution system.Distri-
butionsystemplanninghasbeenformulatedasaproblemof quadraticmixedintegerprogramming(QMIP).
A two-stageiterative solution techniquehasbeenproposedwherethe first stagedeterminesthe optimum
substationsitesand the secondstagedeterminesthe optimum network configurations.To reducethe di-
mensionalityproblem,the integer constraintsarefirst related,thusconverting the quadraticmixed integer
programmingprobleminto a quadraticprogramming(QP)problem.After thesolutionof theQPproblem,
integerconstraintsareimposedusingheuristictechniques.
D. Bienstock.Computationalstudyof a family of mixed-integerquadraticprogrammingprob-lems. MathematicalProgramming, 74(2), 121–140,1996. Seealso, Integer Program-ming andCombinatorialOptimization.4th InternationalIPOCConference,Proceedings(Balas,E. andClausen,J.,eds.),Springer-Verlag,Berlin, Germany, pages90–94,1995.
Abstract. Wepresentcomputationalexperiencewith abranch-and-cutalgorithmto solvequadraticprogram-
ming problemswherethereis anupperboundon thenumberof positive variables.Suchproblemsarisein
financialapplications.Thealgorithmsolvesthelargestreal-life problemsin a few minutesof run-time.
A. Billionnet andA. Sutter. Persistency in quadratic0–1optimization.MathematicalProgram-
ming, 54(1), 115–119,1992.
A. Bjorck. Constrainedleast-squaresproblems. In ‘Numerical Methodsfor LeastSquares
Problems’,chapter5, pp.187–213.SIAM, Philadelphia,USA, 1996.
E. Blum and W. Oettli. Direct proof of the existencetheoremfor quadraticprogramming.OperationsResearch, 20(1), 165–167,1972.
Abstract. A directanalyticalproof is givenfor thefollowing theorem:If theinfimumof aquadraticfunction
onanonempty(possiblyunbounded)polyhedralsetRcontainedin IRn is finite, thentheinfimumis assumed
somewhereonR, thusbeingaminimum.
P. T. Boggs,P. D. Domich,andJ.E. Rogers.An interior point methodfor generallarge-scalequadraticprogrammingproblems.Annalsof OperationsResearch, 62, 419–437,1996a.
Abstract. Presentsan interior point algorithmfor solvingbothconvex andnonconvex quadraticprograms.
The method,which is an extensionof the authors‘ interior point work on linear programmingproblems,
efficiently solvesa wide classof large-scaleproblemsandforms the basisfor a sequentialquadraticpro-
gramming(SQP)solver for generallarge scalenonlinearprograms.The key to the algorithm is a three-
dimensionalcostimprovementsubproblem,which is solvedat every iteration.Theauthorshave developed
anapproximaterecenteringprocedureandanovel, adaptive big-M PhaseI procedurethatareessentialto the
successof thealgorithm.Theauthorsdescribethebasicmethodalongwith therecenteringandbig-M Phase
I procedures.Detailsof theimplementationandcomputationalresultsarealsopresented.
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P. T. Boggs,P. D. Domich,J. E. Rogers,andC. Witzgall. An interior-point methodfor linear
andquadraticprogrammingproblems.COAL Newsletter, 19, 32–40,1991.
P. T. Boggs,P. D. Domich,J.E. Rogers,andC. Witzgall. An interior point methodfor generallargescalequadraticprogrammingproblems.Annalsof OperationsResearch, 62, 419–437,1996b.
Abstract. In this paperwe presentan interior point algorithm for solving both convex and nonconvex
quadraticprograms.The method,which is an extensionof our interior point work on linear program-
ming problems,efficiently solvesa wide classof largescaleproblemsandformsthebasisfor a sequential
quadraticprogramming(SQP)solver for generallarge scalenonlinearprograms.The key to the algorithm
is a 3-dimensionalcost-improvementsubproblem,which is solvedat every iteration.Wehave developedan
approximaterecenteringprocedureanda novel, adaptive big-M PhaseI procedurethatareessentialto the
success.We describethebasicmethodalongwith therecenteringandbig-M PhaseI procedures.Detailsof
theimplementationandcomputationalresultsarealsopresented.
N. L. Boland.A dual-active-setalgorithmfor positivesemidefinitequadraticprogramming.In
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Abstract. Becauseof themany importantapplicationsof quadraticprogramming,fastandefficient methods
for solving quadraticprogrammingproblemsare valued.Goldfarb and Idnani (1983) describeone such
method,Well known to beefficientandnumericallystable,theGoldfarbandIdnanimethodsuffersonly from
therestrictionthatin its original form it cannotbeappliedto problemswhicharepositivesemi-definiterather
thanpositive definite.In this paper, we presenta generalizationof the Goldfarb andIdnani methodto the
positive semi-definitecaseandprove finite terminationof thegeneralizedalgorithm.In our generalization,
wepreserve thespirit of theGoldfarbandIdnanimethod,andextendtheirnumericallystableimplementation
in anaturalway.
I. M. BomzeandG. Danninger. A global optimizationalgorithmfor concave quadraticpro-
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J. F. Bonnansand M. Bouhtov. The trust region affine interior point algorithm for convexandnonconvex quadraticprogramming.RAIRO-RechercheOperationnelle—OperationsResearch, 29(2), 195–217,1995.
Abstract. Westudyfrom a theoreticalandnumericalpoint of view aninterior point algorithmfor quadratic
QPusingatrustregion idea,formulatedby YeandTse(1989).Weshow that,underanondegeneracy hypoth-
esis,thealgorithmconvergesglobally in theconvex case.For anonconvex problem,underamild additional
hypothesis,thesequenceof pointsconvergesto a stationarypoint.Weobtainalsoanasymptoticlinearcon-
vergenceratefor thecostthatdependsonly on thedimensionof theproblem.Whenweshow that,provided
somemodificationsareaddedto thebasicalgorithm,themethodhasagoodnumericalbehaviour.
J. C. G. Boot. Noteson quadraticprogramming:The Kuhn-Tucker andThiel-Van de Panne
conditions,degeneracy andequalityconstraints.ManagementScience, 8, 85–98,1961.
J. C. G. Boot. Binding constraintproceduresof quadraticprogramming. Econometrica,
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J.C. G. Boot. On sensitivity analysisin convex quadraticprogrammingproblems.Operations
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J.C. G. Boot. Quadratic Programming. Algorithms—Anomalies—Algorithms. North Holland,
Amsterdam,theNetherlands,1964.
J.C. G. Boot andH. Theil. A procedurefor integermaximizationof a definitequadraticform.
In G. KrewerasandG. Morlat, eds,‘Proceedingsof the3rd InternationalConferenceon
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V. L. Borre andS. G. Kapoor. A multi-stagequadratic-programmingoptimizationtechnique
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J.M. Borwein. Necessaryandsufficient conditionsfor quadraticminimality. NumericalFunc-
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R. J.Bosch,Y. Ye,andG. G. Woodworth. A convergentalgorithmfor quantileregressionwithsmoothingsplines.ComputationalStatisticsandDataAnalysis, 19, 613–630,1995.
Abstract. An importantpracticalproblemis thatof determininganonparametricestimateof theconditional
quantileof y givenx. If webalancefidelity to thedatawith asmoothnessrequirement,theresultingquantile
functionis a cubicsmoothingspline.Wereformulatethis estimationprocedureasa quadraticprogramming
problem,with associatedoptimality conditions.A recentlydevelopedinterior point algorithmwith proven
convergenceis extendedto solve the quadraticprogram.This solutioncharacterizesthe desirednonpara-
metric conditionalquantilefunction. Thesemethodsare illustratedin a study of audiologicperformance
following cochlearimplants.
A. Bouzaher. SymmetricQP andlinearprogrammingunderprimal-dualuncertainty. Opera-tionsResearch Letters, 6(5), 221–225,1987.
Abstract. A saddle-pointformulationof linearprogrammingproblemswith randomobjective functionand
RHS coefficients is proposed.Under a certaintyequivalent criterion, a pair of primal-dualdeterministic
equivalentsis derived. Theseproblemsaresymmetricdual quadraticprograms,andcanbe interpretedas
generalizationsof theclassicalmean-variancemodel.
J. Bouzitat. On Wolfe’s methodand Dantzig’s methodfor convex quadraticprogramming.RAIRO RechercheOperationelle, 13(2), 151–184,1979.
Abstract. Both Wolfe’s and Dantzig’s methodssolve linear-constrainedconvex quadraticprogramming
problemsby simplex-like algorithms.They usetheKuhn-Tucker conditions,which arenecessaryandsuf-
ficient for suchproblems.The authorpresentsthosetwo methods,with completetheoreticalproofs the
greaterpart of which, to the author’s knowledge,is new. Wolfe’s methodreceivesherea complementand
is thenfound to bemoreefficient thanit previously appeared,on accountof ’blocking’ phenomenawhich
areprovednot to stoptheconvergentprocessof thecompletedalgorithm.The’shearform’ of themethodis
consequentlyapplicableto solve any nonparametricproblem,andthe’ long form’ maybereservedfor para-
metricproblemsonly. Thepresentproofof Dantzig’s algorithmconvergenceis notbasedonthedirectstudy
of computationschemata,but usestheconvexity of thequadraticfunctionto beminimized,which leadsto
quitesimpleproof.Thegeneralpresentationof Wolfe’s andDantzig’s methodsis illustratedby a numerical
problemwhich is solvedby bothof them,soasto permitacomparison.
R. J. Braitsch. A computercomparisonof four quadraticprogrammingalgorithms. Manage-mentScience, 18(11),632–643,1972.
Abstract. This papercomparesthecomputationalperformanceof four quadraticprogrammingalgorithms.
Problemsaregeneratedandsolvedon thecomputerwith iterationcountservingastheprincipalmethodof
comparison.The effect of certainproblemparameterson rateof convergenceis consideredandcomputer
timeandstoragerequirementsof thefour algorithmsarediscussed.
18 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
N. J.Breytenbach.A structuredquadraticprogram.In J.A. Snyman,ed.,‘Proceedingsof the
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problemsareformulatedin suchaway thatrelevanceto realproblemsis readilynoted.
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Abstract. Presentsanalgorithmfor thequadraticprogrammingproblemof determininga localminimumof
f x 1 2xT Qx cT x suchthatATx b whereQ is asymmetricmatrixwhichmaynotbepositive definite.
C. A. Burdet. Generalquadraticprogramming. Technicalreport, Carnegie Mellon Univ ,Pittsburgh,PA, USA, 1971.
Abstract. An algorithmis presentedfor thegeneral(not necessarilyconvex or concave) quadraticprogram-
mingproblemoveralinearlyconstrainedset.Thealgorithmis finitely convergentandmakesuseof aconvex
quadraticprogrammingmethodasa subroutine(like thequadraticsimplex for instance).Thebasictool for
thismethodis a facialdecompositionfor polyhedralsets.
S.J.Byrne.Solutionof quadraticprogrammingproblems.New ZealandOperationalResearch,12(2), 73–89,1984.
Abstract. Quadraticprogrammingproblemsarisein anumberof situations.Typicalexamples,e.g.portfolio
selection,economicmodelling,regressionanalysis,andsolutionof non-linearprogrammingproblems,are
briefly described.Themethodsthathavebeenproposedfor solvingthisproblemarereviewed.Theapproach
usinga linear complementarityprogramis selected,andan efficient, numericallywell-behaved procedure
is developed,basedon Lemke’s algorithmandusingorthogonalfactorizations.A principalpivot versionis
alsopresented,which parallelsDantzigandCottle’s solutionprocedure,but is applicableto thesamevery
wideclassof matricesprocessedby Lemke’salgorithm.A restartfacility is developedfor thisversion,which
acceleratesthesolutionof asequenceof relatedquadraticprograms,by proceedingfrom theoptimumof the
lastproblem.Themethodshavebeencodedin FortranIV andperformwell. Theprocessis easilyextendedto
handlebothupperandlowerboundsonconstraintsandhasprovedacceptablefor usein ageneralnon-linear
programmingalgorithm.
R. CaballeroandA. Santos. A new dual methodfor solving strictly convex quadraticpro-
grams. Technicalreport,Departmentof Applied Economics(Mathematics),University
of Malaga,Spain,1998.
N. I. M. GOULD & PH.L. TOINT 19
A. V. CabotandR. L. Francis.Solvingnonconvex quadraticminimizationproblemsby ranking
theextremepoints.OperationsResearch, 18, 82–86,1970.
L. M. Cabral. An efficient algorithmfor the quadraticprogrammingproblemwith inequalityconstraints.In ‘Proceedingsof the1982AmericanControlConference.IEEE,New York,NY, USA’, Vol. 3, pp.1016–1017,1982.
Abstract. The solution to the generallinear problemAx y with side conditionscan be interpretedas
a quadraticoptimizationproblem.Sucha requirementarisesin the solution of illposed problemswhere
the methodof regularizationis exploited.A compactquadraticoptimizationprocedureis presentedwhich
obviatesthe requirementto invert large matrices,encounteredin typical applications,and therebyreduce
computationtime anderrorsdueto numericalround-off.
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quadraticbilevel programmingtestproblems.ACM Transactionson MathematicalSoft-
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P. H. Calamai,L. N. Vicente,and J. J. Judice. A new techniquefor generatingquadratic-programmingtestproblems.MathematicalProgramming, 61(2), 215–231,1993.
Abstract. Thispaperdescribesanew techniquefor generatingconvex, strictly concave andindefinite(bilin-
earor not) quadraticprogrammingproblems.Theseproblemshave a numberof propertiesthatmake them
usefulfor testpurposes.For example,strictly concave quadraticproblemswith theirglobalmaximumin the
interiorof thefeasibledomainandwith anexponentialnumberof localminimawith distinctfunctionvalues
andindefiniteandjointly constrainedbilinearproblemswith nonextremeglobalminima,canbegenerated.
Unlikemostexistingmethodsourconstructiontechniquedoesnotrequirethesolutionof any subproblemsor
systemsof equations.In addition,theauthorsknow of no othertechniquefor generatingjointly constrained
bilinearprogrammingproblems.
A. J. Calise. Statisticaldesignof sampled-datacontrol systemsvia quadraticprogramming.Technicalreport,Univ Pennsylvania,PA, USA, 1968.
Abstract. Quadraticprogrammingis appliedto the statisticaldesignof Linear sampled-datacontrol sys-
tems.Given a fixed plant, a compensatoris chosenwhich minimizesthe mean-squarevalue of an error
sequencesubjectto a setof constraints.The systeminputsaredescribedin termsof sampledvaluesfrom
autocorrelationandcross-correlationfunctions,eliminatingtheneedfor theanalyticalexpressionsrequired
in theWiener-Hopf equation.Constraintswhichcharacterizetheclosed-loopresponseto deterministicinputs
andconstraintswhich limit thecomplexity of thecompensatingnet-work canbesimultaneouslyemployed.
Thesolutiongeneratestheparametersof theoptimumcompensator. Thecasewheretheerrorsignalis not
sampledandwherethemean-squarevalueof theerroris theindex of theperformanceis alsoconsidered.
E. K. Can. A quadratic-programmingsolution to cost-timetrade-off for CPM. In ‘Applied
SimulationandModelling—ASM ’85’, Vol. 3, pp.253–256,1985.
W. CandlerandR. J.Townsley. Themaximizationof a quadraticfunctionof variablessubject
to linearinequalities.ManagementScience, 10, 515–523,1964.
20 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
E. Canestrelli,S.Giove,andR. Fuller. Stability in possibilisticquadraticprogramming.FuzzySetsandSystems, 82(1), 51–56,1996.
Abstract. Weshow thatpossibilisticquadraticprogramswith crispdecisionvariablesandcontinuousfuzzy
numbercoefficientsarewell-posed,i.e. small changesin the membershipfunction of the coefficientsmay
causeonly a smalldeviation in thepossibilitydistribution of theobjective function.
J.CanisiusandJ.L. vanHemmen.A polynomialtimealgorithmin generalquadraticprogram-ming andground-statepropertiesof spin glasses.EurophysicsLetters, 1(7), 319–326,1986.
Abstract. An algorithmis presentedwhich findsa surprisinglygoodapproximationto theglobalminimum
of a not necessarilyconvex, quadraticfunction in N variables,restrictedto a N-dimensionalcube.For in-
stance,all Ising Hamiltonianswith pair interactionsbelongto this class.In general,the time complexity of
thealgorithmis O N4 . For nearest-neighbourinteractions,it reducesto O N3 . Thealgorithmis usedto find
thegroundstateof variousIsingspinglassmodelsandto studythezero-temperaturebehavior of themagne-
tizationasafunctionof theexternalfield h in bothandthreedimensions.It is foundthatthetwo-dimensional
+or-J modelhasa nonzeromagnetizationash 0.
M. D. CanonandJ.H. Eaton.A new algorithmfor aclassof quadraticprogrammingproblems
with applicationto control. SIAMJournalon Control, 4, 34–45,1996.
J.M. Cao.Necessaryandsufficientconditionfor localminimaof aclassof nonconvex quadratic
programs.MathematicalProgramming, 69(3), 403–411,1995.
M. Capurso. A quadraticprogrammingapproachto the impulsive loading analysisof rigidplasticstructures.Meccanica, 7(1), 45–57,1972.
Abstract. This paperdiscussesthedynamicproblemof rigid plasticstructuressubjectedto impulsive load-
ing. A coupleof ’dual’ extremumtheoremsreducesthe problemto the optimizationof convex quadratic
functionssubjectto linearequalitiesandequations:thefirst theoremtakesasvariablesstressandaccelera-
tions, thesecondaccelerationsandplasticmultiplier rates.The problemis discussedin matrix notationon
the basisof finite elementdiscretizationof the structureandpiecewise linear approximationof the yield
surfaces,usingsomequadraticprogrammingconcepts.The procedureis illustratedby a simplenumerical
example.
R. J.Caron.Parametricquadraticprogramming.WindsorMathematicsReport86-01,Depart-
mentof Mathematics,Universityof Windsor, Ontario,Canada,1986.
T. J. CarpenterandD. F. Shanno.An interior point methodfor quadraticprogramsbasedonconjugateprojectedgradients.ComputationalOptimizationandApplications, 2(1), 65–28,1993.
Abstract. We proposean interior point methodfor large-scaleconvex quadraticprogrammingwhereno
assumptionsaremadeaboutthe sparsitystructureof the quadraticcoefficient matrix Q. The interior point
methoddescribedis a doubly iterative algorithmthat invokes a conjugateprojectedgradientprocedureto
obtain the searchdirection.The effect is that Q appearsin a conjugatedirection routine ratherthan in a
matrix factorization.By doing this, thematricesto be factoredhave thesamenonzerostructureasthosein
linear programming.Further, onevariantof this methodis theoreticallyconvergent with only onematrix
factorizationthroughouttheprocedure.
T. J.Carpenter, I. J. Lustig, J. M. Mulvey, andD. F. Shanno.Higher-orderpredictor-corrector
interior point methodswith applicationto quadraticobjectives. SIAMJournal on Opti-
mization, 3(4), 696–725,1993a.
N. I. M. GOULD & PH.L. TOINT 21
T. J.Carpenter, I. J.Lustig,J.M. Mulvey, andD. F. Shanno.Separablequadraticprogrammingvia a primal-dual interior point methodand its usein a sequentialprocedure. ORSAJournalon Computing, 5(2), 182–191,1993b.
Abstract. Extendsa primal-dualinterior point procedurefor linearprogramsto thecaseof convex separa-
ble quadraticobjectives.Includedareefficient proceduresfor: attainingprimal anddualfeasibility, variable
upperbounding,andfreevariables.A sequentialprocedurethatinvokesthequadraticsolver is proposedand
implementedfor solving linearly constrainedconvex separablenonlinearprograms.Computationalresults
areprovided for several large testcasesfrom stochasticprogramming.Theproposedmethodscomparefa-
vorablywith MINOS, especiallyfor the largerexamples.Thenonlinearprogramsrangein sizeup to 8700
constraintsand22000variables.
J. L. Carpentier, G. Cotto, andP. L. Niederlander. New conceptsfor automaticgenerationcontrolin electricpowersystemsusingparametricquadraticprogramming.In A. Alonso-Concheiro,ed.,‘Real Time Digital ControlApplications.Proceedingsof the IFAC/IFIPSymposium.Pergamon,Oxford,England’,pp.595–600,1984.
Abstract. New conceptsfor automaticgenerationcontrol in electricpower systemsarepresented,where
the two componentsof automaticgenerationcontrol, load frequency control and economicdispatchare
performedat the samerate, i.e. a few seconds,andwhereeconomicdispatchtakes network securityinto
account.This givesnetwork securityandgoodtransients,avoiding contradictoryactionsof load frequency
controlandeconomicdispatchon thegeneratingunits.Thecornerstoneof thesolutionis theuseof a new
faston-lineoptimalpower flow, usinganew parametricquadraticprogrammingmethod,which is presented
in details.
P. Carraresi,F. Farinaccio,andF. Malucelli. Testingoptimality for quadratic0-1 problems.TechnicalReportTR-95-11,Dipartimentodi Informatica,Universitadi Pisa,Italy, 1995.
Abstract. The issuetackledis testingwhethera given solutionof a quadratic0-1 problemis optimal.The
paperpresentsanalgorithmbasedonthenecessaryandsufficientoptimalityconditionintroducedby Hirriart-
Urruty for generalconvex problems.A measureof the quality of the solutionis provided. Computational
resultsshow theapplicabilityof themethod.Themethodis extendedto constrainedquadratic0-1 problems
suchasquadraticassignmentandquadraticknapsack.
E. Casasand C. Pola. An algorithm for indefinite quadraticprogrammingbasedon a par-tial Cholesky factorization. RAIRO-Recherche Operationnelle-Operations Research,27(4), 401–426,1993.
Abstract. A new algorithm is describedfor quadraticprogrammingthat is basedon a partial Cholesky
factorizationthatusesa diagonalpivoting strategy andallows computationof thenull of negative curvature
directions.The algorithm is numericallystableand hasshown efficiency in solving positive-definiteand
indefiniteproblems.It is speciallyinterestingin indefinitecasesbecausetheinitial pointdoesnotneedto be
a vertex of the feasibleset.The authorsthusavoid introducingartificial constraintsin the problem,which
turnsout to bevery efficient in parametricprogramming.At thesametime, techniquesfor updatingmatrix
factorizationsareused.
Y. ChabrillacandJ.-P. Crouzeix.Definitenessandsemidefinitenessof quadraticformsrevisited.
LinearAlgebra andits Applications, 63, 283–292,1984.
T. F. Chan,J.A. Olkin, andD. W. Cooley. Solvingquadraticallyconstrainedleastsquaresusing
blackboxsolvers.BIT, 32, 481–495,1992.
S. W. Chang. A methodfor quadraticprogramming. Naval Research Logistics Quarterly,33(3), 479–487,1986.
22 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. A solutionto thequadraticprogrammingis presentedwith theconstraintof theform Ax b using
thelinearcomplementaryproblemapproach.
Y. Y. ChangandR. W. Cottle. Least-index resolutionof degeneracy in quadraticprogramming.MathematicalProgramming, 18(2), 127–137,1980.
Abstract. Combinesleast-index pivot selectionruleswith Keller’s algorithmfor quadraticprogrammingto
obtainafinite methodfor processingdegenerateproblems.
A. CharnesandJ. Semple.Practicalerrorboundsfor a classof quadraticprogrammingprob-lems. Informatica, 2(3), 352–366,1991.
Abstract. The absoluteerror betweenan approximatefeasiblesolution,generatedvia a dual formulation,
andthetrueoptimalsolutionis measured.Theseerrorboundsinvolve considerablylesscomputationalwork
thanexisting estimates.
B. ChenandP. T. Harker. A noninteriorcontinuationmethodfor quadraticandlinearprogram-
ming. SIAMJournalon Optimization, 3(3), 503–515,1993.
M. ChenandJ.A. Filar. Hamiltoniancycles,quadraticprogramming,andrankingof extreme
points. In C. FloudasandP. Pardalos,eds,‘Global Optimization’, pp. 32–9.Princeton
UniversityPress,USA, 1992.
Y.-H. Chenand S. C. Fang. Neurocomputingwith time delay analysisfor solving convex
quadraticprogrammingproblems.IEEETransactionsonNeural Networks, p. (to appear),
1999.
Y. H. Chenand S. C. Fang. Neurocomputingwith time delay analysisfor solving convexquadraticprogrammingproblems.IEEE Transactionson Neural Networks, 11(1), 230–240,2000.
Abstract. This paperpresentsa neural-network computationalschemewith time-delayconsiderationfor
solvingconvex quadraticprogrammingproblems.Basedonsomeknown results,adelaymargin is explicitly
determinedfor the stability of the neuraldynamics,underwhich the statesof the neuralnetwork doesnot
oscillate.The configurationof the proposedneuralnetwork is provided. Operationalcharacteristicsof the
neuralnetwork aredemonstratedvia numericalexamples.
Z. ChenandN. Y. Deng.Somealgorithmsfor theconvex quadraticprogrammingproblemvia
theABS approach.OptimizationMethodsandSoftware, 8(2), 157–170,1997.
F. T. Cheng, T. H. Chen, and Y. Y. Sun. Efficient algorithm for resolving manipulator
redundancy—the compactQP method. In ‘1992 IEEE InternationalConferenceon
RoboticsandAutomation: Proceedings’,Vol. 1–3,pp.508–513,1992a.
F. T. Cheng,T. H. Chen,Y. S. Wang,andY. Y. Sun. Efficient algorithmfor resolvingmanip-ulator redundancy-thecompactQP method. In ‘Proceedings.1992IEEE InternationalConferenceonRoboticsAnd Automation.IEEEComput.Soc.Press,LosAlamitos,CA,USA’, Vol. 1, pp.508–513,1992b.
Abstract. Dueto hardwarelimitations,physicalconstraints,suchasjoint rateboundsandjoint anglelimits,
alwaysexist. In thepresentwork, theseconstraintsareincludedin thegeneralformulationof theredundant
inversekinematicproblem.To take into accountthesephysicalconstraints,the computationallyefficient
compactQP (quadraticprogramming)methodis derived to resolve the kinematicredundancy problem.In
addition,thecompact-inverseQPmethodis developedto remedythesingularityproblem.ThecompactQP
N. I. M. GOULD & PH.L. TOINT 23
(compactandinverseQP)methodmakesuseof thecompactformulationto obtainthegeneralsolutionsand
to eliminatethe equalityconstraints.As such,the variablesaredecomposedinto basicandfree variables,
andthe basicvariablesareexpressedby the free variables.Thus, the problemsize is reducedandit only
requiresanoptimizationalgorithm,suchasQP, for thefreevariablessubjectto pureinequalityconstraints.
Thisapproachwill expeditetheoptimizationprocessandmake real-timeimplementationpossible.
F. T. Cheng,T. H. Chen,Y. S.Wang,andY. Y. Sun.Obstacleavoidancefor redundantmanipu-latorsusingthecompactQPmethod.In ‘ProceedingsIEEE InternationalConferenceonRoboticsandAutomation.IEEE Comput.Soc.Press,Los Alamitos,CA, USA’, Vol. 3,pp.262–269,1993.
Abstract. ThecompactQP(quadraticprogramming)methodis proposedto resolve theobstacleavoidance
problemfor a redundantmanipulator. The drift-free criterion is consideredwhena redundantmanipulator
performsa repeatedmotion.Dueto thecomputationalefficiency andversatilityof thecompactQPmethod,
real-timeimplementationis ableto beachieved,andphysicallimitationssuchasjoint rateboundsandjoint
anglelimits canbeeasilytaken into account.An exampleis given to demonstratethat this methodis able
to avoid the throatof a cavity, andto remedythedrift problemwhile a primarygoalof themanipulatorsis
carriedout.Simulationresultsshow thatmultiplegoalscaneasilybefulfilled by thismethod.
F. T. Cheng,R. J. Sheu,T. H. Chen,Y. S. Wang, and F. C. Kung. The improved com-pactQP methodfor resolvingmanipulatorredundancy. In ‘IROS ’94. ProceedingsoftheIEEE/RSJ/GIInternationalConferenceonIntelligentRobotsandSystems.AdvancedRoboticSystemsandtheRealWorld. IEEE,New York,NY, USA’, Vol. 2,pp.1368–1375,1994.
Abstract. The compactQP methodis an effective and efficient algorithm for resolvingthe manipulator
redundancy underinequalityconstraints.In this paper, a morecomputationallyefficient schemewhich will
improve the efficiency of the compactQP method-theimproved compactQP method-is developed.With
the techniqueof workspacedecomposition,the redundantinversekinematicsproblemcanbe decomposed
into two subproblems.Thus,thesizeof theredundancy problemcanbereduced.For ann degree-of-freedom
spatialredundantmanipulator, insteadof a 6n matrix, only a 3 n 3 matrix is neededto be manipulated
by Gaussianeliminationwith partialpivoting for selectingthe freevariables.Thesimulationresultson the
CESARmanipulatorindicatethatthespeedupof thecompactQPmethodascomparedwith theoriginalQP
methodis about4.3.Furthermore,thespeedupof theimprovedcompactQPmethodis about5.6.Therefore,
it is believed that the improvedcompactQPmethodis oneof themostefficient andeffective optimization
algorithmfor resolvingthemanipulatorredundancy underinequalityconstraints.
C. C. N. ChuandD. F. Wong.A quadraticprogrammingapproachto simultaneousbuffer inser-tion/sizingandwire sizing. IEEE Transactionson ComputerAidedDesignof IntegratedCircuitsandSystems, 18(6), 787–798,1999.
Abstract. In this paper, we presenta completelynew approachto the problemof delayminimizationby
simultaneousbuffer insertionandwire sizing for a wire. We show that the problemcanbe formulatedas
a convex quadraticprogram,which is known to be solvable in polynomial time. Nevertheless,we explore
somespecialpropertiesof our problemandderive anoptimalandvery efficient algorithm,modifiedactive
setmethod(MASM), to solve the resultingprogram.Given m buffers anda setof m discretechoicesof
wire width, the runningtime of our algorithmis O mn2 andis independentof thewire lengthin practice.
For example,aninstanceof 100buffersand100choicesof wire width canbesolvedin 0.92s. In addition,
we extendMASM to considersimultaneousbuffer insertion,buffer sizing, andwire sizing. The resulting
algorithmMASM-BS is againoptimal andvery efficient. For example,with six choicesof buffer sizeand
10 choicesof wire width, the optimalsolutionfor a 15000µ m long wire canbe found in 0.05s. Besides,
our formulationis soversatilethat it is easyto considerotherobjectiveslike wire areaor power dissipation,
or to addconstraintsto thesolution.Also, wire capacitancelookuptables,or very generalwire capacitance
modelswhichcancaptureareacapacitance,fringing capacitance,couplingcapacitance,etc.canbeused.
24 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
C. S. ChungandD. Gale. A complementarityalgorithmfor optimal stationaryprogramsin
growth modelswith quadraticutility. TechnicalReportORC81-10,OperationsResearch
Center, Universityof California,Berkeley, CA, USA, 1981.
S. J.ChungandK. G. Murty. Polynomiallyboundedellipsoidalgorithmfor convex quadratic
programming.In O. L. Mangasarian,R. R. Meyer andS. M. Robinson,eds,‘Nonlinear
Programming,4’, pp.439–485,AcademicPress,LondonandNew York, 1981a.
S.J.ChungandK. G. Murty. Polynomiallyboundedellipsoidalgorithmsfor convex quadraticprogramming.Methodsof OperationsResearch, 40, 63–66,1981b.
Abstract. LetB, b berespectively agivensquarenonsingularintegermatrixof ordern, andanintegercolumn
vectorin IRn. It is requiredto find thenearestpoint to b in theConePos B x : x Bz z 0 . This leads
to thelinearcomplementarityproblem:find w w1 wn , z z1 zn satisfyingW BTB z BTb,
w 0 z 0, wTz 0.
T. T. Chung. Analysisof platebendingby the quadraticprogrammingapproach.Technicalreport,WashingtonUnivesity, St.Louis,MO, USA, 1974.
Abstract. Finite elementanalysisof platebendingis interpretedasa quadraticprogrammingproblem.The
total potentialenergy, expressedin termsof thecoefficientsof theapproximatingpolynomialsis theobjec-
tive function theminimumof which is soughtsubjectto linearequalityconstraints.Theconstraintsrequire
satisfactionof all kinematicboundaryconditionsandinter-elementcontinuityconditions.Convergencechar-
acteristicsof this approachwith respectto increasingordersof polynomialapproximation,aswell aswith
respectto progressively reducedelementsizes,arediscussedand illustratedwith a numberof examples.
Advantagesof theproposedapproacharediscussed,andtopicsrequiringfurtherinvestigationsareoutlined.
L. Churilov, D. Ralph,andM. Sniedovich. A noteon compositeconcave quadraticprogram-ming. OperationsResearch Letters, 23(3–5),163–169,1998.
Abstract. Wepresentapivotal-basedalgorithmfor theglobalminimizationof compositeconcave quadratic
functionssubjectto linearconstraints.It is shown thatcertainsubclassesof this family yield easy-to-solve
line searchsubproblems.Sincethe proposedalgorithmis equivalent in efficiency to a standardparametric
complementarypivoting procedure,the implication is thatconventionalparametricquadraticprogramming
algorithmscannow be usedastools for the solutionof muchwider classof complex global optimization
problems.
T. F. Colemanand L. A. Hulbert. A direct active set algorithm for large sparsequadratic
programswith simplebounds. MathematicalProgramming, SeriesB, 45(3), 373–406,
1989.
T. F. ColemanandL. A. Hulbert.A globallyandsuperlinearlyconvergentalgorithmfor convex
quadraticprogramswith simplebounds.SIAMJournal on Optimization, 3(2), 298–321,
1993.
T. F. ColemanandY. Li. A reflective Newton methodfor minimizing a quadraticfunctionsubjectto boundson someof thevariables.SIAMJournal on Optimization, 6(4), 1040–1058,1996.
Abstract. We proposea new algorithm,a reflective Newton method,for the minimizationof a quadratic
functionof many variablessubjectto upperandlower boundsonsomeof thevariables.Themethodapplies
to a general(indefinite)quadraticfunction for which a local minimum subjectto boundsis requiredand
is particularlysuitablefor the large-scaleproblem.Our new methodexhibits strongconvergenceproperties
andglobalandsecond-orderconvergenceandappearsto havesignificantpracticalpotential.Strictly feasible
N. I. M. GOULD & PH.L. TOINT 25
pointsaregenerated.We provide experimentalresultson moderatelylarge andsparseproblemsbasedon
bothsparseCholesky andpreconditionedconjugategradientlinearsolvers.
T. F. ColemanandJ.G. Liu. An interior Newton methodfor quadraticprogramming.Mathe-maticalProgramming, 85(3), 491–523,1999.
Abstract. We proposea new (interior) approachfor the generalquadraticprogrammingproblem.We es-
tablishthat thenew methodhasstrongconvergenceproperties:thegeneratedsequenceconvergesglobally
to a point satisfyingthesecond-ordernecessaryoptimality conditions,andtherateof convergenceis 2-step
quadraticif thelimit point is astronglocalminimizer. Publishedalternative interiorapproachesdonotshare
suchstrongconvergencepropertiesfor the nonconvex case.We also report on the resultsof preliminary
numericalexperiments:theresultsindicatethattheproposedmethodhasconsiderablepracticalpotential.
T. F. ColemanandJ.G. Liu. An exteriorNewtonmethodfor strictly convex quadraticprogram-ming. ComputationalOptimizationandApplications, 15(1), 5–32,2000.
Abstract. WeproposeanexteriorNewtonmethodfor strictly convex quadraticprogramming(QP)problems.
Thismethodis basedonadualformulation:asequenceof pointsisgeneratedwhichmonotonicallydecreases
thedualobjective function.Weshow thatthegeneratedsequenceconvergesgloballyandquadraticallyto the
solution(if theQPis feasibleandcertainnondegeneracy assumptionsaresatisfied).Measuresfor detecting
infeasibilityareprovided.Themajorcomputationin eachiterationis to solveaKKT-like system.Therefore,
givenaneffective symmetricsparselinearsolver, theproposedmethodis suitablefor largesparseproblems.
Preliminarynumericalresultsarereported.
D. C. Collins. Terminalstatedynamicprogramming:quadraticcosts,lineardifferentialequa-tions. Journalof MathematicalAnalysisandApplications, 31(2), 235–253,1970.
Abstract. A fairly generalclassof controlproblemscanbeposedin termsof minimizing a costfunctional
involving thestateof thesystemto becontrolledandthecontrolexertedover a fixed interval of time. The
stateandcontrolvariablesarerelatedby a stateequation,oftenwith additionalconstraintsof variousforms
uponthecontrolor state.That is, it is desiredto find miny tεY p x T !"# To q x t $ dt , wherex andy are
theN-dimensionalstatevectorandtheM-dimensionalcontrolvector, respectively, while p andq arescalar-
valuedfunctionsof their arguments.The stateequationis dx dt h x y , x 0 c, with y constrainedto
someclassof admissiblecontrols,Y. This paperdiscussesa specializationof theabove problem,a terminal
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dual stepanda Newton-like modifiedbarrierstepin orderto ensuredescenton a suitablemerit function.
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lem into realandreactive subproblems.Therealandreactive subproblemsaresolvedalternatelyuntil they
converge.Quadraticor linearprogrammingis utilized to solve thetwo subproblems.If thecostcurveof each
generatoris approximatedasa quadraticfunctionthenthecostfunctionof theOLF problemis in quadratic
form andquadraticprogrammingis appliedfor the solution. If the effect of valve point loading is to be
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form of the fareequation.A linear approximationto the demandcurve at thebasecasevaluesresultsin a
quadraticprogrammingproblem.Threealternative modesof usingthemodelsystemaredemonstratedusing
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computationaleffort andstoragerequirements,whenlinear andquadraticprogrammingmethodsareused
for solving large size power systemproblems.The effectivenessof theseprocedureshasbeentestedby
implementingthemin a new computercode,speciallydevelopedfor solving economicdispatchproblem.
Theapproved IEEE testsystemsup to 118buseswereusedfor thetestpurposes.Theresultsshow that the
improvementin thecaseof 118bussystemis ratherdramatic.Numberof nonzeroentriesin thebasisinverse,
while usingthenew code,arefoundto beonly 25%of thosereportedearlier.
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Abstract. Thefinite analysisproblemwith piecewise linearconstitutive laws is formulatedasa linearcom-
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gradientmethods,is proposedwhichovercomesthecomputationaldifficultiesthatarisewhenthereis a large
numberof variables.Throughaphysicalinterpretationof thegradientof theobjective function,eachmathe-
maticalstepof theproposedoptimizationtechniqueis translatedinto acorrespondingphysicaloperationon
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coefficients of the linear combinationaredeterminedby a quadraticprogrammingtechnique.The author
givesanexplicit examplefor systemswith sphericalsymmetry, andsubsequentlyappliesthemethodto the
ComaClusterof galaxies.Statisticallysignificantfits areobtainedusingonly asmallnumberof components.
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that theproblemof selectinganoptimalwaveformreducesto a quadratic0–1 integer program.It is shown
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problems.Theideais illustratedonaclassof extremelylargenonlinearprogrammingproblemsarisingfrom
traffic equilibrium calculationsusingboth the Frank-Wolfe andPARTAN algorithmsto partially solve the
QP subproblems.Computationalresultsindicatethat the convergencerate of the underlyingalgorithm is
indeedenhancedsignificantlywhenFrank-Wolfe is usedto solve theQPsubproblemsbut only marginally
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N. I. M. GOULD & PH.L. TOINT 31
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penaltiesareplacedon oscillationsof theunit graphratherthanon thesizeof its ordinates.Applicationof
themethodologyto realrainfall-runoff datais providedandcomparisonswith existingapproachesaremade.
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32 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
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iterationis proportionalif thenormof violation of theKuhn-Tucker conditionsat active variablesdoesnot
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directiondeterminesa descentdirection that enablesthe releasedvariablesto move far enoughfrom the
boundaryin astepcalledproportioning.An algorithmthatusestheconjugategradientmethodto explorethe
faceof theregion definedby thecurrentiterateuntil a disproportionaliterationis generatedis proposed.It
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nondegeneratethenthe algorithmfinds the solutionin a finite numberof steps.Moreover, a simplelower
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incrementaldisplacement.The performancesof several QP algorithms,including two new versionsof a
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minimizationproblem.The constrainedminimizationproblemconsideredinvolves the minimizationof a
quadraticfunctionalsubjectto linearequalityconstraints.Amongthisclassof convergentinteractive schemes
aregeneralizationsof therelaxedJacobi,Gauss-Seidel,andsymmetricGauss-Seidelschemes.
34 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
B. C. Eaves.Onquadraticprogramming.ManagementScience, 17(11),698–711,1971.
Abstract. A procedurebasedonLemke’salgorithmis developedwhicheithercomputesstationarypointsfor
generalquadraticprogramsor elseshows that theprogramhasno optimum.If a generalquadraticprogram
hasanoptimumandsatisfiesanondegeneracy conditionthenit is demonstratedthatthereareanoddnumber
of stationarypoints.
B. C. Eaves. A finite procedurefor determiningif a quadraticform is boundedbelow on a
closedpolyhedralconvex set.MathematicalProgramming, 14(1), 122–124,1978.
J.G. Ecker. Computationalproceduresin quadraticprogrammingand ) p- approximation.Bul-letin of theOperationsResearch Societyof America, 17(2), B237–238,1969.
Abstract. Abstractonly given, substantiallyas follows. In the framework of a duality theoryof extended
geometricprogramming,computationalproceduresaregiven for obtainingoptimal solutionsto quadratic
programswith quadraticconstraintsandprogramswhich minimizethe * p-normof thedifferencebetweena
fixedvectorandvariablelinearcombinationof otherfixedvectors,subjectto inequalityconstraintsexpressed
by meansof * p-norms.To apply theprocedures,onebasicallyneedonly solve a finite ’Duffin sequence’of
linearly-constrained,concave, uppersemi-continuousprograms.Thegeneralclassof programsto which the
computationalproceduresapplyproperlycontainsavarietyof specialproblems.
U. Eckhardt.Quadraticprogrammingby successiveoverrelaxation.TechnicalReportJul-1064-MA, Kernforschungsanlage,Juelich,WestGermany, 1974.
Abstract. Theideaof solvingthedefinitelinearcomplementarityproblemby successive overrelaxationwas
originally proposedby Cryer. A detaileddiscussionof Cryer’s methodappliedto quadraticprogramming
problemsis given.Theconvergencebehaviour is treatedwithout assumptionsonsolvability of theproblem.
Numericalexamplesindicatetheefficiency of themethod.
U. Eckhardt.Iterative losungquadratischeroptimierungsaufgaben.Zeitschrift fur Angewandte
MathematikundMechanik, 55, T236–T237,1975.
U. Eckhardt.Semi-infinitequadraticprogramming.ORSpektrum, 1(1), 51–55,1979.
Abstract. A methodis presentedfor minimizing a definitequadraticfunction underan infinite numberof
linearinequalityrestrictions.Specialfeaturesof themethodarethatit generatesasequenceof feasiblesolu-
tionsandasequenceof basicsolutionssimultaneouslyandthat it hasvery favourablepropertiesconcerning
numericalstability.
U. Eckhardt. Linear inequalitiesandquadraticprogramming-someapplications.MethodsofOperationsResearch, 53;, 67–81,1986.
Abstract. Someapplicationsof linear inequality systemsand quadraticprogrammingproblemsare pre-
sented.It is not intendedto go into theoreticaldetailsor to investigatenumericalspecialities.
M. M. El Metwally andZ. M. Al Hamouz. Transmissionnetworks planningusingquadraticprogramming.ElectricMachinesandPowerSystems, 18(2), 137–148,1990.
Abstract. Theproblemof transmissionnetworksexpansionhasbeensolvedby consideringthecostof losses
aswell asthecostof investmentin theobjective function.Theproblemis solvedusinganexactquadratic
programmingtechnique.Thisnew formulationhasbeenappliedto a6-bussystem.Thefinal configurations,
which arecharacterizedby minimum costof lossesshow that as the costof the kWh increases,the total
systemcostdecreases.
M. M. El Metwally andA. M. Harb. Transmissionplanningusingadmittanceapproachandquadraticprogramming.ElectricMachinesandPowerSystems, 21(1), 69–83,1993.
N. I. M. GOULD & PH.L. TOINT 35
Abstract. A methodfor transmissionnetworks planningis proposedusingquadraticprogrammingandthe
admittanceapproach.Thecostof investmentandcostof losses,load flow andsecurityconstraintsandthe
interestandinflation ratesareincluded.Thedevelopedmethodcanbeusedfor staticanddynamicmodesof
transmissionplanning.
M. A. H. El Sayed,T. M. Abdel Rahman,andM. O. Mansour. A fastquadraticprogrammingapproachfor large-scalereactivepoweroptimization.Electric MachinesandPowerSys-tems, 20(1), 17–23,1992. Seealso,EuropeanTransactionson ElectricalPower Engi-neering,volume2, number4, pages253–257,1992.
Abstract. A fastquadraticprogrammingapproachhasbeendevelopedto control thereactive power (VAR)
generationin leaddispatchingcenters.The main objective of VAr control is to minimize the transmission
lossestakinginto considerationthevoltageconstraintsateachnodeof thenetwork. Thedevelopedapproach
utilizes the decompositionof the whole systeminto smallersubsystemsandthe coordinationof different
subsystemsolutionsto obtainoptimalVAr control in the largescalesystems.Thesmallestpracticalsizeof
thedecomposedsubsystemsis theone-generator-bussubsystem.Thefastandnon-iterative VAr optimization
of the decomposedsubsystemsis obtainedbasedon consideringonly one active constraintat a time by
solving the quadraticprogrammingsub-problems.The numericaltest resultsof IEEE-118and200 buses
power systemhave indicatedthat the speedandaccuracy of the proposedapproachareadequatefor real-
time applicationson large-scalesystems.This approachis alsoapplicablefor largesystemdeviationsfrom
its normalconditionsandneedssmallmemorysize.
M. A. El Shibini andE. S. Ibrahim. Quadraticprogrammingapproachto reactive power opti-mizationon theprimaryfeeders.Archiv fur Elektrotechnik, 68(4), 267–271,1985.
Abstract. Suggeststheuseof thequadraticprogrammingtechniqueto determinetheoptimumsizeandlo-
cationof shuntcapacitorsonradialdistribution feederssoasto maximizeoverall savings,includingthecost
of capacitors.The saving function which is of quadraticform is maximizedfor a setof linear inequality
constraintsby usingquadraticprogramming.For quadraticprogramming,efficient algorithmshave beende-
velopedwhichcaneasilybeimplementedondigital computers.Theapproachis illustratedby anapplication
to a typicaldistribution feederof 23kV.
N. Elia andM. A. Dahleh. A quadraticprogrammingapproachfor solving the ) 1 multiblockproblem. IEEE Transactionson AutomaticControl, 43(9), 1242–1252,1998. Seealso,Proceedingsof the 35th IEEE Conferenceon DecisionandControl, IEEE, New York,NY, USA, volume4, pages4028–4033,1996.
Abstract. We presenta new methodto computesolutionsto the generalmulti-block l1 control problem.
Themethodis basedon solvinga standardH2 problemanda finite-dimensionalsemidefinitequadraticpro-
grammingproblemof appropriatedimension.The new methodhasmostof the propertiesthat separately
characterizemany existing approaches,in particular, asthedimensionof thequadraticprogrammingprob-
lemincreases,thismethodprovidesconverging upperandlowerboundsontheoptimal * 1 normand,for well
posedmulti-block problems,ensurestheconvergencein normof thesuboptimalsolutionsto anoptimal * 1solution.Thenew methoddoesnot requirethecomputationof theinterpolationconditions,andit allows the
directcomputationof thesuboptimalcontroller.
D. EndresandP. Foldiak. Quadraticprogrammingfor learningsparsecodes.In ‘Proceedings
of ICANN99, Edinburgh,1999’,p. (to appear),1999.
S. S. ErengucandH. P. Benson.An algorithmfor indefiniteintegerquadraticprogramming.ComputersandMathematicswith Applications, 21(6–7),99–106,1991.
Abstract. Presentsanalgorithmfor finding theglobalminimumof anindefinitequadraticfunctionover the
integerscontainedin acompact,convex set.To find thisminimum,thealgorithmfirst transformstheproblem
into anequivalentproblemwith aseparableobjective function.It thenusesabranchandboundsearchonthe
valuesof theconstraints,ratherthanthevariables,of thetransformedproblem.
36 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
S. E. EriksenandP. D. Berger. A quadraticprogrammingmodel for productconfigurationoptimization. Zeitschrift fur Operations Research, SerieB (Praxis), 31(6), 143–159,1987.
Abstract. The variousfeaturesof any productare differentially appealingto the variousportionsof the
consumerpopulationandhavedifferentialcostsof productionandmarketing.Thispaperconsiderstheprob-
lemof ’overall productoptimization’,or morespecifically, ’optimal productconfiguration’.Productpriceis
includedasapartof thisconfiguration.
S. M. Faber. Quadraticprogrammingappliedto the problemof galaxypopulationsynthesis.AstronomyandAstrophysics, 20(3), 361–374,1972.
Abstract. Thetechniqueof quadraticprogrammingasappliedto theproblemof galaxypopulationsynthesis
is described.Themethodoffers significantadvantagesover the trial-and-errorapproachusuallyemployed.
This techniqueis appliedto 38-colordataon the nuclei of M 31, M 32 andM 81 and to integrated10-
colorphotometryfor elliptical galaxies.Theresultsindicatethatestimatesof meanline strengthsin external
galaxiesby meansof populationsynthesisarewell determined.Agesbasedon the main-sequenceturnoff
point are uncertainby a factor of two. The methodis not sensitive to the numberof starson the main
sequencebetweenK 0 V andM 7 V. Consequentlytheslopeof theluminosityfunctionbelow turnoff cannot
bedetermined.Themass-to-lightratioof thecomputedpopulationfor M 31is uncertainby a factorof 4. The
correspondingfactorsfor M 32 andM 81 are50 and16 respectively. Modelsfor elliptical galaxiessuggest
thatthemeanmetalabundanceof thestellarpopulationincreaseswith increasinggalaxyluminosity.
B. J. Falkowski. Risk analysisusingperceptronsandquadraticprogramming. In ‘Computa-tional Intelligence.TheoryandApplications.InternationalConference,6th FuzzyDays,Springer-Verlag,Berlin, Germany’, pp.543–547,1999.
Abstract. A heuristicmethod(computationof weightedaverages)is consideredfor risk analysis.It is im-
proved uponby utilizing the perceptionlearningtheoremandquadraticprogramming.Experimentalwork
shows that both techniquesgive good results,the former onebeingsomewhat moreefficient in termsof
CPU-timeused.In spiteof certaintheoreticalshortcomingsit is arguedthat the familiar paradigmoffers
considerablepotentialfor practicalapplications.
J. Y. FanandL. Zhang. Real-timeeconomicdispatchwith line flow andemissionconstraintsusingquadraticprogramming. IEEE Transactionson Power Systems, 13(2), 320–325,1998.
Abstract. Thepresenceof multiple constraintsdueto network line flow limits andemissionallowancesin
theeconomicdispatchof modernpowersystemsmakestheconventionalLambda-Deltaiterative approachno
longereffective. This paperproposesa practicalstrategy basedon quadraticprogramming(QP) techniques
to solve thereal-timeeconomicdispatchproblem.It formulatestheproblemwith aquadraticobjective func-
tion basedon the unit’s cost curves in quadraticor piecewise-quadraticforms. The operationconstraints
aremodeledaslinear equality/inequality equations,resultingin a typical QPproblem.Goal programming
techniquesarealsoincorporatedin theformulationwhich guaranteesthebestavailablesolutionevenunder
infeasibleconditions.In addition,theproposedstrategy formulatestheproblemin thesecondphasedispatch
in real-timeby includingasetof emergency controlvariablesto provideeffective controlstrategiesfor prop-
erly relieving constraintviolationsif they exist. Theeffectivenessof theproposedstrategy is demonstrated
by anexamplepower dispatchproblem.
L. Fan,I. Chatterton,andL. Walker. Quadraticprogrammingfor grid supplypointmw demandcompositionanalysis.In ‘34-th UniversitiesPower EngineeringConference.Universityof Leicester, UK’, Vol. 1, pp.245–248,1999.
Abstract. This paperreportswork of analysingthe grid supplypoint (GSP)load compositionacrossthe
entireNationalGrid system.A quadraticprogrammingbasedalgorithmhasbeendevelopedfor the nodal
load compositionanalysis.The static load compositionat GSPlevel is the primary informationcrucial to
N. I. M. GOULD & PH.L. TOINT 37
further load analysis,including load dynamics,reactive load composition,and GSPload modelling and
forecasting.
S.C. FangandS.C. Puthenpura.Affinescalingfor convex quadraticprogramming.In ‘Linear
OptimizationandExtensions:TheoryandAlgorithms’, chapter9. PrenticeHall, Engle-
woodCliffs, New Jersey, USA, 1993.
S. C. FangandH. S. J. Tsao. An unconstrainedconvex programmingapproachto solvingconvex quadraticprogrammingproblems.Optimization, 27(3), 235–243,1993.
Abstract. Derivesan unconstrainedconvex programmingapproachto solving convex quadraticprogram-
ming problemsin standardform. Relatedduality theoryis establishedby usingtwo simpleinequalities.An
ε-optimalsolutionis obtainedby solvinganunconstraineddualconvex program.A dual-to-primalconver-
sion formula is alsoprovided.Somepreliminarycomputationalresultsof usinga curved searchmethodis
included.
S.C. Fang,T. M. Huang,C. H. Lin, andW. W. Lin. A relaxedinterior pathfollowing primal–
dualalgorithmfor convex quadraticprogramming.MathematicsToday, SpecialIssueon
MathematicalProgramming, XII-A , 115–144,1994.
H. FarhatandS. From. A quadraticprogrammingapproachto estimatingthe testabilityandrandomor deterministiccoverageof a VLSI circuit. VLSIDesign, 2(3), 223–231,1994.
Abstract. The testabilitydistribution of a VLSI circuit is modeledasa seriesof stepfunctionsover the in-
terval (0, 1). Themodelgeneralizespreviousrelatedwork on testability. Unlike previouswork, however, we
includeestimatesof testabilityby randomvectors.Quadraticprogrammingmethodsareusedto estimatethe
parametersof the testabilitydistribution from fault coveragedata(randomanddeterministic)on a sample
of faults.The estimatedtestabilityis thenusedto predictthe randomanddeterministicfault coveragedis-
tributionswithout theneedto employ testgenerationor fault simulations.Thepredictionof fault coverage
distribution cananswerimportantquestionsaboutthe”goodness”of a designfrom a testingpoint of view.
Experimentalresultsaregivenon thelargeISCAS-85andISCAS-89circuits.
H. Farhat,S.From,andA. Lioy. A quadraticprogrammingapproachto estimatingthetestabilityandcoveragedistributionsof a VLSI circuit. MicroprocessingandMicroprogramming,35(1–5),479–483,1992.
Abstract. Thetestabilitydistributionof aVLSI circuit is modeledasaseriesof stepfunctionsover theinter-
val (0,1).The modelgeneralizesprevious relatedwork on testability. Quadraticprogrammingmethodsare
usedto estimatetheparametersof thetestabilitydistribution from fault coveragedataona sampleof faults.
Theestimatedtestability is thenusedto predictthe fault coveragedistribution without the needto employ
testgenerationor fault simulations.Thepredictionof fault coveragedistribution cananswerimportantques-
tionsaboutthe’goodness’of a designfrom testingpoint of view. Experimentalresultson threeof thelarge
ISCAS-89benchmarkcircuitsreflecttheaccuracy of themodel.
A. M. FaustinoandJ. J.Judice. A slcpalgorithmfor bilinearandconcave quadraticprogram-
ming. InvestigacionOperacional, 8(2), 67–87,1988.
A. M. FaustinoandJ.J.Judice.Solutionof large-scaleconvex quadraticprogramsby Lemke’s
method. In M. A. TurkmanandM. L. Carvalho,eds,‘Actasda 1aConferenciaem Es-
tatisticae Optimizaaao’, pp.681–695,1992.
A. M. FaustinoandJ. J. Judice. Minimization of a concave quadraticfunctionsubjectto box
constraints.InvestigacionOperativa, 4, 49–68,1994.
38 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
L. Y. Faybusovich. Wolfe’s algorithmfor infinite-dimensionalquadraticprogrammingprob-lems.EngineeringCybernetics, 20(3), 20–30,1982.
Abstract. Wolfe’s algorithmis generalizedto infinite-dimensionalquadraticprogrammingproblemswith a
finite numberof inequalityconstraints.A coordinate-freeapproachis developedenablingoneto consider
infinite-dimensionalanaloguesof computationalproceduresof thesimplex type.
L. Y. Faybusovich andJ.B. Moore. Long-steppath-following algorithmfor convex quadraticprogrammingproblemsin a Hilbert space.Journal of OptimizationTheoryand Appli-cations, 95(3), 615–635,1997. Seealso,Proceedingsof the 34th IEEE ConferenceonDecisionandControl,IEEE,New York, NY, USA, volume2, pages1109–1114,1995.
Abstract. We developaninterior-point techniquefor solvingquadraticprogrammingproblemsin a Hilbert
space.As an example,we consideran applicationof theseresultsto the linear-quadraticcontrol problem
with linear inequalityconstraints.It is shown that the newton stepin this situationis basicallyreducedto
solvingthestandardlinear-quadraticcontrolproblem.
K. A. Fegley, S.Blum, J.O. Bergholm,A. J.Calise,J.E. Marowitz, G. Porcelli,andL. P. Sinha.Stochasticanddeterministicdesignandcontrol via linear andquadraticprogramming.IEEE Transactionson AutomaticControl, AC-16(6), 759–766,1971.
Abstract. Theapplicationof linearandquadraticprogrammingto optimalcontrolproblemsandto stochastic
or deterministicsystemdesignproblemsis discussedandillustratedwith examples.
L. Fernandes,A. Fischer, J.J.Judice,C.Requejo,andJ.Soares.A blockactivesetalgorithmforlarge-scalequadraticprogrammingwith boxconstraints.Annalsof OperationsResearch,81, 75–95,1998.
Abstract. An algorithmfor computingastationarypointof aquadraticprogramwith boxconstraints(BQP)
is proposed.Eachiterationof thisprocedurecomprisesaguessingstrategy whichforecaststheactive bounds
at a stationarypoint, thedeterminationof a descentdirectionby meansof solvinga reducedstrictly convex
quadraticprogramwith box constraintsandan exact line search.Global convergenceis establishedin the
sensethatevery accumulationpoint is stationary. Moreover, it is shown that thealgorithmterminatesafter
a finite numberof iterations,if at leastoneiterateis sufficiently closeto a stationarypoint which satisfiesa
certainsufficientoptimalitycondition.Thealgorithmcanbeeasilyimplementedfor sparselarge-scaleBQPs.
Furthermore,it simplifiesfor concave BQPs,asit is not requiredto solvestrictly convex quadraticprograms
in this case.Computationalexperiencewith large-scaleBQPsis includedandshows theappropriatenessof
this typeof methodology.
M. C. Ferris. Parallel constraintdistribution in convex quadraticprogramming.Mathematicsof OperationsResearch, 19(3), 645–658,1994.
Abstract. We considerconvex quadraticprogramswith large numbersof constraints.We distribute these
constraintsamongseveral parallelprocessorsandmodify theobjective function for eachof thesesubprob-
lemswith Lagrangemultiplier informationfrom theotherprocessors.New Lagrangemultiplier information
is aggregatedin a masterprocessorandthewholeprocessis repeated.Linearconvergenceis establishedfor
stronglyconvex quadraticprogramsby formulatingthe algorithmin an appropriatedual space.The algo-
rithm correspondsto astepof aniterative matrix splittingalgorithmfor asymmetriclinearcomplementarity
problemfollowedby aprojectionontoa subspace.
J. A. Filar. Quadraticprogrammingand the single-controllerstochasticgame. Journal of
MathematicalAnalysisandApplications, 113(1), 136–147,1986.
J.A. Filar. TheHamiltoniancycleproblem,controlledMarkov chainsandquadraticprogram-
ming. In ‘Proceedingsof ASOR12thNationalConference’,Vol. 15,pp.263–281,1993.
N. I. M. GOULD & PH.L. TOINT 39
J. A. Filar, M. Oberije,andP. M. Pardalos. Hamiltoniancycle problem,controlledMarkov
chainsandquadraticprogramming.In ‘Proceedingsof the12thNationalConferenceof
theAustralianSocietyfor OperationsResearch’,pp.263–281,1993.
F. L. Filippelli, M. Forti, andS.Manetti.New linearandquadraticprogrammingneuralnetwork.ElectronicsLetters, 30(20),1693–1694,1994.
Abstract. A neuralnetwork is proposedfor solving linearandquadraticprogrammingproblems.Themain
featureis thattherequiredconditionsof symmetryandasymmetryin theinterconnectionsareautomatically
metin practicalimplementations,sothatstability is guaranteed.
B. FinkbeinerandP. Kall. Direct algorithmsin quadraticprogramming.Zeitschrift fur Opera-tionsResearch, SerieA (Theorie), 17(1), 45–54,1973.
Abstract. Accordingto theliteraturethereseemto besomedifficultiesin thesemidefinitecase.A numerical
exampleof asemicomplementarysolutionis presented,whereCottle’s algorithmfails. It will beprovedthat
at leastin Zangwill’s versionof Cottle’s algorithmthis situationcannotoccurtheoretically. Since,however,
in practicalcomputationssuchcasesmayoccurdueto roundoff errors.aslightmodificationof thealgorithm
is proposed,for whichmonotonicityandfinitenessareproved.
F. D. Fischer. To the solutionof the contactproblemof elasticbodieswith extendedcontactareasby quadraticprogramming.Computing, 13(3–4),353–384,1974.
Abstract. An elasticbodyin contactwith anelasticor rigid subgradeis representedby finite elements.The
total potentialenergy of thesystemunderconsiderationof linearly elasticmaterialandsmalldeformations
is now a quadraticfunctionof thenodaldeformationsandthenodalvaluesof thecontactpressurewhich is
approximatedby a polynomial.Only a partof thesurfaceof thebodymustbeproposedwhich includesthe
realcontactsurface.After evaluationof theequilibriumequationsandthecontactconditionin aninequality
anda linear transformationof thenodalvariables,all relationsarenow so formulated,theminimisationof
totalpotentialenergy canbeexpressedasaquadraticprogramin theunknown nodaldeformationsandnodal
valuesof contactpressure.Standardprogramscannow beusedfor solution.
J. Fischer. Stackelberg solutions in constrainedquadratic programmingproblems. InK. ReinischandM. Thoma,eds,‘LargeScaleSystems:TheoryandApplications1989.SelectedPapersfrom the5thIFAC/IFORS/IMACSSymposium.Pergamon,Oxford,Eng-land’, pp.241–246,1990.
Abstract. Dealswith two-personstatic Stackelberg gameswith quadraticobjectives and commonlinear
constraintsfor the leaderandthe follower. Importantpropertiesof therationalreactionfunctionof the fol-
lowercanbederivedfrom theoutlinedrelationsto thetheoryof parametricoptimization.Fromtheresultthe
solutionof thespecifiedStackelberg problemis reducedto thesequentialsolutionof afinite numberof stan-
dardconstrainedquadraticprogrammingproblemsandthe iterative selectionof solutionstructuresfor the
follower reaction.The proposedalgorithmhasbeenimplementedon a large-scalenonlinearprogramming
solver.
N. J.Fisher. Gravity interpretationwith theaidof quadraticprogramming—reply. Geophysics,
46(3), 341–342,1981.
N. J. Fisher. A quadraticprogrammingalgorithmfor geophysicalgravity inversionandotherapplications.Bulletin of theAustralian MathematicalSociety, 25(1), 159–160,1982.
Abstract. The geophysicalgravity methodis describedandthe correspondinginverseproblemstateasan
integral equation.A survey of previous work associatedwith gravity datainversionis given followed by
anaccountof the inherentdifficulties associatedwith gravity inversion;this is doneby a discussionof the
integralequationalreadygiven,theinversionof whichconstitutesanill-posedproblem.Theintegralequation
givesriseto anill-conditionedsystemof linearequations;existingmethodsfor thesolutionsof suchsystems
40 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
are reviewed. Then follows an accountof the previous useof quadraticprogrammingtechniquesin the
solutionof ill-posedproblems.
N. J. FisherandL. E. Howard. Gravity interpretationwith theaid of quadraticprogramming.Geophysics, 45(3), 403–419,1980.
Abstract. The inversegravity problemis posedasa linear least-squaresproblemwith the variablesbeing
densitiesof two-dimensionalprisms.Upperandlowerboundsonthedensitiesareprescribedsothattheprob-
lem becomesa linearly constrainedleast-squaresproblem,which is solvedusinga quadraticprogramming
algorithmdesignedfor upperandlowerbound-typeconstraints.Thesolutionto any problemis smoothedby
damping,usingthesingularvaluedecompositionof thematrix of gravitationalattractions.If thesolutionis
requiredto bemonotonicallyincreasingwith depth,thenthis featurecanbeincorporated.Themethodis ap-
plied to bothfield andtheoreticaldata.Theresultsareplottedfor (1) undamped,nonmonotonic,(2) damped,
nonmonotonic,and(3) damped,monotonicsolutions;theseconditionsillustratethecompositeapproachof
interpretationwherebothdampingtechniquesandlinearconstraintsareusedin refininga solutionwhich at
first is unacceptableongeologicgroundswhile fitting theobserveddatawell.
R. Fletcher. A FORTRAN subroutinefor generalquadraticprogramming. Technicalreport,UKAEA, Harwell,Berks,England,1970.
Abstract. A FORTRAN subroutineis describedandlistedfor minimizingaquadraticfunctionof n variables
subjectto m linear equalityand inequality constraints.The methodis a generaloneso that thereare no
restrictionson the typesof quadraticfunction which canbeminimized.The solutionis usuallyfound in a
moderatemultiple of n3 (or at worst n2m computeroperations.Thesubroutineis very flexible andallows
optionswhichcanconsiderablyshortenthetime takento solve any particularproblem.
R. Fletcher. A generalquadraticprogrammingalgorithm.Journalof theInstituteof Mathemat-icsandits Applications, 7, 76–91,1971.
Abstract. An effective algorithmis presentedfor quadraticprogrammingwhich is of generalapplicability,
but whichis notdependentupontheavailability of a linearprogrammingcodefor its implementation.It is an
algorithmof exchangetype,theexchangesbeingchosensoasto avoid theaccumulationof errorto aslarge
anextentaspossible.
R. Fletcher. Quadraticprogramming. In ‘Practical Methodsof Optimization’, chapter10,
pp.229–258.J.Wiley andSons,Chichester, England,secondedn,1987a.
R. Fletcher. Recentdevelopmentsin linear and quadraticprogramming. In A. IserlesandM. J. D. Powell, eds, ‘State of the Art in Numerical Analysis. Proceedingsof theJoint IMA/SIAM Conference’,pp.213–243.Oxford UniversityPress,Oxford,England,1987b.
Abstract. Describesrecentdevelopmentsin linearprogramming,includingtheellipsoidalgorithm,theKar-
markaralgorithm,new strategiesfor updatingLU factorsin thesimplex method,andmethodswith guaran-
teedterminationin the presenceof degeneracy andround-off errors.Variousnew algorithmsfor quadratic
programmingarediscussed,andthechoiceof matrix factorizationsandtheirupdatesis considered.Theuse
of * 1 penaltyfunctionsin linearandquadraticprogrammingis alsomentionedbriefly.
R. Fletcher. Resolvingdegeneracy in quadraticprogramming.Annalsof OperationsResearch,46–47(1–4),307–334,1993.
Abstract. A techniquefor theresolutionof degeneracy in anactive setmethodfor quadraticprogramming
is described.TheapproachgeneralisesFletcher’s method(1988)which appliesto theLP case.Themethod
is describedin termsof a linearcomplementarityproblemtableau,which is seento provide usefulinsights.
It is shown thatthedegeneracy procedureonly needsto operatewhenthedegenerateconstraintsarelinearly
dependenton thosein theactive set.No significantoverheadsareincurredby thedegeneracy procedure.It is
N. I. M. GOULD & PH.L. TOINT 41
readily implementedin a null spaceformat,andno complicationsin thematrix algebraareintroduced.The
guaranteesof terminationprovidedby Fletcher’s method,extendingin particularto thecasewhereround-off
erroris present,arepreservedin theQPcase.It is arguedthatthetechniquegivesstrongerguaranteesthanare
availablewith otherpopularmethodssuchasWolfe’s method(1963)or themethodof GoldfarbandIdnani
(1983).
R. Fletcher. Stablereducedhessianupdatesfor indefinitequadraticprogramming.Mathemati-cal Programming, 87(2), 251–264,2000.
Abstract. Stabletechniquesareconsideredfor updatingthereducedHessianmatrixthatarisesin anull-space
active setmethodfor quadraticprogrammingwhenthe Hessianmatrix itself may be indefinite.A scheme
for defining andupdatingthe null-spacebasismatrix is describedwhich is adequatelystableandallows
advantageto be taken of sparsityin the constraintmatrix. A new canonicalform for the reducedHessian
matrix is proposedthat canbe updatedin a numericallystableway. Someconsequencesfor the choiceof
minor iterationsearchdirectionaredescribed.
R. FletcherandM. P. Jackson.Minimization of a quadraticfunction of many variablessub-
ject only to lower and upperbounds. Journal of the Institute of Mathematicsand its
Applications, 14(2), 159–174,1974.
O. E. Flippo andA. H. G. Rinnooy Kan. A noteon Bendersdecompositionin mixed-integer
quadraticprogramming.OperationsResearch Letters, 9(2), 81–83,1990.
C. A. FloudasandV. Visweswaran. Quadraticoptimization. In R. Horst andP. M. Pardalos,eds,‘Handbookof GlobalOptimization’,Kluwer AcademicPublishers,Dordrecht,TheNetherlands,1995.
Abstract. Quadraticoptimizationcomprisesone of the most importantareasof nonlinearprogramming.
Numerousproblemsin realworld applications,includingproblemsin planningandscheduling,economies
of scale,andengineeringdesign,andcontrolarenaturallyexpressedasquadraticproblems.Moreover, the
quadraticproblemis known to be NP-hard,which makesthis oneof the most interestingandchallenging
classof optimizationproblems.In this chapter, we review variouspropertiesof thequadraticproblem,and
discussdifferenttechniquesfor solvingvariousclassesof quadraticproblems.Someof themoresuccessful
algorithmsfor solvingthespecialcasesof boundconstrainedandlargescalequadraticproblemsareconsid-
ered.Examplesof variousapplicationsof quadraticprogrammingarepresented.A summaryof theavailable
computationalresultsfor thealgorithmsto solve thevariousclassesof problemsis presented.
F. Forgo. Relationshipbetweenmixed zero-oneinteger linear programmingand certain
quadraticprogrammingproblems. Studia ScientiarumMathematicarumHungarica,
4, 37–43,1969.
A. ForsgrenandG. Sporre.Onweightedlinearleast-squaresproblemsrelatedto interiormeth-odsfor convex quadraticprogramming.ReportTRITA-MAT-2000-OS11,DepartmentofMathematics,Royal Instituteof Technology, Stockholm,Sweden,2000.
Abstract. It is known that thenormof thesolutionto a weightedlinear least-squaresproblemis uniformly
boundedfor the setof diagonallydominantsymmetricpositive definiteweight matrices.This result is ex-
tendedto weightmatricesthatarenonnegative linearcombinationsof symmetricpositive semidefinitema-
trices.Further, resultsare given concerningthe strongconnectionbetweenthe boundednessof weighted
projectiononto a subspaceand the projectiononto its complementarysubspaceusing the inverseweight
matrix. In particular, explicit boundsaregiven for the Euclideannorm of the projections.We apply these
resultsto theNewton equationsarisingin a primal-dualinterior methodfor convex quadraticprogramming
andprove boundednessfor thecorrespondingprojectionoperator.
42 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
A. L. Forsgren,P. E. Gill, andW. Murray. On theidentificationof local minimizersin inertia-controllingmethodsfor quadraticprogramming.SIAMJournal on Matrix AnalysisandApplications, 12(4), 730–746,1991.
Abstract. Theverificationof a localminimizerof ageneral(i.e.,nonconvex) quadraticprogramis in general
anNP-hardproblem.Thedifficulty concernstheoptimality of certainpoints(which we call deadpoints)at
which the first-ordernecessaryconditionsfor optimality aresatisfied,but strict complementaritydoesnot
hold. Inertia-controllingquadraticprogramming(ICQP) methodsform an importantclassof methodsfor
solvinggeneralquadraticprograms.Wederive acomputationalschemefor proceedingatadeadpoint thatis
appropriatefor ageneralICQPmethod.
M. Forti andA. Tesi.New conditionsfor globalstabilityof neuralnetworkswith applicationtolinearandquadraticprogrammingproblems.IEEETransactionsonCircuitsandSystemsI: FundamentalTheoryandApplications, 42(7), 354–366,1995.
Abstract. In this paper, we presentnew conditionsensuringexistence,uniqueness,andGlobalAsymptotic
Stability (GAS) of the equilibrium point for a large classof neuralnetworks.The resultsareapplicableto
bothsymmetricandnonsymmetricinterconnectionmatricesandallow for theconsiderationof all continuous
nondecreasingneuronactivation functions.Suchfunctionsmay be unbounded(but not necessarilysurjec-
tive), mayhave infinite intervalswith zeroslopeasin a piece-wise-linearmodel,or both.Theconditionson
GAS rely on the conceptof Lyapunov DiagonallyStable(or Lyapunov DiagonallySemi-Stable)matrices
andareproved by employing a classof Lyapunov functionsof thegeneralizedLur’e-Postnikov type.Sev-
eralclassesof interconnectionmatricesof applicative interestareshown to satisfyour conditionsfor GAS.
In particular, the resultsareappliedto analyzeGAS for the classof neuralcircuits introducedfor solving
linear andquadraticprogrammingproblems.In this application,the principal resulthereobtainedis that
thesenetworks areGAS alsowhenthe constraintamplifiersaredynamical,as it happensin any practical
implementation.
M. FrankandP. Wolfe. An algorithmfor quadraticprogramming.Naval Research Logistics
Quarterly, 3, 95–110,1956.
P. D. Frank,M. J.Healy, andR. A. Mastro. A range-spaceimplementationfor largequadratic
programswith small active sets. Journal of OptimizationTheory and Applications,
69(1), 109–127,1991.
R. M. Freund. Dual gaugeprograms,with applicationsto quadraticprogrammingand theminimum-normproblem.MathematicalProgramming, 38(1), 47–67,1987.
Abstract. A gaugefunction f + is a nonnegative convex functionthat is positively homogeneousandsatis-
fiesf(0)=0. Normsandpseudonormsarespecificinstancesof a gaugefunction.Thepaperpresentsa gauge
duality theoryfor a gaugeprogram,which is theproblemof minimizing thevalueof a gaugefunction f + over a convex set.The gaugedual programis alsoa gaugeprogram,unlike the standardLagrangedual.
Theauthorpresentssufficient conditionson f + thatensuretheexistenceof optimalsolutionsto thegauge
programandits dual, with no duality gap.Thesesufficient conditionsarerelatively weakandareeasyto
verify, andareindependentof any qualificationsontheconstraints.Thetheoryis appliedto aclassof convex
quadraticprograms,andto theminimum * p normproblem.Thegaugedualprogramis shown to provide a
smallerduality thanthestandarddual,in acertainsensediscussed.
A. FriedlanderandJ.M. Martınez. On themaximizationof a concave quadraticfunctionwith
box constraints.SIAMJournalonOptimization, 4(1), 177–192,1994.
A. Friedlander, J.M. Martınez,andM. Raydan.A new methodfor large-scalebox constrained
convex quadraticminimizationproblems.OptimizationMethodsandSoftware, 5(1), 57–
74,1995.
N. I. M. GOULD & PH.L. TOINT 43
K. R. Frisch. Quadraticprogrammingby the multiplex methodinthe generalcasewherethe
quadraticform maybesingular. Memorandum,UniversityInstitutefor Economics,Oslo,
1960.
M. Fu, Z. Q. Luo, andY. Ye. Approximationalgorithmsfor quadraticprogramming.Journalof CombinatorialOptimization, 2(1), 29–50,1998.
Abstract. We considerthe problemof approximatingtheglobal minimum of a generalquadraticprogram
(QP)with n variablessubjectto mellipsoidalconstraints.For m 1,werigorouslyshow thatanε-minimizer,
whereerrorε , 0 1 , canbeobtainedin polynomialtime,meaningthatthenumberof arithmeticoperations
is a polynomialin n, m, andlog 1 ε . For m 2, we presenta polynomial-time 1 1 m2 -approximation
algorithmaswell asasemidefiniteprogrammingrelaxationfor thisproblem.In addition,wepresentapprox-
imationalgorithmsfor solvingQPundertheboxconstraintsandtheassignmentpolytopeconstraints.
S. T. Fu andR. T. Wang. Power-systemsecurityanalysisandenhancementusingFletcher’s
quadratic-programmingmethod. In ‘Proceedingsof theEighthPower SystemsCompu-
tationConference’,pp.439–445,1984.
S. T. Fu, E. Yu, andX. Zhang. A decoupledoptimal power flow approachusingFletcher’squadraticprogrammingmethod.Proceedingsof theChineseSocietyof Electrical Engi-neering, 6(1), 1–11,1986. Seealso,BridgeBetweenControlScienceandTechnology.Proceedingsof the Ninth Triennial World Congressof IFAC, PergamonPress,Oxford,England,volume4, pages2145–2150,1985.
Abstract. A decoupledmodelwith active andreactive optimizationis derived. In a P-model,a very sparse
schemeis usedastheequalityconstraints.In a Q-model,transmissionlossesareexpressedasdeviation of
swing generation.The active power at generatorbuses(exceptswing generator)is chosenasindependent
variablein P-optimization.Thevoltageamplitudeof PV nodesandV θ node,andthe ratio of transforma-
tion of all tap-changingtransformersarechosenasindependentvariablesin Q-optimization.Theprocedure
is carriedout only in thesubspacedeterminedby independentvariables.Theequalityconstraintsareelim-
inated,and the relationshipsbetweendependentand independentvariablesareestablishedby solutionof
a fastdecoupledload flow method.During P-optimization,all bus voltagesandreactive injectionsareas-
sumedconstantandtheobjective function is to minimize the total fuel consumptionin thesystem.During
Q-optimization,all active generations(except swing generator)andall voltagephaseanglesareassumed
constantandthe objective function is to minimize total transmissionlossesin the network. This approach
hasbeenprogrammedin FORTRAN languageandtestedon a CLASSIC7835computer. Thesparsematrix
programmingtechniquebeingused,thememoryrequirementof a network with 150busesis only 100kB.
For comparisonpurposes,the computationtime andresultsof the sameIEEE 30 bus test systemusinga
differentmethodaretabulated.A Chinese133bussystemwith 164 lineswassimulatedusingthis method.
Therewere2 linesoverloadedand3 busvoltagesout of limit in theinitial state.An OPFresultis tabulated.
All line overloadsarerelievedandall busvoltagesarewithin limit.
T. Fujie andM. Kojima. Semidefiniteprogrammingrelaxationfor nonconvex quadraticpro-grams.TechnicalReportResearchReporton InformationSciencesB-298,Tokyo Insti-tuteof Technology, 1995.
Abstract. Any quadraticinequalityin then-dimensionalEuclideanspacecanberelaxedinto a linearmatrix
inequalityin 1 n- 1 n symmetricmatrices.Basedon this principle,we extendtheLovasz-Schrijver
SDP(semidefiniteprogramming)relaxationdevelopedfor a 0–1 integer programto a generalnonconvex
QP(quadraticprogram),andpresentsomefundamentalcharacterizationof theSDPrelaxationincludingits
equivalenceto a relaxationusingconvex-quadraticvalid inequalitiesfor thefeasibleregion of theQP.
Y. FujimotoandA. Kawanura.Bipedwalking controlwith optimal foot forcedistribution byquadraticprogramming.IEEE/ASMEInternationalConferenceon AdvancedIntelligentMechatronics’97. Final ProgramandAbstractsIEEE,New York, NY, USA, p. 108,1997.
44 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. Summaryform only given.This paperdescribesa novel bipedwalking systembasedon optimal
foot forcedistribution by quadraticprogramming(QP).Thehierarchicalcontrolsystemconsistsof a robust
force controller for the supportingleg, a robust positioncontroller for the non-supportingleg, an attitude
controllerfor thebody, andafree-leg motionplanner. Theattitudeof thebodyof thebipedrobotis controlled
usingthereactive forceasthe input. Themainresultof this paperis an introductionof QPto distribute the
forcerequiredby theattitudecontrollerto eachfoot. Themethodcanstabilizetheattitudeof therobotand
thecontactbetweenthefoot andtheground,in spiteof theexistenceof thelimitation of thefrictional force
betweenthefoot andtheground.Also anew walkingpatterngeneratorbasedonaninvertedpendulummodel
is proposed,which realizesgloballystabletrackingof thecenterof massof thebipedrobot.
K. FukudaandA. Kawanaka. Adaptive processingwith quadraticprogrammingandlpf forreducingblockingartifactsin DCT imagecoding. Transactionsof the Instituteof Elec-tronics,InformationandCommunicationEngineers, J82-A(1), 142–150,1999.
Abstract. This paperpresentsanadaptive post-processingmethodto reducetheblockingartifactsin DCT
imagecoding.In thismethod,theblocksareclassifiedinto thesmoothgroupandthevariationgroupaccord-
ing to thereceived DCT coefficients.For theblocksin thesmoothgroup,we estimatetheDCT coefficients
by solving a quadraticprogrammingproblemconsideringthe continuity acrossthe block boundariesand
thequantizationmatrixusedon thecoder. Furthermore,theDCT coefficientsaresequentiallyobtainedfrom
lower order to higher. For the blocks classifiedinto the variation group, the low passfilters are applied
for smoothingtheblock boundaries.Experimentalresultsfor someimagescodedat low bit ratesshow the
proposedmethodimprovesthereconstructedimagequalityobjectively aswell assubjectively.
R. Gabasov, F. M. Kirillo va, andV. M. Raketskii. Methodsfor solving the generalconvexquadraticprogrammingproblem. Doklady AkademiiNauk SSSR, 258(6), 1289–1293,1981.
Abstract. A theoreticaland numericalinvestigationis reportedof 4 methods(Bil’ s, the direct, dual and
adaptive methods)for solvingthe(mxn) convex quadraticprogrammingproblem: F x. cTx xT Dx 2 to
min, Ax b, d/ x d / , whereD D J J 0 O, A A I J , rankA m, I 1 21 m$ J 1 2 n .E. Galligani and V. Ruggiero. Numerical solution of equality-constrainedquadraticpro-
grammingproblemson vector-parallelcomputers.OptimizationMethodsandSoftware,
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E. Galligani and L. Zanni. Error analysisof elimination methodsfor equality constrained
quadratic-programmingproblems. In ‘MathematicalResearch’,Vol. 89, pp. 107–112,
1996.
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Abstract. Thenumericalstabilityof theorthogonalfactorizationmethod(R.Fletcher, 1987)for linearequal-
ity constrainedquadraticprogrammingproblemsis studiedusinga backwarderroranalysis.A perturbation
formula for theproblemis analyzed;theconditionnumbersof this formulaareexaminedin orderto com-
parethemwith the conditionnumbersof the two matricesof the problem.A classof testproblemsis also
consideredin orderto show experimentallythebehaviour of themethod.
E. Galligani,V. Ruggiero,andL. Zanni. Splitting methodsfor constrainedquadraticprograms
in dataanalysis.ComputersMath.Applic., 32, 1–9,1996.
E. Galligani,V. Ruggiero,andL. Zanni. Splitting methodsandparallelsolutionof constrained
quadraticprograms.Rend.Circ. Matem.PalermoSeriesII , 48, 121–136,1997a.
N. I. M. GOULD & PH.L. TOINT 45
E. Galligani,V. Ruggiero,andL. Zanni. Splitting methodsfor quadraticoptimizationin data
analysis.InternationalJournalof ComputerMathematics, 63, 289–307,1997b.
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grammingStudies, 12, 132–149,1980.
C. E. GarciaandA. M. Morshedi.Quadratic-programmingsolutionof dynamicmatrixcontrol
(QDMC). ChemicalEngineeringCommunications, 46(1–3),73–87,1986.
M. K. Gavurin andV. N. Malozemov. Foundationsof quadraticprogrammingtheory. Vest-nik LeningradskogoUniversiteta,SeriyaMatematikaMekhanikai Astronomiya, 1, 9–16,1980.
Abstract. A compactdescriptionof quadraticprogrammingtheoryis given. It includestheexistencetheo-
rem different formsof optimality criterionandduality theorems.Only compatibility conditionfor a linear
algebraicsystemanda necessaryoptimality conditionin the linear programmingproblemaresupposedto
beknown. Thegeneralconvex programmingtheoryis not used.
J. A. George. Assessmentof errorsin approximatingthe objective function of a quadraticprogrammingproblem.AsiaPacificJournalof OperationalResearch, 5(1),21–36,1988.
Abstract. Dealswith the effectivenessof variousdecisionrulesto approximatetheobjective functionof a
quadraticprogram(QP).It is assumedthat theconstraintsetof theQPis known preciselybut theobjective
functionis known only from thedataof observationsmadeof thatfunction.Decisionrules,somelinearand
somenonlinear, areusedto derive functionsto approximatethe true unknown function from the data.A
seriesof problemsaregeneratedto testthesedecisionrules.In additionto comparingthemwith eachother,
thestudyalsoconsiderstheeffectsof thetypeof quadraticfunction (matrix type), thesizeof theproblem,
thenumberof observations,thescatterpatternof theobservationsandthecurvature(degreeof nonlinearity)
of thequadratic.
V. Georgescu. Estimationof fuzzy regressionmodelswith possibilistic constraints,usingquadraticprogramming.EconomicComputationandEconomicCyberneticsStudiesandResearch, 31(1–4),105–123,1997.
Abstract. We introducea formal criterion, the decouplingprinciple, which allows us to naturallyextend
the projectiontheoreminto a fuzzy regressionframework. This criterion justifiesa radical revision of the
usualestimationmethods.Actually dominantin the fuzzy regressionanalysisis anotherapproach,which
resortsto linearprogrammingandfuzzy arithmeticin orderto minimizethefuzzinessof themodel,subject
to somepossibilisticconstraints.Our criticism relatingto this approachis concludedby the proposalof a
new strategy, wherewe give up usingfuzzy arithmetic(identifiedasa sourceof estimationdistortions)and
we insist in the reestablishmentof thenaturalframework for solvinga minimum normproblem.Sincethe
orthogonalityconceptis relatedto thechoiceof a norm inducedby an innerproduct,our fuzzy estimation
procedureinevitably leadsto a quadraticprogrammingproblemandnot to a linear one.The decoupling
principle consistsin a set of rules,which allow us to expressthe fuzzy regressionmodelasa systemof
two classicalequationsandto choosethe correspondingprojectionsubspaces.We explore the algorithmic
consequences,whichfollow from suchtheoreticalconsiderationsfor varioussituationsandconcretetypesof
problems,andwe proposesomeMATLAB implementations.
46 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
F. Giannessiand G. Tomasin. Nonconvex quadraticprogramming,linear complementarity
problemsandintegerprogramming.In F. GiannessiandG. Tomasin,eds,‘Linear com-
plementarityproblemsandintegerProgramming’,pp.162–201,North-Holland,Amster-
dam,1974.
P. E. Gill andW. Murray. Numericallystablemethodsfor quadraticprogramming.Mathemati-cal Programming, 14(3), 349–372,1978.
Abstract. Numericallystablealgorithmsfor quadraticprogrammingarediscussed.A new algorithmis de-
scribedfor indefinitequadraticprogrammingwhich utilizes methodsfor updatingpositive-definitefactor-
izationsonly. Consequentlyall the updatingproceduresrequiredare commonto algorithmsfor linearly-
constrainedoptimization.Thenew algorithmcanbeusedfor thepositive-definitecasewithout lossof effi-
ciency.
P. E. Gill, N. I. M. Gould,W. Murray, M. A. Saunders,andM. H. Wright. Range-spacemethods
for convex quadraticprogrammingproblems.TechnicalReportSOL 82-9,Department
of OperationsResearch,StanfordUniversity, California,USA, 1982a.
P. E. Gill, N. I. M. Gould, W. Murray, M. A. Saunders,andM. H. Wright. A range-space
quadraticprogrammingalgorithm for problemswith a mixture of boundsandgeneral
constraints.TechnicalReportSOL 82-10,Departmentof OperationsResearch,Stanford
University, California,USA, 1982b.
P. E. Gill, N. I. M. Gould, W. Murray, M. A. Saunders,and M. H. Wright. A weightedGramm-Schmidtmethodfor convex quadraticprogramming. MathematicalProgram-ming, 30(2), 176–195,1984.
Abstract. Range-spacemethodsfor convex quadraticprogrammingimprove in efficiency asthenumberof
constraintsactive at the solutiondecreases.In this paperthe authorsdescribea rangespacemethodbased
uponupdatingweightedGram-Schmidtfactorizationof theconstraintsin theactive set.Theupdatingmeth-
odsdescribedareapplicableto bothprimalanddualquadraticprogrammingalgorithmsthatuseanactive-set
strategy. Many quadraticprogrammingproblemsincludesimpleboundson all thevariablesaswell asgen-
eral linear constraints.A featureof theproposedmethodis that it is ableto exploit the structureof simple
boundconstraints.This allows themethodto retainefficiency whenthenumberof generalconstraintsactive
at thesolutionis small.Furthermore,theefficiency of themethodimprovesasthenumberof active bound
constraintsincreases.
P. E. Gill, W. Murray, andM. A. Saunders.User’sguidefor QPOPT1.0: A Fortranpackagefor
quadraticprogramming.TechnicalReportSOL ??,Departmentof OperationsResearch,
StanfordUniversity, California,USA, 1995.
P. E. Gill, W. Murray, andM. H. Wright. Quadraticprogramming.In ‘PracticalOptimization’,
chapter5.3.2–5.4.1,pp.177–184.AcademicPress,London,England,1981.
P. E. Gill, W. Murray, D. B. Ponceleon, andM. A. Saunders.Solving reducedKKT–systems
in barriermethodsfor linear andquadraticprogramming.TechnicalReportSOL 91-7,
Departmentof OperationsResearch,StanfordUniversity, California,USA, 1991a.
P. E. Gill, W. Murray, M. A. Saunders,andM. H. Wright. User’s guidefor SOL/QPSOL:A
Fortranpackagefor quadraticprogramming.TechnicalReportSOL86-2,Departmentof
OperationsResearch,StanfordUniversity, California,USA, 1986.
N. I. M. GOULD & PH.L. TOINT 47
P. E. Gill, W. Murray, M. A. Saunders,and M. H. Wright. A Schur-complementmethod
for sparsequadraticprogramming.In M. G. Cox andS. J. Hammarling,eds,‘Reliable
ScientificComputation’,pp.113–138,OxfordUniversityPress,Oxford,England,1990.
P. E. Gill, W. Murray, M. A. Saunders,andM. H. Wright. Inertia-controllingmethodsforgeneralquadraticprogramming.SIAMReview, 33(1), 1–36,1991b.
Abstract. Active-setquadraticprogramming(QP)methodsusea working setto definethesearchdirection
andmultiplier estimates.In themethodproposedby Fletcherin 1971,andin severalsubsequentmathemat-
ically equivalent methods,the working set is chosento control the inertia of the reducedHessian,which
is never permittedto have more than one nonpositive eigenvalue. (Suchmethodswill be called inertia-
controlling.) This paperpresentsan overview of a genericinertia-controllingQP method,including the
equationssatisfiedby the searchdirectionwhenthe reducedHessianis positive definite,singularand in-
definite.Recurrencerelationsare derived that definethe searchdirectionand Lagrangemultiplier vector
throughequationsrelatedto the Karush-Kuhn-Tucker system.Discussionis includedof connectionswith
inertia-controllingmethodsthatmaintainanexplicit factorizationof thereducedHessianmatrix.
D. Givoli andI. Doukhovni. Finiteelement-quadraticprogrammingapproachfor contactprob-lemswith geometricalnonlinearity. ComputersandStructures, 61(1), 31–41,1996.
Abstract. Thefinite element-quadraticprogramming(FE-QP)approachfor problemsinvolving frictionless
contactbetweenan elasticbody anda rigid obstacleis presentedin a generalsetting.The validity of this
approachis first provedin thecontext of smalldeformationproblems.Thenaway to extendit to thecaseof
largedeformationproblemsis proposed,andthevalidity andimplementationof thisprocedurearediscussed.
Numericalexamplesinvolving Euler-Bernoulli beamsarepresentedto demonstratetheseissues,andto show
theregimeswherethemethodis successfulor failing.
C. R. Glassey. Analysisof the federalenergy agency programby a quadraticprogrammingmodel. In ‘Proceedingsof LawrenceSymposiumon SystemsandDecisionSciences.WesternPeriodicalsCo,NorthHollywood,CA, USA’, pp.146–150,1977.
Abstract. A 21 sectorinput-outputmodel of the United Stateseconomy, in which exports, imports and
consumerdemandsaredeterminedendogenouslyby linearprice-demandfunctions,is usedto examinethe
impact of increasesin world market oil prices.Equilibrium pricesand flows in the model economyare
computedby maximizing a quadraticprogram.Policiesof the FederalEnergy Agency to mitigate these
impactsareanalysedby suitablemodificationsof themodel.
C.R. Glassey. Pricesensitiveconsumerdemandsin energy modeling-aquadraticprogrammingapproachto theanalysisof somefederalenergy agency policies. ManagementScience,24(9), 877–886,1978a.
Abstract. A 21 sectorinput-outputmodelof the 1972U.S. economyis extendedto includeconsumerde-
mands,imports,andexportsasendogeneousvariables,andusedto analyzesomeconsequencesof theprice
controlpolicy adoptedto mitigatetheimpactof thequadruplingof world crudeoil pricesin 1973–1974.It is
assumedthathouseholdconsumptionof goodsis linearlyrelatedto prices.An equilibriumof themodelecon-
omy is thencomputedby solvingaquadraticprogram.It is shown thatthepricecontrolpolicy is equivalent
to subsidizingimportedoil with revenuesfrom a taxondomesticcrudeproduction.Equilibriaarecomputed
underthis policy andin its absence.Thecomparisonindicatesthepolicy waseffective in reducingtheprice
index increaseandGNPreductionthatwould otherwisehave occurred,but at thecostof adverselyaffecting
thebalanceof payments.
C.R.Glassey. A quadraticnetwork optimizationmodelfor equilibriumsinglecommoditytrade
flows. MathematicalProgramming, 14(1), 98–107,1978b.
J. H. Glick, P. M. Pardalos,and J. B. Rosen. Global minimization of indefinite quadratic
problems.Computing, 39, 281–291,1987.
48 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
M. S.Goheen.Largescaleboundedvariablequadratic programming. PhDthesis,Department
of OperationsResearch,StanfordUniversity, Stanford,California,USA, 1976.
R. Goldbach.Somerandomizedalgorithmsfor convex quadraticprogramming.AppliedMath-ematicsandOptimization, 32, 121–142,1999.
Abstract. We adaptsomerandomizedalgorithmsof Clarkson[3] for linearprogrammingto theframework
of so-calledLP-typeproblems,whichwasintroducedby SharirandWelzl [10]. This framework is quitegen-
eralandallows aunifiedandelegantpresentationandanalysis.Wealsoshow thatLP-typeproblemsinclude
minimizationof a convex quadraticfunction subjectto convex quadraticconstraintsasa specialcase,for
which thealgorithmscanbeimplementedefficiently, if only linearconstraintsarepresent.Weshow thatthe
expectedrunningtimesdependonly linearlyonthenumberof constraints,andillustratethisby somenumer-
ical results.Eventhoughtheframework of LP-typeproblemsmayappearratherabstractat first, application
of themethodsconsideredin this paperto a givenproblemof that type is easyandefficient. Moreover, our
proofsarein fact rathersimple,sincemany technicaldetailsof moreexplicit problemrepresentationsare
handledin a uniform mannerby our approach.In particular, we do not assumeboundednessof thefeasible
setasrequiredin relatedmethods.
D. Goldfarb. Analogsof Newton’smethodfor quadraticprogramming.Noticesof theAmerican
MathematicsSociety, 15(2), 400,1968.
D. Goldfarb. Extensionsof Newton’s methodandSimplex methodsfor solvingquadraticpro-grams.In F. A. Lootsma,ed.,‘Numericalmethodsfor nonlinearoptimization’,pp.255–263.AcademicPress,London,England,1972.
Abstract. Two closely relatedmethods,that may be viewed either asextensionsof Newtons’ methodto
handlelinear equalitiesand inequalitiesor as quadraticanaloguesof the gradientprojectionmethod,are
presentedfor solvingstrictly convex quadraticprograms.Oneof thesemethodsandtheSimplex methodfor
quadraticprogrammingareshown to follow thesamesolutionpathif startedatavertex. A usefulrelationship
betweentheLagrangemultiplier variablesfor anestedsetof constraintbasesis alsoshown
D. Goldfarb. Efficient primal algorithmsfor strictly convex quadraticprograms.In J.P. Hen-
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D. Goldfarb. Strategiesfor constrintdeletionin active setalgorithms. In D. F. Griffiths and
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D. Goldfarb and A. U. Idnani. Dual and primal-dualmethodsfor solving strictly convex
quadraticprograms.In J.P. Hennart,ed.,‘NumericalAnalysis’,number909 in ‘Lecture
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1982.
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D. Goldfarb andS. C. Liu. Interior point potentialfunction reductionalgorithmfor solving
convex quadraticprogramming.Technicalreport,Departmentof IndustrialEngineering
andOperationsResearch,ColumbiaUniversity, New York, USA, 1990.
N. I. M. GOULD & PH.L. TOINT 49
D. Goldfarb andS. C. Liu. An On3L primal interior point algorithmfor convex quadratic
programming.MathematicalProgramming, 49(3), 325–340,1991.
Abstract. Wepresentaprimal interiorpointmethodfor convex quadraticprogrammingwhichis basedupon
a logarithmicbarrierfunctionapproach.This approachgeneratesa sequenceof problems,eachof which is
approximatelysolvedby takingasingleNewtonstep.It is shown thatthemethodrequiresO nL iterations
andO n3 ( 5L arithmeticoperations.By usingmodifiedNewton stepsthe numberof arithmeticoperations
requiredby thealgorithmcanbereducedto O n3L .D. GoldfarbandS. C. Liu. An O
n3L primal dual potentialreductionalgorithmfor solving
convex quadraticprograms.MathematicalProgramming, 61(2), 161–170,1993.
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nite quadraticprogramming.Technicalreport,Centerfor AdvancedProcessDecision-
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G.H. GolubandM. A. Saunders.Linearleastsquaresandquadraticprogramming.In J.Abadie,ed.,‘Integerandnonlinearprogramming’,pp.229–256,North Holland,Amsterdam,theNetherlands,1970.
Abstract. Oneof themostcommonproblemsin any computationcenteris thatof findinglinearleastsquares
solutions.Theseproblemsarisein a varietyof areasandin a varietyof contexts. For instance,thedatamay
bearriving sequentiallyfrom asourceandtheremaybesomeconstrainton thesolution.Linearleastsquares
problemsareparticularlydifficult to solve becausethey frequentlyinvolve largequantitiesof data,andthey
areill-conditionedby their very nature.This paperpresentsseveralnumericalalgorithmsfor solving linear
leastsquaresproblemsin ahighly accuratemanner. In addition,it givesanalgorithmfor solvinglinearleast
squaresproblemswith linearinequalityconstraints.
M. A. Gomez.An on2 activesetalgorithmfor thesolutionof aparametricquadraticprogram.
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Abstract. In this paper, an O n2 active setmethodis presentedfor minimizing the parametricquadratic
function f rac12xT Dx aTx λmax c γTx 0 subjectto l x b, for all nonnegative valuesof the pa-
rameterlambda. Here,D is a positive diagonalnxnmatrix, a andγ arearbitraryn-vectors,c is anarbitrary
scalar, l andb arearbitraryn-vectors,suchthat l b. An extensionof this algorithmis presentedfor min-
imizing theparametricfunction 12xT Dx aTx λ 2 γT x c 2 subjectto l x b. It is alsoshown that these
problemsarisenaturallyin a tax programmingproblem.
A. S.Goncalves.A generalizedprimal-dualtechniquefor quadraticprogrammingin parametric
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Abstract. A primal-dualalgorithmfor quadraticprogrammingwith boundedvariablesis establishedhere,
whichhasthesamecharacteristicsasthemethodby Eisemann(1963)for thelinearprogrammingcase,and
which may be consideredasan extensionof that method.However, distinct from the linear programming
case,therestrictedproblemsconstructedherehave no boundsin their variables.Thecomputationsmaybe
performedby pivotal operationsona tableau.Programmingof thealgorithmis facilitatedby its efficient use
of theproductform of theinversemechanismavailablein mostcommerciallinearprogrammingsystems.
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Abstract. Considersthe problemof maximizinga convex quadraticobjective. The algorithmdevisedfirst
’normalizes’ the given problem,i.e., substitutesit by anotherequivalent problemwhosesolutionscanbe
found very easilyby a L.P. columngenerationprocedure.Extensionsto the generalnon-convex quadratic
programmingarealsoconsidered.
J.Gondzio.Hopdmversion2.30: Benchmarkresults:Solvingconvex quadraticprogramming
problems. Logilab technicalnote,Departmentof ManagementSciences,Universityof
Geneva,Geneva,Switzerland,1998.
V. N. Gordeev. The finitenessof methodsof solving a problemin quadraticprogramming.
Cybernetics, 7(1), 110–114,1971.
V. N. Gordeev. Simplemethodof solving an auxiliary quadraticprogrammingproblem. Cy-
bernetics, 16(4), 616–619,1980.
N. I. M. Gould. Numericalmethodsfor linear and quadratic programming. D. Phil. thesis,
Oxford University, England,1982.
N. I. M. Gould. Thestability of thesolutionof generalquadraticprograms.TechnicalReport
CORR83-11,Departmentof CombinatoricsandOptimization,Universityof Waterloo,
Ontario,Canada,1983.
N. I. M. Gould.Onpracticalconditionsfor theexistenceanduniquenessof solutionsto thegen-eral equalityquadraticprogrammingproblem. MathematicalProgramming, 32(1), 90–99,1985.
Abstract. The authorpresentspracticalconditionsunderwhich the existenceand uniquenessof a finite
solution to a given equality quadraticprogrammay be examined.Different setsof conditionsallow this
examinationto take placewhennull-space,range-spaceor Lagrangianmethodsareusedto find stationary
pointsfor thequadraticprogram.
N. I. M. Gould. An algorithmfor large-scalequadraticprogramming.IMA Journalof Numeri-cal Analysis, 11(3), 299–324,1991.
Abstract. The paperdescribesa methodfor solving large-scalegeneralquadraticprogrammingproblems.
Themethodis basedupona compendiumof ideaswhich have their originsin sparsematrix techniquesand
methodsfor solvingsmallerquadraticprograms.A discussionis includedonresolvingdegeneracy, onsingle
phasemethodsandonsolvingparametricproblems.Somenumericalresultsareincluded.
N. I. M. GouldandPh.L. Toint. A quadraticprogrammingbibliography. NumericalAnalysis
Group InternalReport2000-1,RutherfordAppletonLaboratory, Chilton, Oxfordshire,
England,2000.See”http://www.numerical.rl.ac.uk/qp/qp.html”.
N. I. M. Gould and Ph. L. Toint. Numericalmethodsfor large-scalenon-convex quadratic
programming.TechnicalReportRAL-TR-2001–017,RutherfordAppletonLaboratory,
Chilton,Oxfordshire,England,2001.
N. I. M. Gould,M. E. Hribar, andJ.Nocedal.Onthesolutionof equalityconstrainedquadratic
problemsarisingin optimization.TechnicalReportRAL-TR-98–069,RutherfordApple-
ton Laboratory, Chilton,Oxfordshire,England,1998.
N. I. M. GOULD & PH.L. TOINT 51
F. Granotand J. Skorin-Kapov. Someproximity andsensitivity resultsin quadraticinteger
programming.MathematicalProgramming, 47(2), 259–268,1990a.
F. GranotandJ. Skorin-Kapov. Towardsa stronglypolynomialalgorithmfor strictly convex
quadraticprograms—anextensionof Tardos’algorithm. MathematicalProgramming,
46(2), 225–236,1990b.
F. Granot,J. Skorin-Kapov, andA. Tamir. Usingquadraticprogrammingto solve high multi-plicity schedulingproblemson parallelmachines.Algorithmica, 17(2), 100–110,1997.
Abstract. We introduceandanalyzeseveral modelsof schedulingn different types(groups)of jobs on m
parallelmachines,wherein eachgroupall jobsareidentical.Our maingoal is to exhibit the usefulnessof
quadraticprogrammingapproachesto solve theseclassesof highmultiplicity schedulingproblems,with the
total weightedcompletiontime astheminimizationcriterion.We develop polynomialalgorithmsfor some
models,andstronglypolynomialalgorithmsfor certainspecialcases.In particular, themodelin which the
weightsare job independent,as well as the generallyweightedmodel in which processingrequirements
are job independent,can be formulatedas an integer convex separablequadraticcost now problem,and
thereforesolvedin polynomialtime.Whenwespecializefurther, stronglypolynomialboundsareachievable.
Specifically, for theweightedmodelwith job-independentprocessingrequirementsif werestricttheweights
to bemachineindependent(while still assumingdifferentmachinespeeds),anO mn nlogn algorithmis
developed.If it is alsoassumedthatall themachineshave thesamespeed,thecomplexity of thealgorithm
canbe improved to O mlogm nlogn . Theseresultscanbeextendedto relatedunweightedmodelswith
variableprocessingrequirementsin whichall themachinesareavailableat timezero.
R. L. Graves. A principal pivoting Simplex algorithmfor linear andquadraticprogramming.
OperationsResearch, 15, 482–494,1967.
R. L. Graves. Quadraticprogrammingin Hilbert space.Bulletin of the OperationsResearchSocietyof America, 20, B–78,1972.
Abstract. Problemsareanalysedwhich arisein optimisingover infinite horizonswith discretetime, while
satisfyingafinite numberof inequalityconstraintsin eachtime period.Theseproblemsaretransformedinto
quadraticprogramson a separableHilbert spaceandtheexistenceof solutionsto the resultingprogramsis
established.Waysto calculatedapproximatesolutionsby solvingproblemsover a finite horizonareshown.
Someresults(not complete)for theexistenceof dualvariablesaregiven.
R. L. GravesandL. G. Telser. An infinite-horizondiscrete-timequadraticprogramasapplied
to a monopolyproblem.Econometrica, 35(2), ??,1967.
M. GrigoriadisandK. Ritter. A parametricmethodfor semidefinitequadraticprogramming.
SIAMJournalon Control andOptimization, 7(4), 559–577,1969.
L. GrippoandS.Lucidi. A differentiableexactpenaltyfunctionfor boundconstrainedquadraticprogrammingproblems.Optimization, 22(4), 557–578,1991.
Abstract. Definesacontinuouslydifferentiableexactpenaltyfunctionfor thesolutionof boundconstrained
quadraticprogrammingproblems.The authorsprove that thereexists a computablevalue of the penalty
parametersuchthat global and local minimizersof the penaltyfunction yield global and local solutions
to the original problem.This permitsthe constructionof Newton-typealgorithmsbasedon consistentap-
proximationsof theNewton’s directionof thepenaltyfunction.Conditionsthatensurefinite terminationare
established.
N. Grudinin. Combinedquadratic-separableprogrammingOPFalgorithmfor economicdis-patchandsecuritycontrol. IEEE Transactionson Power Systems, 12(4), 1682–1688,1997.
52 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. Thispaperpresentsanew algorithmfor optimalpowerflow (OPF),whichis basedonP/Qdecom-
positionof OPFproblemandon combinedapplicationof quadraticandseparableprogrammingmethods.
Initially theunit costcurvesareapproximatedby thequadraticfunctionsandthequadraticprogrammingal-
gorithmis appliedasastartingmethod.This givesagoodinitial point for optimizationandreducesthetotal
computationtime.Themodificationof a separableprogramming(SP)algorithmwith generationof approx-
imatingintervals is considered.A new quadratic-separablealgorithmfor OPFis proposed,which combines
themainadvantagesof quadraticandseparableprogrammingmethods.A bi-criterionformulationof security
controlproblemon thebaseof economicandsecurityobjective functions(OF) is proposed.Thenumerical
resultsof OPFfor large-scalepower systemsaregivenfor differentmethods.
L. Guan. An optimal neuronevolution algorithmfor constrainedquadraticprogramminginimagerestoration.IEEETransactionsonSystemsManandCyberneticsPart A—SystemsandHumans, 26(4), 513–518,1996.
Abstract. An optimalneuronevolutionalgorithmfor therestorationof linearlydistortedimagesis presented
in thispaper. Theproposedalgorithmismotivatedby thesymmetricpositive-definitequadraticprogramming
structureinherentin restoration.Theoreticalanalysisandexperimentalresultsshow that the algorithmnot
only significantlyincreasesthe convergencerateof processing,bat alsoproducesgoodrestorationresults.
In addition,the algorithmprovidesa genuineparallelprocessingstructurewhich ensurescomputationally
feasiblespatialdomainimagerestoration.
L. GuanandX. L. Zhou. Real-timeimagefiltering: from optimalneuronevolution to vectorquadraticprogramming. 1994 IEEE International Conferenceon Systems,Man, andCybernetics.Humans,InformationandTechnology IEEE,New York, NY, USA, pp.694–699,1994.
Abstract. In this paper, a new schemeis introducedfor thepartitioningof imagesin imagefiltering using
neuralnetworks.Theproposedschemetakesinto accountthephysicalnatureof theimageformationprocess,
andthussimplifiesthe orderingschemeassociatedwith the imageprocessingframework basedon neural
networkswith hierarchicalclusterarchitecture.Also presentedin thepaperis avectorprocessingalgorithm.
By utilizing this algorithm,the pixels in the samerow/columnof an imageareprocessedsimultaneously.
Comparedwith thescalarneuronevolution algorithms,thevectoralgorithmprovidesgoodvisualquality in
imagefiltering. Visualexamplesareprovidedto demonstratetheperformanceof thenew approach.
J.Guddat.Stability in convex quadraticparametricprogramming.MathematischeOperations-
forschungundStatistik, 7(2), 223–245,1976.
F. Guder. Pairwisereactive SORalgorithmfor quadraticprogrammingof net import spatial
equilibrium-models.MathematicalProgramming, 43(2), 175–186,1989.
F. GuderandJ. G. Morris. Optimal objective function approximationfor separableconvexquadraticprogramming.MathematicalProgramming, 67(1), 133–142,1994.
Abstract. Wepresentanoptimalpiecewise-linearapproximationmethodfor theobjective functionof sepa-
rableconvex quadraticprograms.Themethodprovidesguidelineson how many grid pointsto useandhow
to position them for a piecewise-linearapproximationif the error inducedby the approximationis to be
boundedapriori.
I. Guevski. An analytic form of solution in quadraticprogramming. TekhnicheskaMisul,9(2), 7–21,1972.
Abstract. A parametricmethodis proposedto find a non-negative vectorx*, maximizing(minimizing) the
goal function F x pTx xTCx, whereA is a diagonalnegatively definedmatrix having dimensionsof
n & n, providedthatdx b p 3 0 d 3 0 b 0 . Themethodis usedto expresstherelationx f p A d bin aclosedform.
N. I. M. GOULD & PH.L. TOINT 53
I. Guevski. Parametricmethodfor analyticalsolution of quadraticprogrammingproblems.Izvestiyana InstitutapoTekhnickeskaKibernetika, 15, 15–28,1973.
Abstract. A parametricmethodfor solutionof a quadraticsquareprogrammingproblem,arising in opti-
mizationof resourcesystems,is used.Thedependencebetweentheoptimumvectorandthecoefficientsin
theaim functionsandthe limitations is obtainedin apparentform. Thepossibility for solutionof problems
with a largenumberof variableswithout significantcomputingdifficulties, is shown. A numericalexample
is given.
T. D. GuoandS.Q. Wu. Predictor-correctoralgorithmfor convex quadraticprogrammingwithupperbounds.Journalof ComputationalMathematics, 13(2), 161–171,1995.
Abstract. Thepredictor-correctoralgorithmfor linearprogramming,proposedby S. Mizuno et al. (1990),
becamethemostwell known of theinteriorpointmethods.Thepurposeof thepaperis to extendtheseresults
in two directions.First,wemodify thealgorithmin orderto solveconvex quadraticprogrammingwith upper
bounds.Second,we replacethe correctorstepwith an iterationof R. C. Monteiro andAdler’s algorithm
(1989).With thesemodifications,thedualitygapis reducedby aconstantfactoraftereachcorrectorstepfor
convex quadraticprogramming.It is shown thatthenew algorithmhasaO nL iterationcomplexity.
T. D. GuoandS. Q. Wu. Interior ellipsoid methodfor convex quadraticprogramming.ActaMathematicaeApplacataeSinica, 19(1), 46–50,1996a.
Abstract. In this paper, a new variantof theinterior ellipsoidmethodfor convex quadraticprogrammingis
developed.Comparedto thealgorithmproposedby YeandTse(1989),thesub-problemof thenew algorithm
is a linearprogrammingproblemwhich is easyto solve,althoughbothhave thesamecomplexity.
T. D. Guo andS. Q. Wu. Propertiesof primal interior point methodsfor QP. Optimization,37(3), 227–238,1996b.
Abstract. Studiesthe propertiesof a primal interior point algorithmfor convex quadraticprograms.The
algorithmcanbeviewedasamodificationof thealgorithmspresentedby MonteiroandAdler (1989),Gold-
farbandLiu (1991),etc.In eachiteration,it computesanapproximatelyprojectedNewton directionhk and
needsO n2 4 5 arithmeticoperationsin average.Moving alonghk with a fixedstepsizewhich will make the
log-barrierfunction a significantdecrease.The algorithmterminatesafter O nL iterations.So the total
complexity of this algorithmis O n3L .Z. GuoandG. Xu. Calculationof economicdispatchof interconnectedsystemusingquadratic
programming.Automationof ElectricPowerSystems, 22(1), 40–44,1998.
Abstract. Basedon the principle of decompositioncoordinationandsensitivity analysis,a new modelof
the economicdispatchof interconnectedpower systemsis presentedin this paper, in which active powers
of tie lines areconsideredascoordinatingvariables.The primal problemis decomposedinto N individual
subproblemsof optimizationandonesimplecoordinationproblem.Thealgorithmis basedonquadraticpro-
grammingwith theideaof parametricprogrammingandrelaxationtechniques.Thefeaturesof theproposed
algorithmare little computationand reliableconvergencefor economicdispatchof interconnectedpower
systemsconcerningsafetyconstraints.Numericaltestshavebeencarriedout for a realinterconnectedpower
systemcomposedof threeindividual systems.The test resultsshow that the modelandthe algorithmare
bothfeasibleandeffective.
A. K. GuptaandJ. K. Sharma. A generalizedSimplex techniquefor solving quadraticpro-grammingproblem.IndianJournalof Technology, 21(5), 198–201,1983.
Abstract. A finite iterationprocedureis presentedfor solvingthequasi-concavequadraticfunctionsubjectto
linearconstraints.In eachiterationtwo basicvariablesarereplacedby two non-basicvariables.Theproblem
is solvedstartingwith a basicfeasiblesolutionandshowing theconditionsunderwhich thesolutioncanbe
improved.An illustrative exampleis given.
54 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
O.K. Gupta.Applicationsof quadraticprogramming.Journalof InformationandOptimizationSciences, 16(1), 177–194,1995.
Abstract. Quadraticprogramming(QP) haslong beenstudiedasan importantOR technique.Many algo-
rithmshave beendevelopedfor solvingQPproblems.QPhasalsobeenvery successfulfor modelingmany
real-life problems.This paperreviews applicationareaswhereQPhasbeeneffectively applied.Most appli-
cationsof QP have beenin finance,agriculture,economics,productionoperations,marketing, andpublic
policy. Applicationsin eachof theseareasarebriefly described.
P. GuptaandD. Bhatia. Multiparametricanalysisof themaximumtolerancein quadraticpro-grammingproblems.Opsearch, 37(1), 36–46,2000.
Abstract. In this paper, we usean alternative approachto study multiparametricsensitivity analysisin
quadraticprogrammingproblemsby usingtheconceptof maximumvolumein thetoleranceregion. A nu-
mericalexampleis givento illustratetheresultsof thepaper.
R. P. Gupta. Symmetricdualquadraticprogramin complex space.Proceedingsof theIndianAcademyof Sciences,SectionA, 72(2), 74–87,1970.
Abstract. Symmetricdualquadraticprogramin complex spaceis presentedandsomeduality theoremsare
proved. Self-duallinear andquadraticprogramsin complex spaceare formedandself-duality theoremis
extendedto thesecases.
C. D. Ha. An algorithmfor structured,large-scalequadraticprogrammingproblems.Technical
report,VirginiaCommonwealthUniversity, Virginia,USA, 1981.
W. W. HagerandD. W. Hearn. Thedualactive setmethodandquadraticnetworks. Research
Report90-7, Departmentof Industrial andSystemsEngineering,Gainesville,Florida,
USA, 1990.
W. W. HagerandY. Krylyuk. Graphpartitioningandcontinuousquadraticprogramming.SIAMJournalon DiscreteMathematics, 12(4), 500–523,1999.
Abstract. A continuousquadraticprogrammingformulation is given for min-cut graphpartitioningprob-
lems. In theseproblems,we partition the verticesof a graphinto a collection of disjoint setssatisfying
specifiedsizeconstraints,while minimizing the sumof weightsof edgesconnectingverticesin different
sets.An optimal solution is relatedto an eigenvector (Fiedlervector)correspondingto the secondsmall-
esteigenvalueof thegraph’s Laplacian.Necessaryandsufficient conditionscharacterizinglocal minimaof
thequadraticprogramaregiven.Theeffect of diagonalperturbationson thenumberof local minimizersis
investigatedusinga testproblemfrom theliterature.
W. W. Hager, T. A. Davis, Y. Krylyuk, andS.-C. Park. Graphpartitioningusingquadraticprogramming. Technicalreport, Computerand Information Scienceand EngineeringDepartment,Gainesville,Florida,USA, 1997.
Abstract. A continuousquadraticprogrammingformulationsaregivenfor min-cutgraphpartitioningprob-
lems.An optimalsolutionis relatedto aneigenvector(Fiedlervector)correspondingto thesecondsmallest
eigenvalueof thegraph’s Laplacian.Necessaryandsufficient conditionscharacterizinglocal minimaof the
quadraticprogramare given. A generalizationof the Kernighanand Lin exchangealgorithm to a block
exchangealgorithmcanbeusedto escapefrom alocalminimumof thecontinuousquadraticprogram.Com-
parisonsto theoptimalcut obtainedby otherapproachesto graphpartitioningarepresented.
W. W. Hager, P. M. Pardalos,I. M. Roussos,andH. D. Sahinoglou. Active constraints,in-
definitequadraticprogramming,andtestproblems.Journalof OptimizationTheoryand
Applications, 68(3), 499–511,1991.
N. I. M. GOULD & PH.L. TOINT 55
H. H. Hall, E. O. Heady, A. Stoecker, andV. A. Sposito.Spatialequilibriumin usagriculture:a quadraticprogramminganalysis.SIAMReview, 17(2), 323–338,1975.
Abstract. A spatialcompetitiveequilibriumfor thecropandlivestocksectorsof theUSagriculturaleconomy
is approximated.Agricultural commoditiesare classifiedinto threemutually exclusive classes:primary,
intermediateanddesired.Primarycommoditiesrepresentavailableresources;intermediatecommoditiesare
producedonly asinputsfor furtherproduction;desiredcommoditiesarewantedeitherfor consumptionor
for otherusesoutsidethe system(export). Productionof cropsand livestockand intermarket commodity
shipmentsarerepresentedby linearactivities.Theobjective functionmaximizesaggregateproducerprofits.
Sincethedemandfunctionsarelinear, total revenueis quadraticin thepricesof desiredcommodities,hence
thequadraticprogrammingformulation.In the results,estimatedpricesfor desiredcommoditiesarelower
thanobservedpricesand,with minorexceptions,estimatedquantitiesexceedobservedquantities.
P. L. HammerandA. A. Rubin. Someremarkson quadraticprogrammingwith 0–1variables.RevueFrancaised’Informatiqueet deRechercheOperationnelle, 4(3), 67–79,1970.
Abstract. The aim of this paperis to show that every bivalent (0,1) quadraticprogrammingproblemis
equivalentto onehaving apositive (negative) semi-definitematrix in theobjective function,to establishcon-
ditionsfor differentclassesof localoptimality, andto show thatany problemof bivalent(0,1)programmingis
equivalent(a) to theproblemof minimizing a realvaluedfunction,partly in (0,1)andpartly in non-negative
variables,(b) to theproblemof finding theminimaxof a realvaluedfunctionin bivalent(0,1)variables.
P. L. Hammer, P. Hansen,andB. Simeone.Roof duality, complementationandpersistency in
quadratic0–1optimization.MathematicalProgramming, 28(2), 121–155,1984.
C. G. Han,P. M. Pardalos,andY. Ye. Computationalaspectsof aninterior point algorithmfor
quadratic-programmingproblemswith box constraints.In T. F. ColemanandY. li, eds,
‘Large-ScaleNumericalOptimization’,pp.92–112,SIAM, Philadelphia,USA, 1990a.
C. G. Han, P. M. Pardalos,andY. Ye. Interior point algorithmsfor quadraticprogramming
problems.In ‘Proceedingsof theConferenceon OptimizationMethodsandtheir Appli-
cations,Nauka,USSR’,pp.194–213,1990b. (In Russian).
C. G. Han, P. M. Pardalos,and Y. Ye. Algorithms for the solution of quadraticknapsack
problems.LinearAlgebra andits Applications, 152, 69–91,1991.
C. G. Han,P. M. Pardalos,andY. Ye. Onthesolutionof indefinitequadraticproblemsusingan
interiorpoint method.Informatica, 3(4), 474–496,1992.
S. P. Han. Solvingquadraticprogramswith anexactpenaltyfunction. In O. L. Mangasarian,
R.R.MeyerandS.M. Robinson,eds,‘NonlinearProgramming,4’, pp.25–55,Academic
Press,LondonandNew York, 1981.
S. P. Han. On the hessianof the Lagrangianandsecond-orderoptimality conditions. SIAM
Journalon Control andOptimization, 24, 339–345,1985.
S. P. Han andO. Fujiwara. An inertia theoremfor symmetricmatricesandits applicationto
nonlinearprogramming.LinearAlgebra andits Applications, 72, 47–58,1985.
S.P. HanandO.L. Mangasarian.Characterizationof positivedefiniteandsemidefinitematricesvia quadraticprogrammingduality. SIAM Journal of Algebraic and DiscreteMethods,5(1), 26–32,1984.
56 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. Positive definiteandsemidefinitematricesinducewell-known duality resultsin quadraticpro-
gramming.Theconverseis establishedhere.Thusif certainduality resultshold for a pair of dualquadratic
programs,thentheunderlyingmatrix mustbepositive definiteor semidefinite.For example,if a strict local
minimumof aquadraticprogramexceedsor equalsastrictglobalmaximumof thedual,thentheunderlying
symmetricmatrixQ is positivedefinite.If aquadraticprogramhasalocalminimum,thentheunderlyingma-
trix Q is positive semidefiniteif andonly if theprimalminimumexceedsor equalsthedualglobalmaximum
andxT Qx 0 implies Qx 0. A significantimplication of theseresultsis that theWolfe dual maynot be
meaningfulfor nonconvex quadraticprogramsandfor nonlinearprogramswithout locally positive definite
or semidefiniteHessians,evenif theprimal secondordersufficient optimality conditionsaresatisfied.
M. T. HannaandM. Simaan. A closedform solution to a quadraticprogrammingproblemin complex variables. In ‘Proceedingsof the 23rd IEEE Conferenceon DecisionandControl.IEEE,New York, NY, USA’, Vol. 2, pp.1087–1092,1984.
Abstract. A specialquadraticprogrammingproblemin complex variablesis investigatedfor a closedform
solution.Two differentapproachesareused.Thefirst is a directapproachthat leadsto a family of solutions
becauseof a singularmatrix encounteredin thesolutionprocess.Thesecondis anindirectapproachbased
on parameterizingtheobjective function. It leadsto a solutionwhich is a memberin theabove family and
which is shown to bebounded.
P. Hansen.Quadratic0–1programmingby implicit enumeration.In F. A. Lootsma,ed.,‘Nu-
mericalmethodsfor non linear optimization’, pp. 265–278.AcademicPress,London,
England,1972.
P. HansenandB. Jaumard.Reductionof indefinitequadraticprogramsto bilinear programs.
Journalof GlobalOptimization, 2, 41–60,1992.
P. Hansen,B. Jaumard,M. Ruiz, andJ. Xiong. Global minimizationof indefinitequadratic
functionssubjectto box constraints. ReportG-91-54,GERAD, Ecole Polytechnique,
UniversiteMcGill, Montreal,Quebec,Canada,1991.
F. HarriganandI. Buchanan.A quadratic-programmingapproachto input-outputestimation
andsimulation.Journalof RegionalScience, 24(3), 339–358,1984.
K. HassanandF. Mahmoud.An incrementalapproachfor thesolutionof quadraticprogram-ming problems.MathematicalModelling, 8, 34–36,1987.
Abstract. An approachfor thesolutionof quadraticprogrammingproblemsis introduced.It is basedon an
incrementalmethod,andthemaximumnumberof incrementsis equalto thenumberof constraintsplus1.
E.J.Haug,R.Chand,andK. Pan.Multibody elasticcontactanalysisbyquadraticprogramming.Journalof OptimizationTheoryandApplications, 21(2), 189–198,1977.
Abstract. A quadraticprogrammingmethodfor contactproblemsis extendedto ageneralprobleminvolving
contactof n elasticbodies.Sharpresultsof quadraticprogrammingtheoryprovide anequivalencebetween
the original n-body contactproblemand the simplex algorithmusedto solve the quadraticprogramming
problem.Two multibodyexamplesaresolvedto illustratethetechnique.
B. S.He. A projectionandcontractionmethodfor a classof linearcomplementarity-problemsandits applicationin convex quadraticprogramming. AppliedMathematicsand Opti-mization, 25(3), 247–262,1992.
Abstract. In this paperwe proposea new iterative methodfor solving a classof linear complementarity
problems:u 0, Mu q , uT Mu q5 0, whereM is a given l l positive semidefinitematrix (not
necessarilysymmetric)andq is agiven l-vector. Themethodmakestwo matrix-vectormultiplicationsanda
N. I. M. GOULD & PH.L. TOINT 57
trivial projectionontothenonnegative orthantat eachiteration,andtheEuclideandistanceof theiteratesto
thesolutionsetmonotonouslyconvergesto zero.Themainadvantagesof themethodpresentedareits sim-
plicity, robustness,andability to handlelargeproblemswith any startpoint. It is pointedout thatthemethod
maybeusedto solve generalconvex quadraticprogrammingproblems.Preliminarynumericalexperiments
indicatethatthismethodmaybevery efficient for largesparseproblems.
G. L. Hefley andM. E. Thomas. A comparisonof two quadraticprogrammingalgorithms.Technicalreport,FloridaUniv , Gainesville,FL, USA, 1970.
Abstract. ThepapercomparesWolfe’squadraticprogrammingalgorithmwith CottleandDantzig’sprinciple
pivot method.It is shown thatWolfe’s algorithmrequiresmoreoperations.
R. Helgason,J. L. Kennington,andH. Lall. A polynomially boundedalgorithmfor a singlyconstrainedquadraticprogram.MathematicalProgramming, 18(3), 338–343,1980.
Abstract. Presentsa characterizationof thesolutionsof a singly constrainedquadraticprogram.This char-
acterizationis thenusedin thedevelopmentof apolynomiallyboundedalgorithmfor thisclassof problems.
C. Helmberg. Fixing variablesin semidefiniterelaxations.SIAMJournal on Matrix AnalysisandApplications, 21(3), 952–969,2000.
Abstract. Thestandardtechniqueof reducedcostfixing from linearprogrammingis not trivially extensible
to semidefiniterelaxationsbecausethecorrespondingLagrangemultipliersareusuallynotavailable.Wepro-
posea generaltechniquefor computingreasonableLagrangemultipliers for constraintsthatarenot partof
theproblemdescription.Its specializationto thesemidefinite67 1 18 relaxationof quadratic0–1program-
ming yieldsanefficient routinefor fixing variables.Theroutineoffers thepossibilityof exploiting problem
structure.Weextendthetraditionalbijective mapbetween6 0 18 and 67 1 18 formulationsto theconstraints
sothat thedualvariablesremainthesameandstructuralpropertiesarepreserved.Consequently, thefixing
routinecanbe appliedefficiently to optimal solutionsof the semidefinite6 0 18 relaxationof constrained
quadratic0–1programmingaswell. Weprovidenumericalresultsshowing theefficacy of this approach.
M. P. HelmeandT. L. Magnanti.Designingsatellitecommunicationnetworksby 0–1quadraticprogramming.Networks, 19(4), 427–450,1989.
Abstract. In satellitecommunicationnetworks,distinctive facilitiescalledhomingstationsperformspecial
transmissionfunctions.Local demandnodesclusteredaroundeachhomingstationcommunicatewith each
othervia a local switch at the homingstation;demandnodesin differentclusterscommunicatewith each
othervia satelliteearthstationsat thehomingstations.Designingsucha communicationnetwork requires
choiceson the locationsof theearthstationsandon theassignmentsof demandnodesto the local clusters
at the earthstations.The authorsformulatethis problemasa 0–1 quadraticfacility locationproblemand
transformit into anequivalent0–1integerlinearprogram.Computationalexperienceonrealdatashows that
a branchandboundprocedureis effective in solving problemswith up to 40 demandnodes(major cities)
andthatthesolutionsthatthis algorithmfindsimprove considerablyuponmanagementgeneratedsolutions.
It is alsoshown thata greedyaddheuristic,asimplementedon anIBM PC,consistentlygeneratesoptimal
or near-optimalsolutions.
C. T. Herakovich andP. G. Hodge.Elastic-plastictorsionof hollow barsby quadraticprogram-ming. InternationalJournalof MechanicalSciences, 11(1), 53–63,1969.
Abstract. A numericalmethodwhichprovidesthecompletehistoryof thestressfunctionfor elastic-plastic
torsionof hollow barsduringquasistatic,monotonictwist is presented.Resultsaregiven for severalcross-
sectionsandarecomparedto otheravailableresults.Plasticunloadingis explicitly shown.
G. T. HermanandA. Lent. A family of iterative quadraticoptimizationalgorithmsfor pairs
of inequalities,with applicationin diagnosticradiology. MathematicalProgramming
Studies, 9, 15–29,1978.
58 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
C. Hildreth. A quadraticprogrammingprocedure.NavalResearch LogisticsQuarterly, 4, 79–
85,1957.Erratum,ibid. p. 361.
K. Hitomi andH. Nagasawa. Note on “Solution of the aggregateproductionplanningprob-lem in a multi-stage-multi-productmanufacturingsystemusingfunctionalspaceanaly-sisandquadraticprogrammingapproaches”.InternationalJournal of SystemsScience,15(11),1257–1262,1984.
Abstract. The paperof Teny andKochhar(seeibid., vol. = 14, num. 3, p. 325, 1983)concludedthat the
solutioncalculatedby using the functional spaceanalysistechniquepresentedby Hitomi andNakamura
(1976)is inferior to thatrenderedby thequadraticprogrammingapproach(Beale,1968).This papershows
thecontradictionincludedin their discussionanddeniestheabove conclusion.Functionalspaceanalysisis
alsodevelopedto obtainamixed-integersolution.
D. S. HochbaumandS. P. Hong. About stronglypolynomial-timealgorithmsfor quadratic
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model.In many cases,thestudenttransitiondataneededto estimatethetransitionprobabilitiesby themaxi-
mumlikelihoodtechniqueis notavailable.Theaggregateenrollmentsareknown, thetransitionprobabilities
canbe estimatedby usinga quadraticprogrammingformulation of the problem.This methodis usedto
forecastelementary, secondary, andhighereducationenrollmentsin acontinuousflow process.
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tableausis a handicap.Hence,an approachwas chosenwherebypenaltytermsas usually employed for
constrainingvariablesareconvertedto straight inequalitiesthusyielding linear programmingtechniques.
The methoddevelopedworks in two steps.The first solvesthe unconstrainedproblemwhich is linear, the
secondsolvessuperimposedLP problem.For computationalreasonstheviolationof constraintsis monitored
andtheLP tableauis built up asviolationsaredetected.Sparsity, forward-backwardsubstitutionandupdat-
ing techniquesareexploited.Economicloaddispatchingandlossminimizationareexamplesto illustratethe
effectivenessof themethod.
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Abstract. Weprove thatageneralconvex quadraticprogram(QP)canbereducedto theproblemof finding
thenearestpointonasimplicial conein O(n(3)+ n log L) steps,wheren andL are,respectively, thedimen-
sion andthe encodinglengthof QP. The proof is quite simpleandusesduality andrepeatedperturbation,
The implication,however, is nontrivial sincethe problemof finding the nearestpoint on a simplicial cone
hasbeenconsidereda simplerproblemto solve in the practicalsensedueto its specialstructure,Also we
N. I. M. GOULD & PH.L. TOINT 59
show that,theoretically, this reductionimpliesthat(i) if analgorithmsolvesQPin a polynomialnumberof
elementaryarithmeticoperationsthatis independentof theencodinglengthof datain theobjective function
thenit canbeusedto solve QPin stronglypolynomialtime,and(ii) if L is boundedby a ’first order’ expo-
nentialfunctionof n then(i) canbestatedeven in strongerterms:to solve QPin stronglypolynomialtime,
it sufficesto find analgorithmrunningin polynomialtime that is independentof theencodinglengthof the
quadratictermmatrixor constraintmatrix.Finally, basedon theseresults,weproposeaconjecture.
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linearprograms.
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ing underuncertainty. The methodimproves upon existing grey linear programming(GLP) methodsby
allowing theconsiderationof theeffectsof economiesof scaleoncostcoefficientsin theobjective function.
The approachalsohasadvantagesover a grey nonlinearprogrammingmethod,sincea global optimumis
obtainableandthe model is moderatelyeasyto solve throughcommerciallyavailablequadraticprogram-
mingpackages.Themodellingapproachis appliedto ahypotheticalproblemof wasteflow allocationwithin
a municipal solid wastemanagementsystem.The resultsindicatethat, comparedwith the GLP method,
GQPprovidesamoreeffective meansfor reflectingsystemcostvariationsandmaythereforegeneratemore
realisticandapplicablesolutions.
G.H. Huang,B. W. Baetz,andG.G.Patry. Wasteflow allocationplanningthroughagrey fuzzyquadratic-programmingapproach.Civil EngineeringSystems, 11(3), 209–243,1994.
Abstract. This paperproposesa grey fuzzy quadraticprogramming(GFQP)approachasa meansfor op-
timization analysisunderuncertainty. The methodcombinesthe ideasof grey fuzzy linear programming
(GFLP)andfuzzyquadraticprogramming(FQP)within ageneraloptimizationframework. It improvesupon
thepreviousGFLPmethodby usingn grey controlvariables,x λ i $ i 1 2 n , for n constraintsinstead
of onex λ for n constraintsin orderto incorporatethe independentpropertiesof thestipulationuncertain-
ties; it alsoimprovesupontheFQPmethodby furtherintroducinggrey numbersfor coefficientsin A andC
to effectively reflectthe lefthandsideuncertainties.Comparedwith theGFLPmethod,theGFQPapproach
is helpful for bettersatisfyingmodelobjective/constraints andproviding grey solutionswith highersystem
certaintyandlower systemcost;comparedwith theFQPmethod,moreinformationof the independentun-
certainfeaturesof not only the stipulationsbut also the lefthandsidecoefficients areeffectively reflected
in theGFQPmethod.TheGFQPmodellingapproachis appliedto a hypotheticalcasestudyof wasteflow
allocationplanningunderuncertainty, with the input model stipulationsfluctuatingwithin wide intervals
andhaving independentuncertaincharacteristics.Theresultsindicatedthat reasonablesolutionshave been
generated.ComparisonsbetweentheGFQPandFQP/GFLPsolutionsarealsoprovided,whichdemonstrate
60 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
that the GFQPmethodcould betterreflectsystemuncertaintiesandprovide morerealisticandapplicable
solutionswith lower systemuncertaintiesandhighersystembenefits.
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proposedmethodoutperformsanaverageextrapolationmethodandthePOCS-basedextrapolationmethod
in anexchangeof thecalculationtime.
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Abstract. A new methodfor theeconomicdispatchof active power in transmissionnetworks is presented.
Formulationof the problemas a quadraticprogramallows the inclusion of a linear network model and
operationalconstraintstogetherwith anexplicit representationof the costincurredby transmissionlosses.
The problemis solved by a linear complementarypivoting algorithm which takes full advantageof the
sparseform of theobjective functionandconstraints.Computationalexperienceindicatesthatthemethodis
highly efficient andis capableof solvingrealisticproblemsin elapsedtimesthatarecompatiblewith online
applicationsusingminicomputerhardware.
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Abstract. Proposesadigital computingmethodto determinetheoptimumsupplyvoltageof substationtrans-
formers.Thesupplyvoltageof substationtransformersis regulatedsothatthesupplyvoltageto consumersis
keptwithin anallowablerangewithoutoverloadingsubstationtransformersanddistribution lines.Thisprob-
lem canbeformulatedinto quadraticprogramming,which canbesolvedmosttypically by Beale’s method.
Crout’s methodis useful for solving large-scalesimultaneouslinear equationsand for finding a feasible
solution.
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Abstract. For pt.I seeibid., vol.95,no.2,p.33(1975).Beale’s methodis usedto determinethetruly optimal
solutionof thequadraticprogramming.A practicaldigital computerprogramis alsodevelopedandapplied
to a large-scaledistribution network containingsix transformerbanks,34 feedersand177 loadpoints.The
total computingtime is about4.0sec.Therequiredmemorysizeis assmallas20kilowords.
N. I. M. GOULD & PH.L. TOINT 61
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years(Markowitz’s efficient portfolioswith minimumrisk) andtheotheramorerecentinnovation(Sharpe’s
style analysiswhich estimatesan implied assetallocationfor an investmentfund). We show how, in the
presenceof inequalityconstraints,Excel’s Solver canbeusedto find theoptimalweightsin bothquadratic
programmingapplications.We alsoimplementa directanalyticsolutionfor generatingtheefficient frontier
whenthereareno inequalityconstraintsusingthe matrix functionsin Excel.Both applicationsuseonly a
smallnumberof assetclassesandrequirerepeateduseof theminimisationtask.Weshow how VisualBasic
for Applications(Microsoft’s macrolanguagefor Excel)canbeusedto programsuchtasks,confirmingthat
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series.Thecontrolvariablesareobtainedfrom thesystemstateequationsasa functionof theapproximated
statevariables.In this method,thereis no needto integratethesystemstateequations,andtheperformance
index is evaluatedby an algorithmwhich is alsoproposedin this paper. This converts the optimal control
probleminto a small sizeparameteroptimizationproblemwhich is quadraticin the unknown parameters,
thereforethe optimal value of theseparameterscanbe obtainedby usingquadraticprogrammingresults.
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crashprocedureto obtaina goodinitial basicsolutionfor the quadraticprogrammingalgorithm.We show
how this solutionmay beusedasa startingsolutionfor thesimplex-basedalgorithm.Besidesits ability to
obtaingoodstartingsolutions,thisprocedurehasseveraladditionalproperties.It canbeuseddirectly to find
anoptimalsolutionto aquadraticprograminsteadof simply finding a goodinitial solution;it providesboth
62 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
upperandlower boundson theobjective functionvalueasthealgorithmproceeds;it reducesthecomplex-
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setof constraintsto accomplishthis formulation.This makesthe equationspresentedheremoregenerally
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ationallocationof thermalunitsaccuratelyandquickly. Most conventionalequalincrementalmethodscan-
not alwaysobtaintheoptimalsolutionfor ELD problemswhentransmissionlossesand/orupperandlower
boundsof generatoroutputsaretakeninto account.Numerousstudieshavebeenmadeto solve theproblems
accuratelywith considerablecomplexities.Theformulationof theproblemconsideredasstandardfor QPBS
is very similar to that of the ELD problemwhentransmissionlossesareneglected.Thereforeto the ELD
calculationwe apply the QPBSalgorithm,especiallythe Pardalos-Kovoor method,whosecomputingtime
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to keepimportantgroupsof cellstogetherasthecellsarespreadthroughouttheplacementarea.Numerical
resultsonasetof benchmarkcircuitsillustratethatthisnew approachproducesstandardcell placementsthat
areup to 17%betterin wire length,14%betterin row lengthandup to 24 timesfasterthana well known
simulatedannealingplacementheuristic.
J.L. KenningtonandD. E.Fyffe. A noteonthequadraticprogrammingapproachto thesolutionof the0–1integerprogrammingproblem.Bulletinof theOperationsResearch SocietyofAmerica, 20, B333–334,1972.
Abstract. The problemmaxcT x subjectto Ax b, andx integer with componentseither 0 or 1, canbe
reformulatedas,maxQ x aT x x;Cx subjectto Ax b, andx < 0 1= , whereQ x is strictly convex. The
work of Ritterhasbeenspecializedto solve thisquadraticproblem.Initial computationalresultsindicatethat
theapproachis not competitive time-wisewith theimplicit enumerationalgorithmof Geoffrion.
G. Keri. On theminimumvalueof a quadraticfunctionunderlinearconstraints.StudiaScien-
tiarumMathematicarumHungarica, 6, 193–196,1971.
G. Keri. On a classof quadraticforms. In A. Prekopa,ed.,‘Survey of MathematicalProgram-
ming,Vol. 1’, pp.231–247,North Holland,Amsterdam,theNetherlands,1979.
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E. Kim, H. J. Kang, andM. Park. Numericalstability analysisof fuzzy control systemsviaquadraticprogrammingandlinear matrix inequalities. IEEE Transactionson Systems,ManandCybernetics,Part A (SystemsandHumans), 29(4), 333–346,1999.
Abstract. This paperproposesa numericalstability analysismethodologyfor thesingleton-typelinguistic
fuzzycontrolsystemsbasedonoptimizationtechniques.First, it demonstratesthatasingleton-typelinguistic
fuzzy logic controller(FLC) canbeconvertedinto a region-wisesector-boundedcontrolleror, moregener-
ally, a polytopic systemby quadraticprogramming(QP). Next, the convex optimizationtechniquecalled
linearmatrix inequalities(LMI) is usedto analyzetheclosedloopof theconvertedpolytopicsystem.Finally,
theapplicabilityof thesuggestedmethodologyis highlightedvia simulationresults.
P. Y. Kim andC. W. Yang. Advertisingdecision-model—aquadratic-programmingapproach.
In ‘1985 Proceedingsof theAnnualMeetingof theAmericanInstitutefor DecisionSci-
ences’,Vol. 1–2,pp.551–553,1985a.
P. Y. Kim and C. W. Yang. A quadraticprogrammingmediachoicemodel with repeatedexposurefunctions. In R. Hanham,W. G. Vogt andM. H. Mickle, eds,‘Modeling andSimulation.Proceedingsof theSixteenthAnnualPittsburgh Conference.ISA, ResearchTrianglePark,NC, USA’, pp.1773–1776,1985b.
Abstract. Highlights the fundamentallimitation of the linearprogrammingmediaselectionmodelthrough
numerouscomputersimulations.As analternative, a quadraticprogrammingmodelwasproposedto over-
comethelimitation of linearprogrammingsolutionsto mediaselectionproblems.
Y. H. Kim, S. Y. Kim, andJ. B. Kim. Adaptive-controlof a binarydistillation columnusing
quadraticprogramming.KoreanJournalof ChemicalEngineering, 6(4), 306–312,1989.
K. C. Kiwiel. A methodfor solvingcertainquadratic-programmingproblemsarisingin nons-
moothoptimization.IMA Journalof NumericalAnalysis, 6(2), 137–152,1986.
K. C.Kiwiel. A dualmethodfor certainpositivesemidefinitequadraticprogrammingproblems.SIAMJournalon ScientificandStatisticalComputing, 10(1), 175–186,1989.
Abstract. Presentsadualactive setmethodfor minimizingasumof piecewiselinearfunctionsandastrictly
convex quadraticfunction,subjectto linearconstraints.It maybeusedfor directionfinding in nondifferen-
tiableoptimizationalgorithmsandfor solvingexactpenaltyformulationsof (possiblyinconsistent)strictly
convex quadraticprogrammingproblems.An efficient implementationis describedextendingtheGoldfarb
andIdnanialgorithm,which includesPowell’s refinements.Numericalresultsindicateexcellentaccuracy of
theimplementation.
K. C. Kiwiel. A Cholesky dual methodfor proximal piecewise linear programming. Nu-merischeMathematik, 68, 325–340,1994.
Abstract. A quadraticprogrammingmethodis given for minimizing a sumof piecewise linear functions
anda proximal quadraticterm, subjectto simpleboundson variables.It may be usedfor searchdirection
finding in nondifferentiableoptimizationalgorithms.An efficient implementationis describedthatupdatesa
Cholesky factorizationof active constraintsandprovidesgoodaccuracy via iterative refinement.Numerical
experienceis reportedfor somelargeproblems.
E. Klafszky and T. Terlaky. Orientedmatroids,quadraticprogrammingand the criss-crossmethod.AlkalmazottMatematikaiLapok, 14(3–4),365–375,1989.
Abstract. Quadraticprogramming,symmetry, andpositive (semi)definitenessweregeneralizedby Morris
andToddto orientedmatroids.Theauthorsslightly modify their definitionsin orderto getmoresymmetric
structures.Somegeneralizationsof Terlaky’s criss-crossmethodarepresentedfor orientedmatroidquadratic
68 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
programming.Thesealgorithmsarebasedon thesmallsubscriptrule andon signpatterns,anddo not pre-
serve feasibility on any subsets.Finally two specialcases(positive definitecaseandorientedmatroidlinear
programming)anda modificationarepresented.
E. Klafszky andT. Terlaky. Somegeneralizationsof thecriss-crossmethodfor quadraticpro-gramming.Optimization, 24(1–2),127–139,1992.
Abstract. Three generalizationsof the criss-crossmethod for quadratic programmingare presented.
Tucker’s, Cottle’s andDantzig’s principal pivoting methodsarespecializedasdiagonalandexchangepiv-
ots for the linearcomplementarityproblemobtainedfrom a convex quadraticprogram.A finite criss-cross
method,basedon least-index resolution,is constructedfor solving the LCP. In proving finiteness,orthog-
onality propertiesof pivot tableausandpositive semidefinitenessof quadraticmatricesareused.In the last
sectionsomespecialcasesandtwo furthervariantsof thequadraticcriss-crossmethodarediscussed.If the
matrixof theLCPhasfull rank,thenasurprisinglysimplealgorithmfollows, whichcoincideswith Murty’s
’Bard typeschema’in theP matrix case.
D. Klatte. OntheLipschitzbehaviour of optimalsolutionsin parametricproblemsof quadratic
programmingandlinearcomplementarity. Optimization, 16, 819–831,1985.
J.M. Kleinhans,G. Sigl, F. M. Johannes,andK. J.Antreich. GORDIAN: VLSI placementbyquadraticprogrammingandslicingoptimization.IEEETransactionsonComputerAidedDesignof IntegratedCircuitsandSystems, 10(3), 356–365,1991.
Abstract. In this paperwe presenta new placementmethodfor cell-basedlayoutstyles.It is composedof
alternatingandinteractingglobaloptimizationandpartitioningstepsthatarefollowedby anoptimizationof
theareautilization.Methodsusingthedivide-and-conquerparadigmusuallylosetheglobalview by gener-
atingsmallerandsmallersubproblems.In contrast,GORDIAN maintainsthesimultaneoustreatmentof all
cellsover all globaloptimizationsteps,therebyconsideringconstraintsthatreflectthecurrentdissectionof
the circuit. The global optimizationsareperformedby solving quadraticprogrammingproblemsthat pos-
sessuniqueglobalminima.Improvedpartitioningschemesfor thestepwiserefinementof theplacementare
introduced.Theareautilization is optimizedby anexhaustive slicing procedure.Theplacementmethodhas
beenappliedto realworld problemsandexcellentresultsin termsof bothplacementqualityandcomputation
timehave beenobtained.
B. Klummer. Globalestabilitat quadratischeroptimierungsauprobleme.Wissenschaftlische
Zeitschrift derrHumboldtUniversitat zuBerlin, 5, 565–569,1977.
T. Koch. Algorithm for resolving manipulatorredundancy—the compactQP method inNewton-Raphsoniterationscheme.In V. ChundyandE. Kurekova, eds,‘ISMCR ’95.Proceedingsof the Fourth InternationalSymposiumon Measurementand Control inRobotics.SlovakTech.Univ, Bratislava,Slovakia’, pp.357–362,1995.
Abstract. Kinematicredundancy occurswhena manipulatorpossessesmoredegreesof freedomthanthe
minimumnumberrequiredto executea given task.Thekinematicredundantrobotoffersgreaterflexibility,
versatility anddexterity. It cansimultaneouslyrealizethe main taskandadditionalformulatedtasks,like
for examplekeepingthe joints movementswithin a given limits, obstacleavoidance,singularconfiguration
avoidance,etc. The principal problemof redundantrobot applicationis to find suitablecontrol algorithm
(kinematicinversetransformationscheme).First, the papersetsup the requirementsthat suchalgorithms
shouldfulfil for industrialapplications.Next, thepaperpresentsanalgorithmthatmeetstheserequirements.
It is the compactQP methodin Newton-Raphsoniteration schemewith someelementsof the so called
configurationcontrol.Finally, simulationresultsfrom testingthis algorithmin simulationsystemfor robot
workcell aregiven.
T. Koch and P. Kowalczewski. Control of redundantrobot using the quadraticprogram-ming method.In ‘PraceNaukowe InstytutuCybernetykiTechnicznejPolitechnikiWro-clawskiej,Seria:Konferencjenumber43’, pp.411–418,1996.
N. I. M. GOULD & PH.L. TOINT 69
Abstract. Thealgorithmfor controlof redundantrobotsis presented.It is thecompactquadraticprogram-
mingmethodin aNewton-Raphsoniterationschemewith someelementsof socalledconfigurationcontrol.
Two optionsof thealgorithmaredescribed.Finally theresultsfrom movementssimulationof therobotswith
openkinematicmultiple redundantchainswith 11 jointsarepresented.
M. Kojima and L. Tuncel. Discretizationand localization in successive convex relaxation
methodsfor nonconvex quadraticoptimizationproblems. TechnicalReportCORR98-
34, Departmentof Combinatoricsand Optimization,University of Waterloo,Ontario,
Canada,1998.
H. Konno.Maximizationof aconvex quadraticfunctionunderlinearconstraints.Mathematical
Programming, 11(2), 117–127,1976.
H. Konno.Maximizingaconvex quadraticfunctionoverahypercube.Journalof theOperations
Research Societyof Japan, 23(2), 171–189,1980.
P. KorhonenandG.Y. Yu. A referencedirectionapproachto multipleobjectivequadratic-linearprogramming.EuropeanJournalof OperationalResearch, 102(3), 601–610,1997.
Abstract. In this paper, weproposeaninteractive procedurefor solvingmultiplecriteriaproblemswith one
quadraticobjective,severallinearobjectives,andasetof linearconstraints.Theprocedureis basedontheuse
of referencedirectionsandweightedsums.Referencedirectionsfor thelinearfunctions,andweightedsums
for combiningthequadraticfunctionwith thelinearonesareusedasparametersto implementthefreesearch
of nondominatedsolutions.The idealeadsto the parametriclinear complementarityproblemformulation.
An approachto dealwith this typeof problemsis givenaswell. Theapproachis illustratedwith anumerical
example.
P. Korhonenand G. Y. Yu. On computingobjective function valuesin multiple objectivequadratic-linearprogramming.EuropeanJournalof OperationalResearch, 106(1),184–190,1998.
Abstract. We considerthe computationof objective function valueswhen a nondominatedfrontier is
searchedin multipleobjective quadratic-linearprogramming(MOQLP).Referencedirectionsandweighted-
sumsconstituteamethodologicalbasisfor thesearch.This idealeadsto aparametriclinearcomplementarity
model formulation.A critical task of making a searchprocedureefficient, is to computethe changesin
quadraticandlinearobjective functionsefficiently whena searchdirectionis changedor a basischangeis
performed.Thosechangesin objective functionscanbecomputedby a so-calleddirector indirectmethod.
The direct methodis a straightforward oneand basedon the useof unit changesin basicdecisionvari-
ables.Instead,theindirectmethodutilizessomeotherbasicvariablesof themodel.Weintroducetheindirect
methodandmake theoreticalandempiricalcomparisonsbetweenthemethods.Basedon the comparisons,
wepointout thattheindirectmethodis clearlymuchmoreefficient thanthedirectone.
F. Korner. Remarkson second-orderconditionsin connectionwith thealgorithmof Bealeforquadraticprogramming.EuropeanJournalof OperationalResearch, 40(1),85–89,1989.
Abstract. Thealgorithmof Bealeis considered.It is discussedfor thecasein which the’solution’ point of
Beale’s algorithmis a localminimizer. In thisalgorithmageneralform of second-orderconditionsarises.
F. Korner. Onthenumericalrealizationof theexactpenaltymethodfor quadraticprogrammingalgorithms.EuropeanJournalof OperationalResearch, 46(3), 404–408,1990.
Abstract. Optimizationproblemswith quadraticnonconvex objective functionsandlinear inequalitiesare
considered.An activeconstraintstrategy isusedto defineasequenceof equalityconstrainedproblems.Under
discussionis how thesubproblemscanbesolvedefficiently.
70 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
F. Korner. A noteon weakly active constraintsin connectionwith nonconvex quadraticpro-gramming.EuropeanJournalof OperationalResearch, 57(3), 409–411,1992.
Abstract. In generalit is very difficult to determinea true local minimizer in nonconvex quadraticpro-
gramming.Themainproblemsariseif wehave so-calledweaklyactive constraints.Theauthordiscussesan
efficient methodfor checkingthelocal optimality or determiningadirectionof descent.
F. KornerandB. Luderer. On the implementationof quadraticprogrammingalgorithms.Sys-temsAnalysisModellingSimulation, 6(9), 699–707,1989.
Abstract. Sequencesof systemsof linearequationsareconsideredwhich ariserecursively by addingor re-
moving oneequation.It is shown how asolutioncanbeefficiently obtainedfrom thesolutionof theprevious
system.Systemsof this kind originatefor examplein thesolutionprocessof generalquadraticoptimization
problemswith linear inequality constraintsvia reducingthemto equality constrainedproblems.Together
with thesolutionof thesystems,thenumberof positive, negative andzeroeigenvaluesis determined.Thus,
thevalidity of second-orderKuhn-Tucker conditionsmaybeeasilychecked.
F. KornerandC. Richter. On the efficient treatmentof the booleanquadraticprogrammingproblem.NumerischeMathematik, 40(1), 99–109,1982.
Abstract. Thedualof theBooleanquadraticprogrammingproblemis considered.Theoptimalvalueof the
objective functiongivesboundsfor thebranchandboundprocesswhichseemto bebetterthanthoseknown
from otherauthors.
L. V. Korsi andV. G. Sokolov. Synthesisof a systemof isotropic radiatorsby the methodof quadraticprogramming.RadioEngineeringandElectronic Physics, 17(3), 358–364,1972.
Abstract. An effective methodof synthesizinga systemof isotropicradiatorsis proposed.Theoptimizing
algorithmis basedon themethodof projectedgradient.Thesolutionsof theproblemsof mixedandphase
synthesisof an array of radiatorsare obtainedby this method.Examplesof numericalcomputationare
presentedillustratingdifferentapplicationsof theproposedmethod.
M. M. Kostreva. Generalizationof Murty’s direct algorithm to linear andconvex quadraticprogramming.Journalof OptimizationTheoryandApplications, 62(1), 63–76,1989.
Abstract. Murty’s algorithmfor the linearcomplementarityproblemis generalizedto solve theoptimality
conditionsfor linear andconvex quadraticprogrammingproblemswith both equalityand inequalitycon-
straints.An implementationis suggestedwhich providesbothefficiency andtight errorcontrol.Numerical
experimentsaswell asfield testsin variousapplicationsshow favorableresults.
O. I. Kostyukova andV. M. Raketskii. A methodof reducingthe suboptimalityestimatein
quadraticprogramming.DokladyAkademiiNaukBelarusi, 29(12),1072–1075,1985.
M. KotaniandI. Nemoto.Theinverseproblemin magnetopneumography—useof hypotheticaldistributionsandquadraticprogramming.NuovoCimentoD, 2(2), 594–607,1983.
Abstract. Theinverseproblemin themagneticmeasurementof thelungwasstudiedwith amethodusinghy-
potheticaldistributionsof magneticdust.Both unconstrainedandconstrainedminimizationsof anobjective
functionwereperformed.Simulationsandanalysisshowedtheefficacy of themethod.
P. F. Kough.Globalsolutionto theindefinitequadraticprogrammingproblem.Technicalreport,WashingtonUniv., St Louis,MO, USA, 1974.
Abstract. Theglobal solutionto the indefinitequadraticproblemis obtainedvia a generalizedBendercut
procedure.TheBendercut methoditeratively addscutsto a masterproblem.Maximising themasterprob-
lem is shown to be equivalent to maximisingseveral convex quadraticsubproblems.Eachcut createsa
N. I. M. GOULD & PH.L. TOINT 71
subproblem.Thisdecompositionprovidesfor thesolutionof themasterproblemby animplicit enumeration
algorithmcombinedwith Tui cuts.In orderto accelerateconvergenceonly a subsetof the subproblemsis
solved. The formal methodis thencombinedwith the acceleratedprocedureto insureconvergenceto the
globaloptimum.
P. F. Kough.Theindefinitequadraticprogrammingproblem.OperationsResearch, 27(3),516–533,1979.
Abstract. Developsseveral algorithmsthatobtaintheglobaloptimumto the indefinitequadraticprogram-
mingproblem.A generalizedBenderscutmethodis employed.Thesealgorithmsall possessε-finite conver-
gence.To obtainfinite convergencetheauthordevelopsexactcuts,whicharelocally preciserepresentations
of a reducedobjective. A finite algorithmis thenconstructed.Introductorycomputationalresultsarepre-
sented.
I. S.Kourtev andE. G. Friedman.Clockskew schedulingfor improvedreliability via quadraticprogramming.1999IEEE/ACM InternationalConferenceon ComputeridedDesignDi-gestof TechnicalPapers.IEEE,Piscataway, NJ, USA, pp.239–243,1999.Also appearedin the Proceedingsof the Twelfth Annual IEEE InternationalASIC/SOCConference,IEEE,Piscataway, NJ,USA, 1999,pp 210–215.
Abstract. Thispaperconsiderstheproblemof determininganoptimalclockskew schedulefor asynchronous
VLSI circuit. A novel formulationof clock skew schedulingasa constrainedquadraticprogramming(QP)
problemis introduced.Theconceptof a permissiblerange,or avalid interval, for theclockskew of eachlo-
caldatapathis key to this QPapproach.Fromareliability perspective, theidealclockschedulecorresponds
to eachclock skew within thecircuit beingat thecenterof the respective permissiblerange.However, this
ideal clock scheduleis nor practically implementablebecauseof limitations imposedby the connectivity
amongthe registerswithin thecircuit. To evaluatethe reliability, a quadraticcostfunction is introducedas
theEuclideandistancebetweentheidealscheduleanda givenpracticallyfeasibleclock schedule.This cost
function is the minimizationobjective of the describedalgorithmsfor the solutionof the previously men-
tionedquadraticprogram.Furthermore,thework describedheresubstantiallydiffersfrom previousresearch
in that it permitscompletecontrolover specificclock signaldelaysor skews within thecircuit. Specifically,
thealgorithmsdescribedherecanbeemployedto obtainresultswith explicitly specifiedtargetvaluesof im-
portantclock delays/skews with a circuit, suchasfor example,theclock delays/skews for I/O registers.An
additionalbenefitis apotentialreductionin clock periodof up to 10%.An efficient mathematicalalgorithm
is derived for thesolutionof theQPproblemwith O r3 run time complexity andO r2 storagecomplex-
ity, wherer is the numberof registersin the circuit. The algorithmis implementedasa C++ programand
demonstratedontheISCAS’89suiteof benchmarkcircuitsaswell asonanumberof industrialcircuits.The
work describedhereyieldsadditionalinsightsinto thecorrelationbetweencircuit structureandcircuit timing
by characterizingthedegreeto which specificsignalpathslimit theoverall performanceandreliability of a
circuit. This informationis directly applicableto logic andarchitecturalsynthesis.
L. B. KovacsandM. Kotel. Indefinitequadraticprogrammingby gradientprojectionmethodand its applicationto optimal control of a chemicalplant. IFAC symposiumon multi-variablecontrol systemsVDI/VDEFachgruppeReqelungstechnik,Dusseldorf, WestGer-many, 1968.
Abstract. As a subproblemof the optimal control of a chemicalplant an arbitrary (possibly indefinite)
quadraticfunction is to beminimizedsubjectto linearconstraints.Theproblemis solvedby meansof two
computerprogramsthefirst of whichgivesalocaloptimumstartingfrom any feasiblepoint.Thisprogramis
basedon thewell-known gradientprojectionmethodutilizing thespecialstructureof theobjective function.
The gradientprojectionmethodhasbeenchosenbecauseof its goodconvergenceproperties.The second
programgenerates’good’ startingpointsfor thefirst programandterminatesthealgorithmif certaincriteria
aresatisfiedmakingvery likey thefactthatthepresentbestsolutionis aglobalmaximumof theproblem.
K. Koyama,K. Minato, S. Eiho, and M. Kuwahara. A methodfor multivariablequadratic
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Abstract. An algorithmis constructedfor theexactsolutionof a quadraticprogrammingproblem,thevol-
ume(in binarysymbols)of computationbeinglimited by thelengthof theinput,i.e. it is shown thatquadratic
programmingbelongsto the classP of problemssolvableon deterministicTuring machines,in a time ex-
pressedasapolynomialof thebinaryinput length.
M. K. Kozlov, S. P. Tarasov, andL. G. Khachiyan.Polynomialresolutionof convex quadraticprogramming.ZhurnalVychislitel’noi Matematikii Matematicheskoi Fiziki, 20(5),1319–1323,1980.
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numberof binarysymbolsnecessaryto describetheinputdata.An exactsolutionalgorithmfor thequadratic
programmingproblemis developedfor which thenumberof elementalcalculationsis boundedby apolyno-
mial in theinput length.It is shown thatquadraticprogrammingproblemsbelongto classPproblems,which
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feasibleregiondefinedby constraintsAx b, β x α is presented.ThealgorithmusestheSUBmethodto
establisha startingfeasiblepoint andto constructthestartingbasismatrix. A feasibledirectionfor thek-th
iterationis chosenfrom a setof row-vectorsof theinverseof thecurrentbasismatrix.After a finite number
of iterationsan optimal solutionis found.A conjugacy propertyof feasibledirections,convergenceof the
algorithm,thequestionof cycling anda comparisonwith otheralgorithmsfrom thesameclassof methods
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minimize(1) thetotalcopperlosses,or (2) thevoltagedeviationsof theloadpoints,areobtainedby quadratic
programmingmethods.
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N. I. M. GOULD & PH.L. TOINT 73
Abstract. Thealgorithmpresentedis basedon a modificationof thecoupleddirectionmethod,but requires
somewhatfewercomputingoperationsandlessmemory. Thereis alsoaprocedurefor eliminatingcumulative
machineerror.
P. Lachout. Degenerationin Wolfe’s algorithm for quadraticprogramming. EkonomickoMatematicky Obzor, 24(4), 463–468,1988.
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grams.This papershows examplesin which, dueto degeneracy, the long form of Wolfe’s algorithmfails
to find the optimal solution.A new algorithmthat solvesevery problemof convex quadraticprogramsis
suggested.
Y. T. Lai, C. C. Kao, andW. C. Shie. A quadraticprogrammingmethodfor interconnectioncrosstalkminimization.In ‘ISCAS’99. Proceedingsof the1999IEEEInternationalSym-posiumonCircuitsandSystemsVLSI . IEEE,Piscataway, NJ,USA’, Vol. 6,pp.270–273,1999.
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cordingly it becomesimportantto considercrosstalkcausedby thecouplingcapacitancebetweenadjacent
wires in thelayoutdesignfor thefastandsafeVLSI circuits.Thecrosstalkis a functionof couplinglength
anddistance.Thecouplinglengthcanbereducedby segmentrearrangementtechnique.This paperpresents
a crosstalkminimizationtechniqueby adjustingthespacebetweenadjacentwires.An exampleis shown in
thispaperto demonstratetheeffectivenessof thetechnique.
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caninfluencetheprocessof optimalallocationof dataamongthenodesin adistributeddatabase.Thefactors
includecommunicationcosts,translationcosts,congestioncostsandstoragecosts.Beale’s methodis used
to solve theresultingquadraticprogram.Somenumericalexamplesarepresentedandthepotentialsof such
anapproachin thedesignandanalysisof distributeddatabasesarediscussed.
A. H. LandandG. Morton. An inverse-basismethodfor Beale’s quadraticprogrammingalgo-rithm. ManagementScience, 19(5), 510–516,1973.
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problemof maximisingaquadraticfunctionunderlinearconstraints.Themodificationdiscussedheremakes
it possibleto retain the ’inversebasis’ tableauwhich hasto be augmentedby additionalconstraintsto be
called’auxiliary.’ Thealgorithmhasbeensuccessfullytestedonacomputer.
A. H. LandandS.Powell. Fortrancodesfor mathematicalprogramming:linear, quadratic and
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Abstract. Two new methodto synthesizea feed array for the illumination of reflectorantennasarepre-
sented.Spillover radiationpatternor total spillover power can be consideredto obtainefficient feedsby
usingquadraticprogrammingor Tikhonov regularization,respectively. Apertureor surfacefield optimiza-
tion is consideredto obtaina prescribedaperturedistribution. Optimizationis ensuredin the globalsense,
andthespillover playstherole of aconstraint.
M. C. Lang. Designof nonlinearphaseFIR digital filters usingquadraticprogramming.1997IEEEInternationalConferenceonAcoustics,Speech,andSignalProcessingIEEECom-put.Soc.Press,LosAlamitos,CA,USA, 3, 2169–2172,1997.
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Abstract. This paperpresentstwo methodsfor thedesignof FIR filters with arbitrarymagnitudeandphase
responsesaccordingto a weightedmeansquarederrorcriterionwith constraintson theresultingmagnitude
andphaseerrors.This constrainedleastsquarecriterion allows for an arbitrary trade-off betweenpureL2
filtersandChebyshev filters.Theresultingnonlinearoptimizationproblemis eitherconvertedinto astandard
quadraticprogrammingproblem(method1) or exactly solvedby asequenceof quadraticprograms(method
2). Thequadraticprogrammingproblemscanbesolvedefficiently usingstandardsoftware.
D. J.Laughhunn.Quadraticbinaryprogrammingwith applicationsto capitalbudgetingprob-
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Abstract. Considersmixed-integer quadraticprogramsin which the objective function is quadraticin the
integer and in the continuousvariables,and the constraintsare linear in the variablesof both types.The
generalizedBenders’decompositionis asuitableapproachfor solvingsuchprograms.However, theprogram
doesnot becomemore tractableif this methodis used,sinceBenders’cuts are quadraticin the integer
variables.A new equivalent formulationthat rendersthe programtractableis developed,underwhich the
dual objective function is linear in the integer variablesandthe dual constraintset is independentof these
variables.Benders’cuts that arederived from the new formulationare linear in the integer variables,and
the original problemis decomposedinto a seriesof integer linear masterproblemsandstandardquadratic
subproblems.The new formulationdoesnot introducenew primary variablesor new constraintsinto the
computationalstepsof thedecompositionalgorithm.
R. Lazimy. Improved algorithm for mixed-integer quadraticprogramsand a computational
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minimizing an indefinitequadraticfunctionover a boundedpolyhedralconvex setwhich is not necessarily
givenexplicitly by a systemof linearinequalitiesand/orequalities.It is requiredthatfor this setthereexists
anefficient algorithmto verify whethera point is feasible,andto find a violatedconstraintif this point is
not feasible.The algorithmis baseduponthe fact that the problemof minimizing an indefinitequadratic
form over anellipsoidcanbeefficiently solvedby someavailable(polynomialandnonpolynomialtime) al-
gorithms.In particular, theD.C. (differenceof convex functions)algorithm(DCA) with restartingprocedure
recentlyintroducedby PhamDinh TaoandL.T. HoaiAn is appliedto globallysolvingthisproblem.DCA is
alsousedfor locally solvingthenonconvex quadraticprogram.It is restartedwith currentbestfeasiblepoints
in thebranch-and-boundscheme,andimprovedthemin its turn.ThecombinedDCA-ellipsoidalbranch-and-
boundalgorithmthenenhancestheconvergence:it reducesconsiderablytheupperboundandtherebya lot
of ellipsoidscanbeeliminatedfrom furtherconsideration.Severalnumericalexperimentsaregiven.
C. Y. C. Lee, D. R. Wiff, and V. G. Rodgers. Calculatinga relaxationspectrumfrom ex-perimentaldatavia quadraticprogrammingwith andwithout regularization.Journal ofMacromolecularScience—Physics, B19(2), 211–225,1981.
Abstract. Theregularizationquadraticprogrammingapproachto infer relaxationspectrafrom experimental
mechanicaldatawasstudied.It wasfoundthattheinferior of thesumof thesquareddifferencebetweenthe
inputdataandtheback-calculatedvaluesdid notyield asatisfactoryrelaxationspectrumastheregularization
weightingparameterα wasincreased.A modificationwasmadeto thefunctionto beminimizedsothatall
datapointswereequallyweighted.Quadraticprogrammingwasthenfoundto besufficient to infer a reliable
relaxationspectrumif theinputexperimentaldatahadahighdegreeof accuracy. Whentheexperimentaldata
werenotsufficiently accurate,regularizationover-regularizedthehigh-valueregion of thesolutionspectrum
N. I. M. GOULD & PH.L. TOINT 75
beforeit could improve the low-valueregion. Decreasingthenumberof sought-afterpointsin thesolution
spectrumcancompensatefor thenoiselevel in theinputexperimentaldataandwill allow inferringareliable
relaxationspectrumfrom theexperimentaldata.
L. J.Lefkoff andS.M. Gorelick. Designandcost-analysisof rapidaquiferrestorationsystems
usingflow simulationandquadraticprogramming.GroundWater, 24(6),777–790,1986.
L. L. Lei andJ. E. Epperson. A regional-analysisof vegetableversusrow crop production
usingquadraticprogramming.AmericanJournalof Agricultural Economics, 69(5),1097,
1987.
C. E. Lemke. A methodof solutionfor quadraticprograms.ManagementScience, 8, 442–453,
1962a.
C. E. Lemke. Orthogonality, duality, andquadratictype problemsin mathematicalprogram-
ming. RPI MathRep56, Departmentof Mathematics,RensselaerPolytechnicInstitute,
Troy, NY, USA, 1962b.
M. L. LenardandM. Minkoff. Randomlygeneratedtestproblemsfor positivedefinitequadraticprogramming.ACM Transactionson MathematicalSoftware, 10(1), 86–96,1984.
Abstract. A procedureis describedfor randomlygeneratingpositive definitequadraticprogrammingtest
problems.The testproblemsareconstructedin the form of linear least-squaresproblemssubjectto linear
constraints.Theprobabilitymeasurefor theproblemssogeneratedis invariantunderorthogonaltransforma-
tions.The procedureallows the userto specifythe sizeof the least-squaresproblem(numberof unknown
parameters,numberof observations,andnumberof constraints),therelative magnitudeof theresiduals,the
conditionnumberof the Hessianmatrix of the objective function, and the structureof the feasibleregion
(thenumberof equalityconstraintsandof inequalitieswhich will beactive at thefeasiblestartingpoint and
at the optimal solution).An exampleis given illustrating how theseproblemscanbe usedto evaluatethe
performanceof asoftwarepackage.
A. LentandY. Censor. Extensionsof Hildreth’srow-actionmethodfor quadraticprogramming.SIAMJournalon Control andOptimization, 18(4), 444–454,1980.
Abstract. An extendedversionof Hildreth’s iterative quadraticprogrammingalgorithmis presented,geo-
metrically interpreted,andproved to producea sequenceof iteratesthat (i) convergesto the solution,and
(ii) hasan importantintermediateoptimality property. This extendedHildreth algorithmis castinto a new
form which morepronouncedlybringsout its primal-dualnature.Theapplicationof thealgorithmmaybe
governedby an index sequencewhich is moregeneralthana cyclic sequence,namely, by analmostcyclic
control,andasequenceof relaxationparametersis incorporatedwithout ruiningconvergence.Thealgorithm
is a row-actionmethodwhich is particularlysuitablefor handlinglarge(or huge)andsparsesystems.
D. J.Leo andD. J. Inman. A quadraticprogrammingapproachto thedesignof active-passivevibrationisolationsystems.Journalof SoundandVibration, 220(5), 807–825,1999.
Abstract. A quadraticprogrammingalgorithmispresentedfor studyingthedesigntradeoffs of active-passive
vibration isolationsystems.The novelty of the techniqueis that the optimal control problemis posedasa
quadraticoptimizationwith linear constraints.The quadraticcost function representsthe meansquarere-
sponseof thepayloadaccelerationandisolatorstroke,andthelinearconstraintsrepresentasymptotictrack-
ing requirementsandpeakresponseconstraints.Posingtheproblemasa quadraticoptimizationguarantees
thataglobaloptimalsolutioncanbefoundif oneexists,andtheexistenceof anoptimalsolutionguarantees
that the vibration isolationsystemsatisfiesthe specifieddesignconstraints.The utility of the techniqueis
demonstratedon a comparisonof passive vibration isolationandactive-passive vibration isolationutilizing
relative displacementfeedback.
76 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
B. H. Leung.Decimationfiltersfor oversampledanalogto digital convertersbasedonquadraticprogramming.1990IEEE InternationalSymposiumon CircuitsandSystemsIEEE,NewYork, NY, USA, 2, 899–901,1990.
Abstract. A descriptionis givenof a quadratic-programming-basedmethodologyfor thedesignof decima-
tion filtersin CMOStechnology. Themethodologypresentedtacklesthefilter designproblemby formulating
a quadraticprogrammingproblemthat minimizesthe integral of the aliasednoisesubjectto the passband
andstopbandconstraints.Thismethodologyofferstheflexibility of incorporatingnonsinusoidalquantization
noisespectraldensityin thedesignprocess.Moreover thefilter responsecanbeoptimizedto meetarbitrary
antialiasrequirements.Becausethe projectedHessianmatrix of the objective function is positive definite,
thequadraticfunctionhasauniqueminimum.
B. H. Leung.Designmethodologyof decimationfiltersfor oversampledadcbasedonquadraticprogramming.IEEETransactionson CircuitsandSystems, 38(10),1121–1132,1991.
Abstract. A designmethodologyfor oversampledanalog-to-digitalconverterdecimationfilters is presented.
Themethodologytacklesthefinite-impulse-response(FIR) filter designproblemby formulatingaquadratic
programmingproblemthatminimizestheintegral of thealiasednoisesubjectto thepassbandandstopband
constraints.Theapproachoffersa designwhoseresponseis optimizedto meetarbitraryquantizationnoise
power spectraldensityandanti-aliasrequirements.Becausethe projectedHessianmatrix of the objective
function is positive definite,the quadraticfunctionhasa uniqueminimum.Themethodologyis appliedto
designfilters for differentrequirementsandtheperformanceis comparedto conventionalapproaches.
A. J. Levy. A fast quadraticprogrammingalgorithm for positive signal restoration. IEEETransactionson Acoustics,Speech andSignalProcessing, 31(6), 1337–1341,1983.
Abstract. Whena signalor pictureis processedby deconvolution, any additionala priori informationis of
prime interestsinceit canpotentiallyleadto an improvementin resultsandto superresolutionby reducing
ambiguity. The authorproposesa deconvolution methodthatassumes,first, that the unknown signalhasa
known finite extent, andsecond,that this signal is positive. The problemis statedin termsof a quadratic
programmingproblemis statedin termsof aquadraticprogrammingproblemwith positivity constraintsand
proposedis a new algorithmdeliveredfrom a conjugategradientmethod,especiallysuitedto this particular
situation,which leadsto a low costsolution.Experimentalresultson two-dimensionalsignals,emphasizing
relevantsuperresolutionarethenpresented.
J. T. Lewis, R. Murphy, andD. W. Tufts. Designof minimum noisedigital filters subjecttoinequalityconstraintsusingquadraticprogramming. IEEE Transactionson Acoustics,Speech andSignalProcessing, ASSP-24(5), 434–436,1976.
Abstract. A numericalmethodis presentedfor designingdigital filters.Themethodallows oneto minimise
themean-squareerroror noisepower over someintervals of frequency, while simultaneouslyconstraining
themaximumerror in otherintervals of frequency. Thus,for example,onecanminimisenoisepower from
a stopbandof frequencieswhile constrainingsignal fidelity in a passbandof frequenciesby limiting the
maximumpassbanddeviation.
W. Li. Error boundsfor piecewiseconvex quadraticprogramsandapplications.SIAMJournalon Control andOptimization, 33(5), 1510–1529,1995.
Abstract. In this paper, we establisha local error estimatefor feasiblesolutionsof a piecewise convex
quadraticprogramandaglobalerrorestimatefor feasiblesolutionsof aconvex piecewisequadraticprogram.
Theseerror estimatesprovide a unified approachfor deriving many old andnew error estimatesfor linear
programs,linear complementarityproblems,convex quadraticprograms,andaffine variationalinequality
problems.Theapproachrevealsthefact thateacherrorestimateis a consequenceof somereformulationof
theoriginalproblemasapiecewiseconvex quadraticprogramor aconvex piecewisequadraticprogram.In a
sense,evenRobinson’s resulton theupperLipschitzcontinuityof a polyhedralmappingcanbeconsidered
asa specialcaseof errorestimatesfor approximatesolutionsof a piecewise convex quadraticprogram.As
N. I. M. GOULD & PH.L. TOINT 77
anapplication,wederive new (global)errorestimatesfor iteratesof theproximalpointalgorithmfor solving
aconvex piecewisequadraticprogram.
W. Li. Differentiablepiecewisequadraticexactpenaltyfunctionsfor quadraticprogramswith
simpleboundconstraints.SIAMJournalonOptimization, 6(2), 299–315,1996.
W. Li andJ.J.deNijs. An implementationof QSPLINEmethodfor solvingconvex quadraticprogrammingproblemswith simpleboundconstraints.Technicalreport,DepartmentofMathematicsandStatistics,Old DominionUniversity, Norfolk, Virginia,U.S.A.,2000.
Abstract. A convex quadraticprogrammingproblemwith simpleboundconstraintscanbereformulatedas
anunconstrainedminimizationproblemwith a convex quadraticspline(i.e., a differentiableconvex piece-
wise quadraticfunction) as the objective function. This leadsto a new paradigmfor solving the original
quadraticprogrammingproblem,in which variousunconstrainedminimizationalgorithmscanbe usedto
find a stationarypoint of the convex quadraticspline.In this paper, we give an implementationof a reg-
ularizedNewton method(calledQSPLINEMethod)for finding a stationarypoint of the convex quadratic
spline.QSPLINEMethodcanalsobeconsideredasan implicit active-setmethodwith two novel features:
(i) a mixedprimal-dualapproachfor identifying active indices,and(ii) a line searchstrategy for a dynamic
balancebetweentheneedof minimizingtheoriginalobjective functionandthatof forcingtheiteratesto stay
in thefeasibleregion.Theimplementedversionof QSPLINEMethodusesa matrix up datingtechniquefor
computingNewton directionsanda correctionstrategy for robust reductionof the convex quadraticspline
whenline searchin aNewtondirectionfails.Wehavetestedthecodeonavariationof thetestproblemsused
by More andToraldooraldo(1989).For our generatedtestproblem,the Hessianof the objective function
couldbeann by n positivesemidefinitematrixwith rank2n 3 , andonethird of theLagrangemultiplierscor-
respondingto active constraintsat ageneratedoptimalsolutioncouldbezero.Thatis, our testproblemsare
degenerateandhave highly singularHessians.Thecodefindsveryaccuratenumericalsolutionsfor all 2800
randomlygeneratedtestproblems(with n 300and500)eventhoughall theseproblemsaredegenerateand
about60%of theseproblemshave infinitely many solutions.
W. Li andJ.Swetits.A Newton methodfor convex regression,datasmoothing,andquadratic
programmingwith boundedconstraints.SIAMJournal on Optimization, 3(3), 466–488,
1993.
W. Li andJ. Swetits. A new algorithmfor solvingstrictly convex quadraticprograms.SIAMJournalon Optimization, 7(3), 595–619,1997.
Abstract. We reformulateconvex quadraticprogramswith simple boundconstraintsand strictly convex
quadraticprogramsasproblemsof unconstrainedminimizationof convex quadraticsplines.Therefore,any
algorithmfor finding a minimizer of a convex quadraticsplinecanbe usedto solve thesequadraticpro-
grammingproblems.In this paper, we proposea Newton methodto find a minimizerof a convex quadratic
splinederived from the unconstrainedreformulationof a strictly convex quadraticprogrammingproblem.
TheNewton methodis a ”naturalmixture” of a descentmethodandanactive-setmethod.Moreover, it is an
iterative method,yet it terminatesin finite operations(in exactarithmetic).
X. Li andZ. Xuan. An interior-point QP algorithm for structuraloptimization. StructuralOptimization, 15(3–4),172–179,1998.
Abstract. A new algorithmfor convex quadraticprogramming(QP)is presented.Firstly, thesurrogateprob-
lem for QPis developed,andtheKarush-Kuhn-Tucker conditionsof the surrogateproblemhold if theun-
constrainedminimumof theobjective function doesnot satisfyany constraints.Then,Karmarkar’s (1984)
algorithmfor linear programming(LP) is introducedto solve the surrogatedual problem.In addition, the
caseof generalconstraintsis alsodiscussed,andsomeexamplesof optimumtrusssizingproblemsshow that
theproposedalgorithmis robustandefficient.
78 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
R.H. LiangandY. Y. Hsu.Hydroelectricgenerationschedulingusingaquadraticprogrammingbasedneuralnetwork. In ‘IPEC ’95. Proceedingsof theInternationalPowerEngineeringConference.NanyangTechnol.Univ, Singapore’,Vol. 2, pp.684–689,1995.
Abstract. A new approachbasedon neuralnetworks is proposedfor hydroelectricgenerationscheduling.
Thepurposeof hydroelectricgenerationschedulingis to determinetheoptimalamountsof generatedpowers
for the hydro units in the systemfor the next N (N 24 in this paper)hoursin the future.The proposed
approachis basicallya two-stagesolutionmethod.In thefirst stage,a quadraticprogrammingbasedneural
network is developedin orderto reacha preliminarygenerationschedulefor the hydro units.Sincesome
practicalconstraintsmaybeviolatedin thepreliminaryschedule,a heuristicrule basedsearchalgorithmis
developedin thesecondstageto reachafeasiblesuboptimalschedulewhichsatisfiesall practicalconstraints.
The proposedapproachis applied to hydroelectricgenerationschedulingof Taiwan power system.It is
concludedfrom theresultsthat theproposedapproachis very effective in reachingproperhydrogeneration
schedules.
N. Limic andA. Mikelic. ConstrainedKriging usingquadraticprogramming.Journal of the
InternationalAssociationfor MathematicalGeology, 16(4), 423–429,1984.
Y. Lin andJ. Pang. Iterative methodsfor large convex quadraticprograms:a survey. SIAM
Journalon Control andOptimization, 25, 383–411,1987.
G. Ling. An optimal neuronevolution algorithm for constrainedquadraticprogramminginimagerestoration.IEEETransactionsonSystems,ManandCybernetics,Part A (SystemsandHumans), 26(4), 513–518,1996.
Abstract. An optimalneuronevolutionalgorithmfor therestorationof linearlydistortedimagesis presented
in thispaper. Theproposedalgorithmis motivatedby thesymmetricpositive-definitequadraticprogramming
structureinherentin restoration.Theoreticalanalysisandexperimentalresultsshow that the algorithmnot
only significantlyincreasesthe convergencerateof processing,but alsoproducesgoodrestorationresults.
In addition,the algorithmprovidesa genuineparallelprocessingstructurewhich ensurescomputationally
feasiblespatialdomainimagerestoration.
C. T. Liu, G. C. Temes,andH. Samueli.FIR filter designfor sigma-deltaa/dconvertersusingquadraticprogramming.IEEE Pacific RimConferenceon Communications,ComputersandSignalProcessingIEEE,New York, NY, USA, 2, 760–763,1991a.
Abstract. Thedesignof overallfinite impulseresponse(FIR) filter for sigma-deltaanalog-to-digitalconvert-
ersis formulatedasa quadraticprogramwith thetotal outputnoiseastheobjective function,subjectto the
passbandandstopbandconstraints.The formulationis soflexible that it cansuit differentrequirementsfor
suppressingthequantizationnoiseaswell asrejectingtheout-of-bandspurioussignal.Bothone-stagesingle
rateandmultistagemultiratefilter structuresareconsidered.For thepurposeof suppressingthequantization
noise,themultistagemultirateFIR filter with combfiltersasprefiltershasmuchlower complexity with little
performancedegradationrelative to theone-stageoptimalfilter.
C. T. Liu, G. C. Temes,and H. Samueli. FIR filter designusing quadraticprogramming.1991IEEEInternationalSymposiumonCircuitsandSystemsIEEE,New York, NY, USA,1, 148–151,1991b.
Abstract. Theoptimumdesignof FIR (finite impulseresponse)filters is formulatedasaquadraticprogram.
An algorithmis describedfor theefficient useof computermemoryandfor guaranteedconvergenceto the
solutionof theoriginal problem.Examplesillustrating thedesignof decimationfilters for sigma-deltaA/D
convertersusingthenew techniquearealsopresented.Thequadraticprogrammingapproachis moreflexible
andoftenyieldsbetterresultsthanthepreviously reportedleast-squaresdesignapproaches.
N. I. M. GOULD & PH.L. TOINT 79
J. Q. Liu, T. T. Song,andD. Z. Du. On the necessaryandsufficient conditionof the local
optimal solutionof quadraticprogramming. ChineseAnnalsof MathematicsSeriesB,
3(5), 625–630,1982.
X. Liu, Y. Sun,andW. Wang.Stabilizingcontrolof robustnessfor systemswith maximumun-certainparameters-aquadraticprogrammingapproach.Control TheoryandApplications,16(5), 729–732,1999.
Abstract. Basedon a sufficient conditionfor stabilization,a way to stabilizelinear discrete-timesystems
with uncertainparametersis developed.By quadraticprogramming,theoptimalcontrollerof fixedorderwe
haveobtainedcanstabilizethesystemwith maximuminfinite normof theuncertainsystemparametervector.
Theconclusionis confirmedby simulation.
K. L. Lo and S. P. Zhu. A decoupledquadraticprogrammingapproachfor optimal powerdispatch.Electric PowerSystemsResearch, 22(1), 47–60,1991.
Abstract. Theauthorspresenta decoupledapproachfor optimalactive andreactive power dispatchfor the
economicoperationof an electricutility. The proposediterative schemereporteddecomposesthe optimal
loadflow probleminto a P-subproblemanda Q-subproblem,wherethecontrolvariablesaretheactive and
reactivepoweroutputsof generators.At eachiterationthesubproblemsaresolvedby quadraticprogramming
using the flexible toleranceoptimizationmethod(FTOM). In this approach,constraintsare linearizedby
usingsensitivity analysisandtheperturbationtechnique,in orderto obtaina wider near-feasibleregion for
this quadraticprogrammingmethod.This approachhasa goodcomputationspeed.The resultsof two test
systemsandthreecasesarepresentedandcomparedwith theresultsof othermethods.
W. LochrieandD. Isaacs.Theapplicationof quadraticprogrammingin adaptive filtering. In‘Proceedingsof the8th annualAllerton conferenceon circuit andsystemtheory. IEEE,New York, NY, USA’, pp.101–110,1970.
Abstract. The problemof designingan adaptive filter for a nonlineardynamicsystemis formulatedasa
constrainedestimationproblemin which themeasurementnoisevariancesareestimated.An adaptive filter
mechanizationwith animbeddedconstrainedestimationalgorithmis thendeveloped.A quadraticprogram-
mingalgorithmis incorporatedin themechanizationdevelopedfor learningthesevariancevalues.
E. Loehman,D. Pingry, andA. B. Whinston.Quadraticprogrammingandestimationproblems.Technicalreport,PurdueUniv , Lafayette,IN, USA, 1970.
Abstract. Thepaperpresentsvariousproblemsrelatingto multivariateregressionin a unifiedfashionusing
thequadraticprogrammingtableau.Thetopicsconsideredaremulticollinearityandthegeneralizedinverse,
picking the bestregressionmodel,andconstrainedestimatesandhypothesistestingfor regressioncoeffi-
cients.
H. Lofgren. LiberalisingEgypt’s agriculture:a quadraticprogramminganalysis. Journal of
African Economies, 2(2), 238–261,1993.
F. A. LootsmaandJ.D. Pearson.An indefinite-quadratic-programmingmodelfor acontinuous-productionproblem.PhilipsResearch Reports, 25(4), 244–254,1970.
Abstract. A model is presentedfor a problemof schedulingthe lengthsof N productionperiodson one
machine,whichwill manufacture1 products.Theproblemis to chooseproductionperiodssoasto minimize
thesumof inventorycostsfor theq productsin thepresenceof givendemands.Mathematically, theproblem
is one of minimizing an indefinite quadraticfunction subjectto linear constraints.A numericalexample
concludesthepaper.
S.L. Louwes,J.C. G. Boot,andS.Wage.A quadraticprogrammingapproachto theproblem
of theoptimaluseof milk in thenetherlands.Journal of Farm Economics, 45, 309–317,
1963.
80 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Z. Lu andZ. Wei. Decompositionmethodfor quadraticprogrammingproblemwith box con-straints.MathematicaNumericaSinica, 21(4), 475–482,1999.
Abstract. A decompositionmethodfor solving quadraticprogramming(QP) with box constraintsis pre-
sentedin this paper. It is similar to the iterative methodfor solving linear systemof equations.The main
ideasof thealgorithmto split theHessianmatrix Q of theQPprobleminto thesumof two matricesN and
H suchthatQ=N+H and(N-H) is symmetricpositive definitematrix ((N,H) is calleda regular splitting of
Q). A new quadraticprogrammingproblemwith Hessianmatrix N to replacethe original Q is easierto
solve thantheoriginal problemin eachiteration.Theconvergenceof thealgorithmis provedundercertain
assumptions,andthesequencegeneratedby thealgorithmconvergesto optimalsolutionandhasalinearrate
of R-convergenceif thematrixQ is positive definite,or astationarypoint for thegeneralindefinitematrixQ,
andthenumericalresultsarealsogiven.
A. Lucia andJ. Xu. ChemicalprocessoptimizationusingNewton-like methods.ComputersandChemicalEngineering, 14(2), 119–138,1990.
Abstract. Variousinterrelatedissuesthat effect the reliability andefficiency of Newton-like methodsfor
chemicalprocessoptimizationarestudied.An algorithmfor solving large, sparsequadraticprogramming
(QP)problemsthat is basedon anactive setstrategy anda symmetric,indefinitefactorizationis presented.
TheQPalgorithmis fastandreliable.A simpleasymmetrictrust region methodis proposedfor improving
the reliability of successive QP methods.Ill-defined QP subproblemsareavoidedby adjustingthe sizeof
thetrustregion in anautomaticway. Finally, it is shown thatreliableinitial valuesof theunknown variables
andmultiplierscanbegeneratedautomaticallyusinggenericprobleminformation,short-cuttechniquesand
simulationtools.Many relevantnumericalresultsandillustrationsarepresented.
A. Lucia, J. Xu, and G. C. D’Couto. Sparsequadraticprogrammingin chemicalprocessoptimization.Annalsof OperationsResearch, 42(1–4),55–83,1993.
Abstract. The quadraticprogrammingaspectsof a full spacesuccessive quadraticprogramming(SQP)
methodaredescribed.In particular, fill-in, matrix factor andactive set updating,numericalstability, and
indefinitenessof the Hessianmatrix arediscussedin conjunctionwith a sparsemodificationof Bunchand
Parlett factorizationof symmetricindefinite(Kuhn-Tucker) matricesof the typeoften encounteredin opti-
mization.A new pivoting strategy, calledconstrainedpivoting, is proposedto reducefill-in andcompared
with complete,partial and thresholdpivoting. It is shown that constrainedpivoting often significantly re-
ducesfill-in andthusthe iterative computationalburdensassociatedwith the factorizationandsolutionof
Kuhn-Tucker conditionswithin theQPsubproblem.Several chemicalprocessoptimizationproblems,with
smallandlargedegreesof freedom,areusedastestproblems.Theseincludeminimumwork calculationsfor
multistageisothermalcompression,minimumareatargetingfor heatexchangernetworksanddistillationop-
timizationinvolving someazeotropicandextractive distillations.Numericalresultsshow uniformly thatboth
theproposedQPandSQPalgorithms,particularlythefull spaceNewton method,arereliableandefficient.
No failureswereexperiencedateitherlevel.
F. T. Luk andM. Pagano.Quadraticprogrammingwith M-matrices. Linear Algebra and ItsApplications, 33, 15–40,1980.
Abstract. Studiestheproblemof quadraticprogrammingwith M-matrices.Theauthorsdescribeaneffective
algorithmfor thecasewherethevariablesaresubjectto alower-boundconstraint,andananalogousalgorithm
for thecasewherethevariablesaresubjectto lower-and-upper-boundconstraints.Theauthorsdemonstrate
thespecialmonotonebehavior of the iterateandgradientvectors.Theresulton thegradientvectoris new.
It leadsusto considera simpleupdatingprocedurewhich preservesthemonotonicityof bothvectors.The
procedureusesthefactthatanM-matrix hasanonnegative inverse.Two new algorithmsarethenconstructed
by incorporatingthisupdatingprocedureinto thetwo givenalgorithms.Theauthorsgivenumericalexamples
whichshow thatthenew methodscanbemoreefficient thantheoriginalones.
L. Luksan.Dualmethodfor solvingaspecialproblemof quadraticprogrammingasasubprob-lem at linearly constrainednonlinearminimaxapproximation.Kybernetika, 20(6), 445–457,1984.
N. I. M. GOULD & PH.L. TOINT 81
Abstract. Describesthedualmethodfor solvinga specialproblemof quadraticprogrammingasa subprob-
lem at linearly constrainednonlinearminimax approximation.Completealgorithmof the dual methodis
presentedandits convergenceafterafinite numberof stepsis proved.
L. Luksan. Dual methodfor solving a specialproblemof quadraticprogrammingasa sub-problemto nonlinearminimax approximation. AplikaceMatematiky, 31(5), 379–395,1986.
Abstract. The dual methodfor solving a specialproblemof quadraticprogrammingas a subproblemto
nonlinearminimax approximationis described.Two casesareanalyzedin detail; they differ in the linear
dependenceof thegradientsof theactive functions.Thecompletealgorithmof thedualmethodis presented
andits finite stepconvergenceis proved.
A. D. Lutzenko andA. V. Martynov. Minimax solutionsof problemsin linear andquadraticprogramming.SovietJournalof ComputerandSystemsSciences, 2, 22–27,1968.
Abstract. A methodisgivenfor obtainingtheoptimumsolutionof monimaxproblemsin linearandquadratic
programming.It is shown thatfor minimaxproblemsin linearprogrammingthesaddle-pointconditionis not
satisfied.
C. Y. MaaandM. A. Shanblatt.Linearandquadraticprogrammingneuralnetwork analysis.IEEE Transactionson Neural Networks, 3(4), 580–594,1992.
Abstract. Neuralnetworksfor linearandquadraticprogrammingareanalyzed.Thenetwork proposedby M.
P. KennedyandL. O. Chua(IEEE Trans.Circuits Syst.,vol. 35, pp.554–562,May 1988)is justified from
the viewpoint of optimizationtheoryandthe techniqueis extendedto solve optimizationproblems,such
asthe least-squaresproblem.For quadraticprogramming,the network convergeseitherto an equilibrium
or to anexact solution,dependingon whetherthe problemhasconstraintsor not. The resultsalsosuggest
ananalyticalapproachto solve the linearsystemBx b without calculatingthematrix inverse.Theresults
aredirectly applicableto optimizationproblemswith C2 convex objective functionsandlinear constraints.
Thedynamicsandapplicabilityof thenetworks aredemonstratedby simulation.Thedistancebetweenthe
equilibriaof thenetworksandtheproblemsolutionscanbecontrolledby theappropriatechoiceof anetwork
parameter.
J. H. Maddocks. Restrictedquadratic-forms,inertia theorems,and the Schurcomplement.
LinearAlgebra andIts Applications, 108, 1–36,1988.
K. MadsenandH. Schjær-Jacobsen.Minimax optimizationusingquadraticprogramming.In‘Proceedingsof theIEEEInternationalConferenceonCircuitsandComputersICCC80.IEEE,New York, NY, USA’, pp.1135–1137,1980.
Abstract. A new methodfor solving thenon-linearminimaxproblemis presented.The problemis solved
througha sequenceof quadraticprogramsandno line searchis used.Themethodmaybeusefulin sparse
problemsandanalgorithmfor this caseis briefly outlined.
K. Madsen,H. B. Nielsen,andM. C. Pinar. A finite continuationalgorithmfor boundcon-strainedquadraticprogramming.SIAMJournalon Optimization, 9(1), 62–83,1998.
Abstract. Thedualof thestrictly convex quadraticprogrammingproblemwith unit boundsis posedasa lin-
ear * 1 minimizationproblemwith quadraticterms.A smoothapproximationto thelinear * 1 functionis used
to obtainaparametricfamily of piecewise-quadraticapproximationproblems.Theuniquepathgeneratedby
theminimizersof theseproblemsyieldsthesolutionto theoriginal problemfor finite valuesof theapprox-
imationparameter. Thus,a finite continuationalgorithmis designed.Theresultsof extensive computational
experimentsarereported.
K. Madsen,H. B. Nielsen,andM. C. Pinar. Boundconstrainedquadraticprogrammingviapiecewisequadraticfunctions.MathematicalProgramming, 85(1), 135–156,1999.
82 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. We considerthestrictly convex quadraticprogrammingproblemwith boundedvariables.A dual
problemis derivedusingLagrangeduality. Thedualproblemis theminimizationof anunconstrained,piece-
wisequadraticfunction.It involvesa lower boundof λ 1 , thesmallesteigenvalueof a symmetric,positive
definitematrix,andis solvedby Newton iterationwith line search.Thepaperdescribesthealgorithmandits
implementationincludingestimationof λ 1 , how to geta goodstartingpoint for theiteration,andup- and
downdatingof Cholesky factorization.Resultsof extensive testingandcomparisonwith othermethodsfor
constrainedQParegiven.
R.A. Maggio.An explorationof theconvex quadratic-programmingblendingmodel.Modeling
andSimulation:RobotsandGeneral Modeling, 18(5), 1617–1621,1987.
M. MahfoufandD. A. Linkens.Constrainedmultivariablegeneralizedpredictivecontrol(GPC)for anaesthesia:the quadratic-programmingapproach(QP). International Journal ofControl, 67(4), 507–527,1997.
Abstract. Thispaperconsiderstheextensionof thestandardGPCalgorithmto includeinputrate,magnitude
andoutputconstraintsusingthequadraticprogramming(QP)approachonaderivednonlinearmultivariable
anaesthesiamodel comprisingsimultaneouscontrol of musclerelaxation(paralysis)andunconsciousness
(in termsof bloodpressuremeasurements).Simulationresults,whicharepresented,analysedanddiscussed,
demonstratethesuperiorityof theextendedversionin thedeterministicandstochasticcasesevenwhenlow
outputpredictionhorizonsarechosen,andalsothegreatflexibility with respectto choosingthelimits on the
manipulatedaswell asthe outputvariables.The studyalsorevealsthat whenheavy externaldisturbances
occur, the algorithm,which combinesinput andoutputconstraints,performsbetterthaneitherthe uncon-
strainedoneor theonethat includesonly input constraints.Underextremeconditions,thesamealgorithm
reducesto an algorithm with only input constraintswhen the phenomenonof constraintsincompatibility
occurs.
G. Maier. A quadraticprogrammingapproachfor certainclassesof nonlinearstructuralprob-
lems.Meccanica, 3, 121–130,1968.
A. Majthay. Optimality conditionsfor quadraticprogramming.MathematicalProgramming,
1(3), 359–365,1971.
A. Majthay, A. B. Whinston,and J. Coffman. Local optimisationfor nonconvex quadraticprogramming.NavalResearch LogisticsQuarterly, 21(3), 465–490,1974.
Abstract. Presentsan algorithmfor determininga local minimum to the following generalquadraticpro-
grammingproblemMinimize q x> pT x 1 2xT Qxsubjectto Ax b x 0,whereQ is ann by n symmetric
matrixA is anmby n matrix p is ann by 1 vectorb is anmby 1 vector. No additionalrestrictionsareplaced
onthematrixQ. And so,theabove problemis theminimizationof apossiblynonconvex quadraticobjective
functionsubjectto asetof linearinequalityconstraints.
K. Malanowski. Onapplicationof aquadraticprogrammingprocedureto optimalcontrolprob-lemsin systemsdescribedby parabolicequations.Control andCybernetics, 1(1–2),43–56,1972.
Abstract. An optimal control problem of a systemdescribedby linear partial differential equationof
parabolictypeis considered.Bothboundaryanddistributedcontrolsareinvestigated.Theperformanceindex
is aquadraticone.It is shown thatthisproblemcanbereducedto somequadraticprogrammingproblemin a
Hilbert space.An iterative procedureof solvingthis problemis proposed.Onedimensionalheatconduction
systemis consideredasanexample.Somenumericalresultsobtainedusinga digital computeraregiven.
K. Malanowski. A quadraticprogrammingmethodin Hilbert spaceandits applicationto opti-mal controlof systemsdescribedby parabolicequations.5th IFIP Conferenceon Opti-mizationTechniques(Abstractsonly received).Univ Rome, Rome, Italy, p. 37,1973.
N. I. M. GOULD & PH.L. TOINT 83
Abstract. Abstractonly givensubstantiallyasfollows. Theproblemof finding theminimumof a quadratic
functionalJ y onaclosed,boundedandconvex setGammain aHilbert spaceis considered.ThesetGamma
maynot begiven in anexplicit form. An iterative procedureis proposed,basedon successive minimization
of the functionalJ y on somesubsetsΓi of the original setΓ . Threemethodsof constructingthesubsets
Γi areproposed.The useof the above procedurefor solving optimal control problemsfor linear systems
describedby parabolicequationswith quadraticperformanceindex is proposed.Somenumericalexamples
of optimalboundarycontrolof onedimensionalheat-transferequationarepresented.On thebasisof these
examplesthethreemethodsof constructingthesubsetsΓi arecompared.Thespeedof convergenceandthe
simplicity of theobtainedcontrolaretakeninto accountin thiscomparison.
K. Man’chakandY. A. Tomashevski. Solvingquadraticprogrammingproblemsonananaloguecomputer. ArchiwumAutomatykii Telemechanika, 16(2), 185–204,1971.
Abstract. In thepapera questionof solving quadraticprogrammingproblemsis exemplifiedby a specific
taskof waterresourceallocation.Gradientmethodsof quadraticprogrammingareexamined.Theprinciple
of theapplicationof ananaloguecomputerfor solvingsuchproblemsis discussed.Resultsof simulationand
experimentalinvestigationsaregiven.Thelatteraredividedinto two series.Thefirst oneis concernedwith
thestaticalcase,thesecondonewith thedynamicalcase.
D. W. Manchala,A. B. Palazzolo,A. F. Kascak,G. T. Montague,andG. V. Brown. Constrainedquadraticprogramming,activecontrolof rotatingmassimbalance.Journalof SoundandVibration, 205(5), 561–580,1997.
Abstract. Jetenginesmayexperienceseverevibrationdueto thesuddenimbalancecausedby bladefailure.
Thecurrentresearchinvestigatesemploymentof piezoelectricactuatorsto suppressthis usingactive vibra-
tion control.This requiresidentificationof thesourceof thevibrationsvia anexpertsystem,determination
of therequiredphaseanglesandamplitudesfor thecorrectionforces,andapplicationof thedesiredcontrol
signalsto thepiezoelectricactuators.Correctionforcesmayexceedthephysicallimitationsof theactuators;
henceresultsof ”constrainedforce” quadraticprogramming,leastsquaresandmulti-point correctionalgo-
rithmswill becompared.It is demonstratedthatsimply scalingdown theleastsquarespredictedcorrection
forcesto satisfythe actuatorsaturationconstraintsdoesnot necessarilyyield optimal reductionsin vibra-
tion. In thispapertestresultsareshown for suddenimbalance,andthecomputationaltimerequirementsand
balancingeffectivenessfor thevariousapproachesarecompared.
O. L. Mangasarian.Duality in quadraticprogramming.In ‘NonlinearProgramming’,chapter
8-2,pp.123–126.McGraw-Hill, New York,USA,1969.ReprintedasClassicsin Applied
Mathematics10, SIAM, Philadelphia,USA, 1994.
O. L. Mangasarian. Locally uniquesolutionsof quadraticprograms,linear and non-linear
complementarityproblems.MathematicalProgramming, 19(2), 200–212,1980.
O. L. Mangasarian.Sparsity-preservingSOR algorithmsfor separablequadraticand linear
programming.ComputersandOperationsResearch, 11, 105–112,1984.
O. L. MangasarianandH. Stone.Two-personnonzero-sumgamesandquadraticprogramming.
Journalof MathematicalAnalysisandApplications, 9, 348–355,1963.
H. B. Mann. Quadraticformswith linear constraints.TheAmericanMathematicalMonthly,
50, 430–433,1943.
B. D. Manos.Leastsquarefit andquadraticprogramming.Diastase, 2, 3–14,1986. In Greek.
B. D. ManosandG. I. Kitsopanidis. A quadraticprogrammingmodelfor farmplanningof aregion in centralmacedonia.Interfaces, 16(4), 2–12,1986.
84 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. Quadraticprogrammingmodelsareusedin farm planningbecauserisk anduncertaintyarein-
volvedin thetechnicalandeconomiccoefficientsusedandthequantitiesandpricesof resources.A special
quadraticprogrammingmodel(the E-V model)wasusedto plan a Greekfarm region, the formerLake of
Giannitsa.The resultingplan is preferredby farmersto thoseresultingfrom the linear andmixed-integer
programmingmodelsandto thepreviously usedplanbecauseit includescropsexpectedto give thehighest
minimumtotalgrossmargin with thesametotalfixedcosts.Thefarmerswantplansthatachieve notonly the
highestbut alsothemoststableeconomicresults.
M. A. Marino andH. A. Loaiciga. Multireservoir operationplanningvia quadraticprogram-
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1985.
H. M. Markowitz. The optimizationof a quadraticfunction subjectto constraints. Naval
Research LogisticsQuarterly, 3, 111–133,1956.
I. MarosandC.Meszaros.A repositoryof convex quadraticprogrammingproblems.Optimiza-tion MethodsandSoftware, 11-12, 671–681,1999.
Abstract. The introduction of a standard set of linear programming problems, to be found in
NETLIB/LP/DATA, hadanimportantimpactonmeasuring,comparingandreportingtheperformanceof LP
solversUntil recentlytheefficiency of new algorithmicdevelopmentshasbeenmeasuredusingthis impor-
tantreferencesetPresently, wearewitnessinganevergrowing interestin theareaof quadraticprogramming
Theresearchcommunityis somewhat troubledby the lack of a standardformat for defininga QPproblem
andalsoby the lack of a standardreferencesetof problemsfor purposessimilar to thatof LP In thepaper
weproposeastandardformatandannouncetheavailability of a testsetof 138collectedQPproblems.
J.E. Marowitz andK. A. Fegley. Systemdesignusingquadraticprogramming.IEEE Transac-tionson AutomaticControl, 16(3), 241–247,1971.
Abstract. Quadraticprogrammingis appliedto theconstraineddesignandcompensationof stochasticand
deterministicsystems.The developedconstraintscontrol systemstability, plant compensatorrealizability,
systemrise time, andbandwidth.The techniqueseemsattractive to caseswherethe designinformationis
givenonly in numericalform.
A. D. Martin. Mathematicalprogrammingof portfolio selections. ManagementScience,1(2), 152–166,1955.
Abstract. PresentsMarkowitz modelasa mathematicalprogramanda solutionprocedurefor thequadratic
program
B. Martos. Quadraticprogrammingwith a quasiconvex objective function. OperationsRe-search, 19(1), 87–97,1971.
Abstract. Givesbothnecessaryandsufficient conditionsfor a quadraticfunction to bequasiconvex in the
nonnegative orthant.Methodsof pseudoconvex programming(suchasthoseof FrankandWolfe) cansolve
linearlyconstrainedquadraticprogrammingproblemswith suchanobjective function.
Y. K. L. MatitskasandG.S.Palubetskis.Graphcuttingand0–1quadraticprogramming.SovietJournalof ComputerandSystemsSciences, 25(1), 100–107,1987.
Abstract. A problemof 0–1 programmingwith a quadraticobjective function andtwo constraintsin the
form of inequalities,whichcanbeinterpretedasaproblemof cuttingagraphinto two parts,is examined.An
effective algorithmfor cuttinga treeandabranch-and-boundalgorithmfor anarbitrarygrapharepresented.
Theresultsof a computationalexperimentaregiven.The feasibility of usinga reduciblelower estimateof
theoptimumfor thepurposeof testingheuristicalgorithmsis indicated.
N. I. M. GOULD & PH.L. TOINT 85
R. McBride andJ. Yormark. Finding all solutionsfor a classof parametricquadraticinteger
programmingproblems.ManagementScience, 26, 784–795,1980.
B. A. McCarl andT. Tice. Shouldquadratic-programmingproblemsbeapproximated.Ameri-
canJournalof Agricultural Economics, 64(3), 585–589,1982.
B. A. McCarl, H. Moskowitz, andH. Furtan. Quadraticprogrammingapplications.Omega,5(1), 43–55,1977.
Abstract. This paperreviews andextendssomeof the methodologicalareaswhereQP is applicable,dis-
cussingandillustratingthecharacteristicsandaspectsof theaccompanying solutions.Someof themethod-
ologicalareasamenableincluderegressionanalysis,decisionanalysis,andquadraticapproximationsto gen-
erally complex functions.In the functionalmanagementareas,QP is applicableto problemsin economics
suchasdemand-supplyresponseandenterpriseselection.In finance,it is usedin portfolio analysis;in agri-
culture,in cropselection.
M. P. Mckenna,J. P. Mesirov, and S. A. Zenios. Data-parallelquadraticprogrammingonbox-constrainedproblems.SIAMJournalon Optimization, 5(3), 570–589,1995.
Abstract. We develop designsfor thedataparallelsolutionof quadraticprogrammingproblemssubjectto
box constraints.In particular, we considertheclassof algorithmsthat iteratebetweenprojectionstepsthat
identify candidateactive setsandconjugategradientstepsthat explore the working space.Using the al-
gorithm of More andToraldo[ReportMCS-p77–0589, ArgonneNationalLaboratory, Illinois, 1989] asa
specificinstanceof thisclassof algorithmsweshow how its componentscanbeimplementedefficiently ona
data-parallelSIMD computerarchitecture.Alternative designsaredevelopedfor botharbitrary, unstructured
Hessianmatricesandfor structuredproblems.Implementationsarecarriedout on a ConnectionMachine
CM-2. They areshown to bevery efficient, achieving a peakcomputingrateover 2 Chops.Problemswith
severalhundredthousandvariablesaresolvedwithin oneminuteof solutiontimeonthe8K CM-2.Extremely
largetestproblems,with up to 2.89million variables,arealsosolvedefficiently. Thedataparallelimplemen-
tationoutperformsa benchmarkimplementationof interior point algorithmson an IBM 3090–6009vector
supercomputerandasuccessive overrelaxationalgorithmonanIntel iPSC/860hypercube.
P. Mcleanmeyinsse.Interregionalflowsof corn—aquadratic-programmingapproach.In ‘Mod-
eling andSimulation’,Vol. 19,pp.419–424,1988.
W. G. Medlin and J. F. Kaiser. Bandpassdigital differentiatordesignusing quadraticpro-gramming.ICASSP91.1991InternationalConferenceon Acoustics,Speech andSignalProcessingIEEE,New York, NY, USA, 3, 1977–1980,1991.
Abstract. A novel techniqueutilizing quadraticprogrammingfor thedesignof bandpassFIR (finite impulse
response)digital differentiatorsof arbitraryorderis presented.Thenew differentiatorshave linearphaseand
aremaximally accurateat the centerof the differentiationband.Their designis basedon a minimization
procedurefor the integratedsquareerror of the frequency responseover designatedapproximationbands.
Theclosed-formsolutionfor the filter coefficients is obtainedby the methodof Lagrangemultipliers.The
inclusionof stopbandin the designprocessis alsodiscussed.This techniquehasbeensuccessfullyused
by the authorsfor thedesignof optimal low passdifferentiators.The new differentiatorsareimportantfor
applicationswherethefirst-, second-,or higher-orderderivative of a digital signalis requiredto beaccurate
atmidrangefrequencies.
K. Meer. On thecomplexity of quadraticprogrammingin realnumbermodelsof computation.TheoreticalComputerScience, 133(1), 85–94,1994.
Abstract. The complexity of linearly constrained(nonconvex) quadraticprogrammingis analyzedwithin
the framework of real numbermodels,namelythe oneof L. Blum, M. Shub,andS. Smale(1989)andits
modificationrecently introducedby Koiran (1993) (”weak BSS-model”).In particularwe show that this
problemis not NP-completein theKoiransetting.Applicationsto the(full) BSS-modelarediscussed.
86 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
N. MegiddoandA. Tamir. Linear time algorithmsfor someseparablequadraticprogrammingproblems.OperationsResearch Letters, 13(4), 203–211,1993.
Abstract. A large classof separablequadraticprogrammingproblemsis presented.The problemsin the
classcanbesolvedin lineartime.Theclassincludestheseparableconvex quadratictransportationproblem
with afixednumberof sourcesandseparableconvex quadraticprogrammingwith nonnegativity constraints
andafixednumberof linearequalityconstraints.
S.MehrotraandJ.Sun.An algorithmfor convex quadraticprogrammingthatrequiresOn3 ? 5L
arithmeticoperations.Mathematicsof OperationsResearch, 15(2), 342–363,1990.
Abstract. A new interior point methodfor minimizing a convex quadraticfunction over a polytopeis de-
veloped.The authorsshow that the methodrequiresO n3 ( 5L arithmeticoperations.In the algorithmthey
constructa sequencePz0 Pz1 . . . Pzk , . . . of nestedconvex setsthatshrink towardsthesetof optimalsolu-
tion(s).During iterationk they take a partial Newton stepto move from anapproximateanalyticcenterof
Pzk @ 1 to an approximateanalyticcenterof Pzk . A systemof linear equationsis solved at eachiteration to
find the stepdirection.The solutionthat is availableafter O mL iterationscanbe convertedto an opti-
mal solution.Theanalysisindicatesthat inexactsolutionsto thelinearsystemof equationscouldbeusedin
implementingthis algorithm.
A. MelmanandR. Polyak. TheNewton modifiedbarriermethodfor QPproblems.AnnalsofOperationsResearch, 62, 465–519,1996.
Abstract. Themodifiedbarrierfunctions(MBF) haveelementsof bothclassicalLagrangians(CL) andclas-
sical barrierfunctions(CBF). The MBF methodsfind anunconstrainedminimizer of somesmoothbarrier
functionin primalspaceandthenupdatetheLagrangemultipliers,while thebarrierparametereitherremains
fixedor canbeupdatedateachstep.Thenumericalrealizationof theMBF methodleadsto theNewtonMBF
method,wheretheprimalminimizeris foundby usingNewton‘smethod.Thisminimizeris thenusedto up-
datetheLagrangemultipliers.In this paper, theauthorsexaminetheNewton MBF methodfor thequadratic
programming(QP)problem.It is shown thatunderstandardsecond-orderoptimality conditions,thereis a
ball aroundtheprimalsolutionandacut conein thedualspacesuchthatfor asetof Lagrangemultipliersin
this cut cone,themethodconvergesquadraticallyto theprimal minimizer from any point in theaforemen-
tionedball, andcontinuesto do soaftereachLagrangemultiplier update.TheLagrangemultipliers remain
within thecut coneandconvergelinearly to their optimalvalues.Any point in this ball will becalleda ”hot
start”. Startingat such”hot start”, at mostO loglogε ' 1 Newton stepsaresufficient to performtheprimal
minimizationwhich is necessaryfor theLagrangemultiplier update.Here,ε 3 0 is thedesiredaccuracy. Be-
causeof the linearconvergenceof theLagrangemultipliers, this meansthatonly O logε ' 1 O loglogε ' 1 Newtonstepsarerequiredto reachanε-approximationto thesolutionfrom any ”hot start”. In orderto reach
the”hot start”,onehasto performO mlogC Newtonsteps,wherem characterizesthesizeof theproblem
andC 3 0 is the conditionnumberof the QP problem.This conditionnumberis characterizedexplicitly
in termsof key parametersof the QP problem,which in turn dependon the input dataandthe sizeof the
problem.
C. Y. Meng,S.Frimpong,andM. J.Zuo. A quadraticprogrammingmodelfor blastscheduling.Journalof Universityof ScienceandTechnologyBeijing, 6(3), 165–167,1999.
Abstract. A quadraticprogrammingmodelis establishedto choosetheblocksto beblastedin a givenpe-
riod. The lengthof this perioddependson theproductionplanningrequirements.During the given period,
theblocks’ parametersareavailablefrom thegeologicaldatabaseof themine.Theobjective is to minimize
the deviation of the averageore gradeof blastedblocksfrom the standardore graderequiredby the mill.
Transportationability constraint,productionquantitydemandconstraint,minimumsafetybenchconstraint,
block sizeconstraintandblock, benchprecedenceconstraintsareconsideredin forming the programming
model.This modelhasmorepracticalobjective functionandreasonableconstraintscomparedwith theex-
istingmodelfor thiskind of problems.
H. M. Merrill. Failurediagnosisusingquadraticprogramming.IEEE Transactionson Relia-bility, R-22(4), 207–213,1973.
N. I. M. GOULD & PH.L. TOINT 87
Abstract. This paperdiscussestheproblemof determiningwhich of a largesetof possiblebut improbable
malfunctionsgave riseto agivensetof measurements.Theclassesof systemsunderconsiderationgenerally
leadto underdeterminedsetsof equations.Threemethodsof formulatingandsolvingthis classof problems
arepresented:1) thepseudoinversemethod:this leadsto aneasily-solved computationalproblembut it is
not physicallyrealisticandit tendsto give poor results;2) a patternrecognitionapproachbasedon a more
realisticproblemformulation: unfortunately, the computationalproblemsassociatedwith this formulation
maybeformidable;and3) aquadraticprogrammingapproach:this is basedonminimizationof aphysically
realisticobjective function.
P. MerzandB. Freisleben.Geneticalgorithmsfor binaryquadraticprogramming.In ‘GECCO-99.Proceedingsof theGeneticandEvolutionaryComputationConference.JointMeetingof theEighthInternationalConferenceonGeneticAlgorithms(ICGA-99)andtheFourthAnnualGeneticProgrammingConference(GP-99).MorganKaufmannPublishers,SanFrancisco,CA, USA’, Vol. 1, pp.417–424,1999.
Abstract. Geneticalgorithmsfor theunconstrainedbinaryquadraticprogrammingproblem(BQP)arepre-
sented.It is shown that for smallproblems,a simplegeneticalgorithmwith uniform crossover is sufficient
to find optimumor best-known solutionsin ashorttime,while for problemswith ahighnumberof variables
(n¿or=200),it is essentialto incorporatelocal searchto arrive at high quality solutions.A hybrid genetic
algorithmincorporatinglocal searchis testedon 40 probleminstancesof sizescontainingbetweenn=200
andn=2500.The resultsof the computerexperimentsshow that the approachis comparableto alternative
heuristicssuchastabu searchfor small instancesandsuperiorto tabu searchandsimulatedannealingfor
largeinstances.New bestsolutionscouldbefoundfor 14 largeprobleminstances.
C. Meszaros. On the sparsityissuesof interior point methodsfor quadraticprogramming.TechnicalReportWP 98-4, Laboratoryof OperationsResearchandDecisionSystems,HungarianAcademyof Sciences,1998a.
Abstract. In this paperwe will investigatehow thesparsityof non-separablequadraticprogrammingprob-
lemsbehavesin interior point methods.We will show that for thenormalequationapproach,two orderings
canbeperformedin independentstepsto reducethefill-in duringinteriorpoint iterations.Oneof thepermu-
tationshasto beperformedonthecolumns,whereastheotheron therowsof theproblem.Weshow thatone
caneasilyattribute the sparsityissuesof non-separablequadraticprogrammingproblemsto that of linear
programmingfor which well–developedtechniquesareavailable.We describehow the fundamentalstruc-
tural propertiesof non–separablequadraticprogrammingproblemscanbe representedby a singlematrix
whosesparsitypatterncanserve to determinetherow permutationandto useheuristicsdevelopedfor linear
programmingfor determiningwhich of the augmentedsystemandnormalequationapproachis moread-
vantageous.Numericalresultsaregivenon a wide varietyof non–separableconvex quadraticprogramming
problems.
C. Meszaros.Theseparableandnon-separableformulationsof convex quadraticproblemsin
interior point methods.TechnicalReportWP 98-3,Laboratoryof OperationsResearch
andDecisionSystems,HungarianAcademyof Sciences,1998b.
C. Meszaros.Steplengthsin interior-point algorithmsof quadraticprogramming.OperationsResearch Letters, 25(1), 39–45,1999.
Abstract. An approachto determineprimal anddual stepsizesin the infeasible-interior-point primal-dual
methodfor convex quadraticproblemsis presented.Theapproachreducestheprimalanddualinfeasibilities
in eachstepand allows different stepsizes.The methodis derived by investigatingthe efficient set of a
multiobjective optimizationproblem.Computationalresultsarealsogiven.
M. C. Meyer. An algorithmfor projectionsonto convex coneswith applicationsto nonpara-
metric regressionandquadraticprogramming.TechnicalreportSta99-12,Department
of Statistics,Universityof Georgia,Athens,Georgia,USA, 1999.
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M. Minoux. A polynomialalgorithmfor minimum quadraticcostflow problems. European
Journalof OperationsResearch, 18, 377–387,1984.
B. K. MishraandC. Das.Maximin problemandaduality theoremfor mixed-integerquadratic
programming.Zeitschrift fur AngewandteMathematikund Mechanik, 65(7), 310–312,
1985.
G. D. Mistriotis. Quadraticprogramming:anefficientalternativeto micro-simulation.Bulletinof theOperationsResearch Societyof America, 20, B328,1972.
Abstract. This papersuggestsan alternative to micro-simulationasa meansof updatinga given sample
representingsomebackgroundpopulation.Theproblemis translatedto a quadraticprogrammingproblem
with linear equality constraintsby the adjustmentof the individual recordweightsof the sample.In this
way thesamesamplewith thenew revisedrecordweightswould adequatelyrepresenttheagedpopulation.
The philosophyof the problemformulationandits advantagesarediscussed.Even for large lengthrecord
problemstheenormouslyrelative low costandthelow corememoryrequirementsareillustrated.Finally, the
comparative limitationsareoutlined.
A. A. Mitsel andA. N. Khvashchevsky. New algorithmfor solvingthequadraticprogrammingproblem.Optoelectronics,InstrumentationandData ProcessingAbstracts, 3, 1999.
Abstract. An algorithmfor solving the quadraticprogrammingproblemis considered.It differs from the
known algorithmsby theabsenceof artificial variables.In someparticularcasesthealgorithmis reducedto
solvingasystemof linearalgebraicequations.
E. Mitsopoulou. Unilateralcontact,dynamicanalysisof beamsby a time-stepping,quadraticprogrammingprocedure.Meccanica, 18(4), 254–265,1983.
Abstract. Discussesa methodfor thedynamicanalysisof aslenderbeam,undergoing’small’ deformations
andcontrastedwithout friction by a supportingprofile. Therelevant contact-impact(unilateralconstrained)
problemis studied,with referenceto adiscretemodelin spaceandtime,after its formulationasa quadratic
programmingproblem(QPP)with signconstraintsonly. Thecontact-impactproblemfor a rigid supporting
profileis first solved,andsubsequentlythecontactproblemfor anelasticprofile.Amongavailablealgorithms
for the numericalsolution,the implicit unconditionallystablealgorithmof Zienkiewicz, WoodandTaylor
(1980)provedto bethemostefficient for thetimediscretization,while amodificationof Hildreth-d’Esopo’s
algorithmwith theintroductionof avariableoverrelaxationfactorwasusedfor thesolutionof theQPP.
E. Mitsopoulou and I. Doudoumis. Unilateral contactof elastic bodiesby finite-element
methodandquadratic-programmingtechnique.Zeitschrift fur AngewandteMathematik
undMechanik, 66(5), T421–T423,1986.
S. Mizuno andK. G. Murty. Computationof theexactsolutionfrom a nearoptimumsolutionfor convex QP. TechnicalReport10/92,Departmentof OR andIE, Cornell,New York,USA, 1992.
Abstract. Considera convex quadraticprogramof sizeL involving m inequalityconstraintsandpossibly
someequality constraintsin n variables.Given a feasiblesolution whoseobjective value is within 2' 2L
of the optimum objective value, we discussa procedurethat finds a true optimum solution in at mostm
iterations,whereeachiterationinvolvesminimizing thequadraticobjective functiononanaffine space(this
takesat mostn conjugategradientsteps).
Y. S.Mo, W. S.Lu, andA. Antoniou.An iterativequadraticprogrammingmethodfor multiratefilter design.In ‘ISCAS ’98. Proceedingsof the1998IEEE InternationalSymposiumonCircuitsandSystems.IEEE,New York, NY, USA’, Vol. 5, pp.41–44,1998.
N. I. M. GOULD & PH.L. TOINT 89
Abstract. An iterative quadraticprogrammingmethodfor multiratefilter designis described.The design
problemis formulatedas a 4th-ordernonlinearoptimizationproblemin which the objective function is
a weightedsum of a reconstructionterm andan aliasing-errorterm. Constraintson the filter’s frequency
responsein thepassbandandstopbandareimposedasa setof linearinequalities.Theoptimizationproblem
issolvedby iteratively minimizingaquadraticfunctionsubjecttoasetof linearconstraints.Explicit formulas
for evaluatingtheminimumsof thesequadraticfunctionsaredescribed,which leadto anefficient andfast
algorithm.An exampleis includedto illustratethedesignmethod.
N. Moal andJ. J. Fuchs. Sinusoidsin white noise: a quadraticprogrammingapproach. In‘Proceedingsof the1998IEEEInternationalConferenceonAcoustics,SpeechandSignalProcessing,ICASSP’98. IEEE,New York, NY, USA’, Vol. 4, pp.2221–2224,1998.
Abstract. We addressthe problemof the estimationandidentificationof real sinusoidsin white Gaussian
noiseusinga correlation-basedmethod.We estimatea partial covariancesequencefrom the dataandseek
a representationof thesenew observationsasa super-positionof a small numberof cosineschosenfrom
a redundantbasisandthe white noisecontribution. We proposeto minimize a quadraticprogramin order
to choosea parsimoniousdecompositionamongthe many that allow the reconstruction.We develop op-
timality conditionsfor the criterion that canbe geometricallyinterpretedandpresenta dual criterion that
hasanappealingphysicalinterpretation.Somesimulatedexamplesarealsopresentedto show theexcellent
performancein resolutionof theapproach.
I. B. Mohd andY. Dasril. Constraintexplorationmethodfor quadraticprogrammingproblem.AppliedMathematicsandComputation, 112(2–3),161–170,2000.
Abstract. In this paper, we representa methodwhich is basedon the violatedconstraintsby the uncon-
strainedminimum of the objective function of the quadraticprogrammingproblemfor exploring, locating
andcomputingtheoptimalsolutionof theproblemwithout usingadditionalinformationashave beendone
in mostof thefavouriteestablishedmethods.
A. Mohri. A computationalmethodfor optimal control of a linear systemby quadraticpro-gramming.InternationalJournalof Control, 11(6), 1021–1039,1970.
Abstract. The numericalmethodderived is applicableto the problem,wherethe control u is considered
to be held at a constantvalueduring the constanttime interval, and the criterion function is given by the
following integral type:J # x; Qx u; Ru dt. But themethodintroducedin thispaperrequiresnonumerical
integrationexceptto obtainthetransitionmatrix.Thereforethecomputationaleffort is considerablysaved.
J.A. Momoh,R. E. L. Adapa,andM. E. Hawary. A review of selectedoptimalpowerflow lit-eratureto 1993.I. Nonlinearandquadraticprogrammingapproaches.IEEETransactionson PowerSystems, 14(1), 96–104,1999.
Abstract. Thepaperpresentsa review of literatureon optimalpower flow tracingprogressin this areaover
from 1962–93.Part 1 dealswith theapplicationof nonlinearandquadraticprogramming.
R.D. C.Monteiro,I. Adler, andM. G.C.Resende.A polynomial-timeprimal-dualaffinescalingalgorithmfor linear andconvex quadraticprogrammingandits power seriesextension.Mathematicsof OperationsResearch, 15(2), 191–214,1990.
Abstract. Describesan algorithmfor linear andconvex quadraticprogrammingproblemsthat usespower
seriesapproximationof the weightedbarrier path that passesthroughthe current iteratein order to find
thenext iterate.If r 1 is theorderof approximationused,theauthorsshow that their algorithmhastime
complexity O n1A 2 B 1C 1r ! L 1 9 1r ! iterationsandO n3 n2r arithmeticoperationsper iteration,wheren is
thedimensionof theproblemandL is thesizeof the input data.Whenr 1, they show that thealgorithm
canbeinterpretedasanaffine scalingalgorithmin theprimal-dualsetup.
R. D. C. Monteiro and I. Adler. Interior path following primal-dualalgorithms.2. Convex
quadraticprogramming.MathematicalProgramming, 44(1), 43–66,1989.
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R. D. C. Monteiro andT. Tsuchiya. Global convergenceof the affine scalingalgorithmforconvex quadraticprogramming.SIAMJournalon Optimization, 8(1), 26–58,1998.
Summary. A global convergenceproof of the second-orderaffine scalingalgorithmfor convex quadratic
programmingproblemsis given,wherethenew iterateis thepoint thatminimizestheobjective functionover
the intersectionof the feasibleregion with theellipsoidcenteredat thecurrentpoint andwhoseradiusis a
fixedfractionβ D 0 1E= of theradiusof the largest“scaled”ellipsoid inscribedin thenonnegative orthant.
Theanalysisis basedon the local Karmarkarpotentialfunction introducedby Tsuchiya.For any β D 0 1andwithout makingany nondegeneracy assumptionon the problem,the sequencesof primal iteratesand
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lar, thatif thequadraticis boundedbelow onthefeasiblesetthenterminationoccursatastationarypoint in a
finite numberof iterations.Moreover, if all stationarypointsarenondegenerate,terminationoccursata local
minimizer. A numericalcomparisonof thealgorithmbasedon thegradientprojectionalgorithmwith astan-
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considerablelessiterationsandtime thantheactive setstrategy. On nondegenerateproblemsthenumberof
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of nodevoltagesin rectangularcoordinates.As a result, the stateestimationproblemis formulatedas a
quadraticprogrammingproblem.The advantageof the proposedmethodis that the obtainedestimatesare
N. I. M. GOULD & PH.L. TOINT 91
very robustto grosserrorsof measurementsor baddatadueto theexactobservationequationin a quadratic
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of S, x O) whereg c x y, andb arecolumnvectorsof appropriatedimension;A B, andD arematrices
of appropriatedimension;andS is a closed,boundedset whoseelementshave integer components.The
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to considerevery line capacityandNOx emissionconstraintwheneachline andthermalunit fail proba-
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satisfythe linescapacityandonly a healthystatecontrolstheemissionexceptingfaulty states.Theuseful-
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thequadraticprogrammingmethod.Thereportdescribestheoutlinesof both theproposedmethodandthe
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needingminimum cost of generationand minimum system-transmissionlosses.Thesehave beensolved
sequentiallyto achieve optimalallocationof realandreactive power generationandtransformertapsettings
with considerationof system-operatingconstraintson generation,busbarvoltageandline-flow limits. The
potentialof the new algorithmfor OPFhasbeendemonstratedthroughsystemstudiesfor two IEEE test
systemsandanIndiansystem.Resultsrevealthattheproposednew algorithmhaspotentialfor onlinesolving
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constraints.The typesconsideredarequite generalandinclude(1) equalityconstraints,(2) nonnegativity
constraints,(3) inequalityconstraints,and(4) any combinationof the first threetypes.In eachcase,it is
assumedthatthematrix of thequadraticobjective functionis at leastpositive semidefinite.Theexpressions
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programwith its cost function given by the mean-squareerror betweenthe arrayresponseanda properly
selectedpatterndescribedby a known mathematicalfunction.This quadraticprogramcanbea constrained
or unconstrainedoptimizationproblemdependingon the requirementsof the desiredarraypattern.In for-
mulatingthequadraticprogram,noassumptionhasbeenmadeon thegain/phaseresponseor characteristics
of theindividual arrayelements.Therefore,onecansynthesizeanarrayof arbitraryshapeto any appropriate
patternwith thecharacteristicof thearrayelementstaken into considerationaslong asoneis ableto model
the arrayaccurately. The proposedmethodis usedto synthesizearraysof different shapes,linear aswell
asplanararrays(including rectangularandcircular planararrays),usinga Chebyshev polynomialor zero
functionasa designtemplate,to illustratethefeasibilityof theproposedmethod.
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An optimal solution for real power dispatchis obtainedby quadraticprogramming,andoptimumalloca-
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voltageandreactive power limits. An ac load flow analysisis incorporatedtogetherwith a load prediction
programbasedonaspectralanalysisof pastloaddata.Thefeasibilityof simulatingtheoverall systemprob-
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marginal costsregardingthehydroenergy arecalculatedfor usein thehydroscheduling.Theallocationof
thehydrodischarges/energies is thenusedfor anothercalculationof hydromarginal coststo beusedin the
next iteration.In theplanningsystemdevelopedat theSwedishStatePower Boardthehydrosubproblemis
modeledwith atime-incrementof oneweekwhile theloadduringeachweekis describedby threeloaddura-
tion curves-onefor eachof day-time,night-timeandweekends.To calculatethecorrectmarginalcostfor the
hydroenergy, theavailableweeklyenergieshasto beallocatedoptimally within eachweek.This optimiza-
tion problemcanbeformulatedasaquadraticprogrammingproblemwith onelinearequalityconstraint:the
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utility’s normaldatacollectionproceduresdoesnot provide this informationandobtainingit, even infre-
quently, hasentailedcostlystatisticalsamplingoperations.A statisticalidentificationmodelwith inequality
constraintson the unknown variableshasbeenconstructedasa meansfor providing estimateson the de-
siredquantities.Model inputsincludemonthlycustomerbilling dataandaggregatepeakdemands.Testcase
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Abstract. TheweightedChebyshev designof two-dimensionalFIR filters is in generalnot uniquesincethe
Haarconditionis not generallysatisfied.However, for a designbasedon thediscretefrequency domain,the
Haarconditionmight befulfilled. Thequestionof uniquenessis, however, ratherextensive to investigate.It
is thereforedesirableto definesomesimpleadditionalconstraintsto theChebyshev designin orderto obtain
a uniquesolution.The weightedChebyshev solutionof minimum Euclideanfilter weight norm is always
unique,andrepresentsa sensibleadditionalconstraintsinceit impliesminimumwhite noiseamplification.
N. I. M. GOULD & PH.L. TOINT 95
It is shown that this uniqueChebyshev solution can always be obtainedby using an efficient quadratic
programmingformulationwith astrictly convex objective functionandlinearconstraints.An examplewhere
aconventionalChebyshev solutionis non-uniqueis discussed.
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Abstract. A methodto solve a generalbroadbandbeamformerdesignproblemis formulatedasa quadratic
program.As a specialcase,theminimaxnear-field designproblemof a broadbandbeamformeris solvedas
a quadraticprogrammingformulationof theweightedChebyshev approximationproblem.Themethodcan
alsobeappliedto thedesignof multidimensionaldigital FIRfilterswith anarbitrarilyspecifiedamplitudeand
phase.For linearphasemultidimensionaldigital FIR filters,thequadraticprogrambecomesalinearprogram.
Examplesaregiventhatdemonstratetheminimaxnear-field behavior of thebeamformersdesigned.
S. Nordebo,I. Claesson,andS. Nordholm. Quadraticprogrammingfor the designof two-
dimensionalweightedChebyshev FIR filters with incompletespecifications.Nonlinear
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Abstract. This letterpresentsa new extendedactive setstrategy for optimumfinite impulseresponse(FIR)
window designby semi-infinitequadraticprogramming.Thewindows maybeasymmetriccorrespondingto
frequency responseswith generalnonlinearphase.Theoptimalitycriterionis to minimizethesidelobeenergy
(L2-norm)subjectto a peaksidelobemagnitude-constraint(L∞-constraint).Additional linearconstraintsare
usedto form the mainlobe(unity DC gain).Numericalexamplesinvolving groupdelayspecificationsare
usedto illustratetheusefulnessof thealgorithm.
A. Nou. An algorithmfor a singly constrainedquadraticprogramsubjectto lower bounds.
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Abstract. Thepurposeof optimalshorttermschedulingis to maximizeelectricalenergy productiondepend-
ing onnaturalwaterinflowsof ariver, accordingto daily loaddurationcurveandoperationof thermalpower
plantswith minimaloscillatingload.Thermalpowerplantshavealongstart-uptimeandcannotfollow rapid
changesin loaddemand,sothey areusedto cover theconstantpartof theload,while peaksin loaddemand
arecoveredby hydrounits.A mathematicalmodelof a hydroplantchainis developedwhich usesonly two
insteadof threevariables.TheoptimizationalgorithmusesNewton’s method,which is appliedon theexact* 1 penaltyfunction,allowing a nonfeasiblestartingpoint sothatno phaseI simplex or projectionmethodis
neededat thebeginningof optimization.Theprogramwasappliedto thehydro-chainon theriver Drava.
I. Nowak.Someheuristicsandtestproblemsfor nonconvex quadraticprogrammingoverasim-plex. Preprintseries,Institut fur Mathematik,Humboldt-UniversitatzuBerlin, Germany,1998.
Abstract. In thispaperwecomparetwo methodsfor estimatingaglobalminimizerof anindefinitequadratic
form overasimplex. Thefirst methodis basedon theenumerationof localminimizersof aso-calledcontrol
polytope.Thesecondmethodis basedon anapproximationof theconvex envelopeusingsemidefinitepro-
gramming.In orderto testthealgorithmsa methodfor generatingrandomtestproblemsis presentedwhere
96 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
theoptimal solutionis known andthe numberof bindingconstraintsis prescribed.Moreover, it is investi-
gatedif somemodificationsof theobjective functioninfluencetheperformanceof thealgorithms.Numerical
experimentsarereported.
I. Nowak. A global optimality criterion for nonconvex quadraticprogrammingover a sim-plex. Preprintseries,Institut fur Mathematik,Humboldt-UniversitatzuBerlin, Germany,1998b.
Abstract. In this paperwe proposea global optimality criterion for globally minimizing a quadraticform
over the standardsimplex, which in additionprovidesa sharplower boundfor the optimal value.The ap-
proachis basedonthesolutionof asemidefiniteprogram(SDP)andaconvex quadraticprogram(QP).Since
thereexist fast(polynomialtime) algorithmsfor solvingSDP’s andQP’s thecomputationaltime for check-
ing theglobaloptimality criterionandfor computingthelower boundis reasonable.Numericalexperiments
on randomtestexamplesup to 30variablesindicatethattheoptimality criterionverifiesaglobalsolutionin
almostall instances.
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timerequiredto optimallysolveaircraftconflictsis known to grow exponentiallywith thenumberof aircraft
involvedandmaybecomeprohibitive whenlargenumbersof aircraftareinvolved.As anattemptto circum-
ventthis issue,aheuristicpolynomial-timeconflict resolutionalgorithmis proposedonthebasisof analysis
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Y. Ohashi,H. Kando,H. Ukai, andT. Iwazumi.Optimaldesignof FIR linearphasedigital filtersvia convex quadraticprogrammingmethod. InternationalJournal of SystemsScience,19(8), 1469–1482,1988.
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techniques.An optimizationdesignmethodvia the convex quadraticprogrammingmethodis investigated
in the paper. The aim of this methodis to designdigital filters with certaindesiredfrequency responses
that keepthe passbandripple to a minimum while maintaininga balancebetweenoppositecharacteristics
of frequency responses,i.e. thenarrow transitionbandandthegreatstopbandattenuation.In orderto verify
the effectivenessof this method,low-passfilters are designedandcomparisonstudieswith other typical
optimizationmethodsaregiven.
S. Ohtaki andK. I. Hata. Planeelastic-plasticstressanalysisusingquadraticprogramming.adoptionof theyield functioncontainingthethird invariantof thedeviatoricstresstensor.Memoirsof theHokkaidoInstituteof Technology, 15, 35–41,1987.
Abstract. Thefinite elementelastic-plasticanalysisbasedon thetheoryof plasticitytakinginto accountthe
secondordereffect usingthe quadraticprogrammingmethodis appliedto the planestressproblem.This
secondordereffect is causedfrom theyield functionderived from Prager’s theory, which is assumedto be
expressedin termsof not only thesecondinvariantof thedeviatoric stresstensorJ;2 but alsoof thethird one
J;3 Matricesconcerningwith thismethodareformulatedundertheplanestressstate.A numericalcalculation
is presentedfor thestressconcentrationproblemof a thin rectangularplatewith semi-circularsidenotches
subjectedto auni-axialtension.Resultsobtainedby usingthesecondorderplastictheoryarecomparedwith
thoseof theordinaryJ2 theory.
N. I. M. GOULD & PH.L. TOINT 97
S.OhtakiandA. Kurimura.An analysisof two-dimensionalelastic-plasticstressproblemusingthequadraticprogramming.(Thecaseof adoptingPrandtl-ReussequationandvonMisesyield function). Bulletin of theJapanSocietyof MechanicalEngineers, 28(243),1864–1867,1985.
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incrementspertainingto the yield surfacein stressspacearenot satisfiedandwork-softeningeffects are
consideredwith theaidof quadraticprogrammingconcepts.Theauthorsappliedthismethodfor solvingthe
elastic-plasticplanestressproblemasa quadraticoptimizationin termsof displacementratesandplastic
multiplier rates.The matricesareformulatedin explicit forms for the problemof a perforatedstrip under
tension.A comparisonis madebetweenthe resultsobtainedby the presentmethodand by the ordinary
matrixmethod.Theelastic-plasticproblemis fairly tractableto thepresentmethod.
A. OhuchiandI. Kaji. Algorithmsfor optimalallocationproblemshaving quadraticobjectivefunction. Journalof theOperationsResearch Societyof Japan, 23(1), 64–79,1980.
Abstract. An optimal allocationproblem(APQ) with a quadraticobjective function,a total resourcecon-
straintandanupperandlowerboundconstraintis considered.TheAPQis averybasicandsimplemodelbut
it canserve asa subproblemin thesolutionof thegeneralizedallocationproblem.Applying the Lagrange
relaxationmethodanexplicit expressionof thedualfunctionassociatedwith theAPQandanequationwhich
theoptimaldualvariablemustsatisfyandderivedfirst. Then,somepropertiesof theequationarediscussed.
Finally, threealgorithmsfor solvingtheequationareproposed,andsomecomputationalresultsfor theAPQ
aregiven.Theseresultsrevealtheeffectivenessof thealgorithm.
A. OhuchiandI. Kaji. An algorithmfor theHitchcocktransportationproblemswith quadraticcost functions. Journal of the OperationsResearch Societyof Japan, 24(2), 170–181,1981.
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hasbeenproposedwhich involvessuccessively minimising the Lagrangianwith respectto eachof its dual
coordinates.Thefactthatthealgorithmsfor optimalallocationproblemshaving quadraticobjective functions
(APQ) canbeusedfor the maximisingprocedurein network transportationproblemswith quadraticcosts
functionswasinvestigated,andAPQstudiedin detail.Theapplicationof algorithmsproposedby theauthors
to solvingHTPQis considered.In particularseveral examplesaregiven of relatively large-scaleproblems,
with computationaltimes.
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mizing a convex quadraticform subjectto upperandlower boundson the variables.This methodexploits
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very effective methodfor analyzingelastoplastictorsionof prismaticbarsposedasquadraticprogramming
problems.Solutionsfor barswith elliptical andSokolovsky’s oval cross-sectionsarepresented.Thesolutions
98 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
for the elliptical barsagreewith the existing elasticand limit plasticsolutionsat the two extremesof the
elastic-plasticrange.ThealgorithmalsoreproducesaccuratelytheSokovsky solutionandextendsit beyond
its limitations.
T. OnodaandH. Sekimoto. A neutronspectrometryunfolding codebasedon quadraticpro-gramming.NuclearInstrumentsandMethodsin PhysicsResearch, SectionA (Accelera-tors,Spectrometers,DetectorsandAssociatedEquipment), 272(3), 844–846,1988.
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timing constraints.The problemis formulatedasa constrainedprogrammingproblemandsolved in two
phases:cut-setminimizationandtiming satisfaction.A mathematicalprogrammingtechniquebasedon it-
erative quadraticprogramming(TPIQ) is usedto find anapproximatesolutionto the constrainedproblem.
Whenthetiming constraintsaretoo strict to have a feasiblesolution,nodereplicationis usedto satisfythe
constraints.Experimentalresultson the ISCAS89benchmarksuiteshow that TPIQ cansolve the timing-
drivenbipartitioningproblemwith little impacton thechipsize.
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Abstract. The authorsdescribethe problemsof steadystateoptimal load sheddingin power systemsand
presenta methodfor minimizing load curtailmentundera given set of emergency operatingconditions.
Electric power consumersare inconveniencedif there is an inadequatelevel of supply voltageand load
curtailment.Theinconvenienceis to beminimizedsubjectto thesystempower-flow equationsandthelimits
on realandreactive power generationalongwith the limits on the line flows. This optimizationproblemis
solvedusingquadraticprogrammingwhichhasbeenfoundeffective in theassignmentof distributionof load
curtailmentover theentiresystem.Themethodis illustratedon a samplesystemfor generation-outageand
line-outageconditions.
V. N. Palyaeva. A linearization method with quadraticprogrammingfor linearly con-
strainedapproximation-problems. VestnikLeningradskogo UniversitetaSeriyaMatem-
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to a certainimplicit formulation of a quadraticprogramgeneratesthe samesequenceof primal feasible
vectorsasdoestheVon Hohenbalken simplicial decompositionalgorithmspecializedto thesameprogram.
Suchanequivalenceof thetwo algorithmsextendsearlierresultsfor a least-distanceprogramdueto Cottle-
Djang.
J.S.Pang.Methodsfor quadraticprogramming:asurvey. ComputersandChemicalEngineer-ing, 7(5), 583–594,1983.
N. I. M. GOULD & PH.L. TOINT 99
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The discussionis centeredon (i) the unificationof several classesof finite methods,(ii) recentdevelop-
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largescaleapplications.
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whethera functionis locally strictly convex is alsoNP-hard.
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applications,however, it turnsout, we found, to have someperformancelimitationssuchasunrealistically
high torquedueto joint anglelimit constraintsandtrackingerrorsdueto joint torquelimit constraintsunder
parametervariationanddisturbance.Remedyingthe limitations, the enhancedcompactQP methodis de-
velopedby usingthepracticalinequalityconstraintswith p-step-aheadpredictorandtime delayestimation.
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provesthecompactQPmethodin termsof efficiency andeffectivenessfor thereal-timecontrolof redundant
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extensionof A. W. Tucker’sCombinatorialTheoryunderlyinglinearprogramsto convex quadraticprograms.
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N. I. M. GOULD & PH.L. TOINT 101
given. On the basisof this theoreticalstudy, two principal solutionmethodsarepresented.An important
applicationof nonconvex quadraticprogrammingis thecomputationof rite stepto anew iteratein theTrust
Region (TR) approachmethodswhich areknown to beefficient for nonlinearoptimizationproblems.Also,
wediscussthemathematicalmodelsof someimportantproblemsencounteredin ComputerVision.Most of
themcanbeformulatedasaminimizationof asumof squaresof nonlinearfunctions.A practicalTR-based
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stability of Lagrangianduality is establishedand completecharacterizationsof a global optimal solution
aregiven.On thebasisof this theoreticalstudy, two principalsolutionmethodsarepresentedAn important
applicationof nonconvex quadraticprogrammingis thecomputationof thestepto a new iteratein theTrust
Region (TR) approachmethodswhich areknown to beefficient for nonlinearoptimizationproblems.Also,
wediscussthemathematicalmodelsof someimportantproblemsencounteredin ComputerVision.Most of
themcanbeformulatedasaminimizationof asumof squaresof nonlinearfunctions.A practicalTR-based
algorithmis proposedfor nonlinearleastsquaresproblemwhichseemsto bewell suitedfor ourapplications.
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by randomnoisearesuggested.Methodoneis basedontheapriori informationthatapicturehasnonnegative
intensityandleadsto quadraticprogramming.In methodtwo, thefinite extentof thepictureis shown to make
Fouriertransformationin thecomplex domainuseful.
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Abstract. Considerstheconvolution equationf & h e d, where f is sought,h is a known ’point spread
function’, e representsrandomerrorsandd is the measureddata.All thesefunctionsare definedon the
integersmod N . A mathematical-statistical formulationof the problemleadsto minf F f & h d F A, where
theA-normis derivedfrom thestatisticaldistributionof e. If f is known to benonnegative, this is aquadratic
programmingproblem.UsingthediscreteFourier transforms(DFTs)F H, andD of f h, andd, theauthor
arrivesat a minimization in anothernorm: minF F F H D F α. A solutionwould be F D H, but H has
zeros.He considersthe theoreticalandpracticaldifficulties that arisefrom thesezerosanddescribestwo
methodsfor calculatingF numericallyalsowhenH haszeros.Numericaltestsof methodsarepresented,in
particulartestswith oneof themethods,called’thederivative method’,whered is ablurredimage.
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Abstract. We considerthe parametricprogrammingproblem(Qp) of minimizing the quadraticfunction
f x p : xT Ax bTx subjectto theconstraintCx d, wherex IRn, A IRn G n, b IRn, C IRm G n, d IRm,
andp : H A b C d is theparameter. Here,thematrix A is not assumedto bepositive semidefinite.Theset
of the global minimizersand the set of the local minimizersto (Qp) are denotedby M p andMl oc p ,respectively. It is provedthat if thepoint-to-setmappingMl oc + is lower semicontinuousat p thenMl oc p
102 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
is anonemptysetwhichconsistsof atmostI m4 n points,whereI m4 n Jm
min6K<m 2=L n8NM is themaximal
cardinalityof theantichainsof distinctsubsetsof 6 1 2 O m8 which have at mostn elements.It is proved
also that the lower semicontinuityof M + at p implies that M p is a singleton.Under someregularity
assumption,thesenecessaryconditionsbecomethesufficient ones.
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nalsis presented.This work extendstheearlierwork of theauthorandhis colleaguesin computingpositive
TFD (1994).This paperdescribesa generalquadraticprogrammingapproachto theproblemof computing
thesesignal-dependentdistributions.Themethodis basedon anevolutionaryspectrumformulationof posi-
tive TFD. Theminimizationproblemreducesto a linearly-constrainedquadraticprogrammingproblem,for
whichstandardsolutionsarewidely available.
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Abstract. We considerthreeparametricrelaxationsof the (0,1)-quadraticprogrammingproblem.These
relaxationsare to: quadraticmaximizationover simple box constraints,quadraticmaximizationover the
sphere,andthemaximumeigenvalueof aborderedmatrix.Whenminimizedover theparameter, eachof the
relaxationsprovidesanupperboundontheoriginaldiscreteproblem.Moreover, theseboundsareefficiently
computable.Our mainresultis that,surprisingly, all threeboundsareequal.
S.Poljak,F. Rendl,andH. Wolkowicz. A recipefor semidefiniterelaxationfor (0,1)-quadraticprogramming.Journalof Global Optimization, 7(1), 51–73,1995.
Abstract. Wereview variousrelaxationsof (0,1)-quadraticprogrammingproblems.Theseincludesemidef-
inite programs,parametrictrust region problemsandconcave quadraticmaximization.All relaxationsthat
weconsiderleadto efficiently solvableproblems.Themaincontributionsof thepaperarethefollowing. Us-
ing Lagrangianduality, we prove equivalenceof therelaxationsin a unifiedandsimpleway. Someof these
equivalenceshave beenknown previously, but our approachleadsto shortandtransparentproofs.Moreover
weextendtheapproachto thecaseof equalityconstrainedproblemsby takingthesquaredlinearconstraints
into theobjective function.We show how this techniquecanbeappliedto theQuadraticAssignmentProb-
lem, theGraphPartition ProblemandtheMax- CliqueProblem.Finally we show our relaxationto bebest
possibleamongall quadraticmajorantswith zerotrace.
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N. I. M. GOULD & PH.L. TOINT 103
M. J.D. Powell. ZQPCVX,aFortransubroutinefor convex quadraticprogramming.Technical
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Abstract. Two implementationsof the algorithmof Goldfarb andIdnani (1983)for convex quadraticpro-
grammingareconsidered.A pathologicalexampleshows that thefasteronecanbeunstable,but numerical
testingon somedifficult problemsindicatesthat both implementationsgive excellentaccuracy. Therefore
theauthor(1983)hasprovided for generalusea FORTRAN subroutinethatappliesthefasterimplementa-
tion.Thissubroutineis comparedwith two widely availablequadraticprogrammingsubroutinesthatemploy
feasiblepoint methods,namelyQPSOL(Gill, Murray, Saunders,andWright, 1983)andVEO2A (Fletcher,
1970).Theauthorconcludesthatthealgorithmof GoldfarbandIdnaniis verysuitablein practicefor convex
quadraticprogrammingcalculations.
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Abstract. An algorithmhasbeendevelopedto solve quadraticprogramsthathave a dynamicprogramming
structure.It hasbeendevelopedfor useaspart of a paralleltrajectoryoptimizationalgorithmandaimsto
achieve significantspeedwithout sacrificingnumericalstability. The algorithmmakesuseof the dynamic
programmingproblemstructureandthedomaindecompositionapproach.It parallelizestheorthogonalfac-
torizationnull-spacemethodof quadraticprogrammingby developingaparallelorthogonalfactorizationand
aparallelCholesky factorization.Testsof thealgorithmona32-nodeINTEL iPSC/2hypercubedemonstrate
speedupfactorsaslargeas10 in comparisonto thefastestknown equivalentserialalgorithm.
R. Pytlak. A range-spacemethodfor piecewise-linearquadraticprogramming:anapplicationto optimalcontrolalgorithms.In ‘Proceedingsof the33rdIEEEConferenceonDecisionandControl.IEEE,New York, NY, USA’, Vol. 2, pp.1462–1463,1994.
Abstract. A new methodfor solvinga convex optimizationproblemwith box constraintsis presented.The
objective functionhasa positive-definitequadratictermanda piecewise-linearterm.Themethodis derived
from a rangespacemethodfor QP problems.The approachis particularlyefficient if the piecewise-linear
termhasfew breakpoints.Numericalcomparisonswith anefficient implementationof anull-spaceactive-set
algorithm(LSSOL)arealsopresented.
A. J. Quist,E. Deklerk,C. Roos,andT. Terlaky. Copositive relaxationfor generalquadraticprogramming.OptimizationMethodsandSoftware, 9(1–3),185–208,1998.
Abstract. We considergeneral,typically nonconvex, QuadraticProgrammingProblems.TheSemi-definite
relaxationproposedby Shorprovidesboundson the optical solution.but it doesnot alwaysprovide suffi-
cientlystrongboundsif linearconstraintsarealsoinvolved.To getrid of thelinearside-constraints.another,
strongerconvex relaxationis derived.This relaxationusescopositive matrices.Specialcasesarediscussed
for whichbothrelaxationsareequal.At theendof thepaper, thecomplexity andsolvability of therelaxations
arediscussed.
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S.R.Radhakrishnan.Capitalbudgetingandmixed0–1integerquadraticprogramming.Bulletinof theOperationsResearch Societyof America, 20, B331,1972.
Abstract. Thecapitalbudgetingproblemis treatedasapartof thegeneraltheoryof choice,whereutility is
to bemaximizedsubjectto theopportunitiesandconstraints.Theutility functionis assumedto bequadratic
to takecareof any risk-aversebehaviour of theshareholdersandaquadraticmixed0–1integerprogramming
modelis developed.Theduality conceptsin discreteprogrammingareusedto derive propertiesof optimal
solutionsto themodel.Specialsolutiontechniquesfor themixed0–1integerquadraticprogrammingmodel
arediscussed.
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Abstract. The authorformulatesa theory of the sufficient conditionsof optimality of a certainclassof
nonlinearequationsof thesecondorder. By theuseof spectralexpansionof anon-self-conjugateoperator, a
solutionof theproblemof linearly-quadraticprogrammingis constructedin closedform.A proof is obtained
of aseriesof theoremsof thepropertiesof thesolutionof aRikkati equationby substantiationof themethod
of dynamicprogramming.Theresultsobtainedarebasedonthepropertiesof aRissbasisformedby theeigen
andadjointelementsof a stationaryoperatorforming theprincipalpartof thedynamicequationexamined.
It is shown thattheapproximatesolutionof theproblemconvergesto anexactone.
M. R. Rakhimov. The use of methodsof spectrumseriesand dynamic-programmingfor
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R. RamabhadranandJ. K. Antonio. Fastsolutiontechniquesfor a classof optimal trajectoryplanningproblemswith applicationsto automatedspraycoating. TechnicalReportTR-EE95–9,PerdueDepartmentof ECE,PerdueUniversity, ???,USA, 1995.
Abstract. Optimal trajectoryplanningproblemsareoften formulatedasconstrainedvariationalproblems.
In general,solutionsto variationalproblemsare determinedby appropriatelydiscretizingthe underlying
objective functionalandsolving the resultingnonlineardifferentialequation(s)and/ornonlinearprogram-
ming problem(s)numerically. Thesegeneralsolutiontechniquesoftenrequirea significantamountof time
to becomputed,andthereforeareof limited valuewhenoptimaltrajectoriesneedto befrequentlycomputed
and/orre-computed.In this paper, a realisticclassof optimal trajectoryplanningproblemsis definedfor
which theexistenceof fastnumericalsolutiontechniquesaredemonstrated.To illustratethepracticalityof
thisclassof trajectoryplanningproblemsandtheproposedsolutiontechniques,threeoptimaltrajectoryplan-
ningproblemsfor spraycoatingapplicationsareformulatedandsolved.Basedontheproposeddiscretization
technique,it is shown thattheseproblemscanbereducedto eithera linearprogramor aquadraticprogram,
which arereadilysolved.In contrast,usingthestandarddiscretizationof theseproblemsgenerallyleadsto
nonconvex nonlinearprogrammingproblemsthatrequirea significantamountof computationto arrive at a
(possibly)locally optimalsolution.
H. V. Ramakrishnan.Largescaleenergy systemplanningusingquadraticprogramming.Jour-nal of theInstitutionof Engineers (India) Electrical EngineeringDivision, 68, 170–176,1988.
Abstract. A quadraticprogrammingapproachto the problemof planninga large scaleenergy systemis
presented.Theapproachdeterminesoptimalplantlocation,typeandcapacityof generatingunitsto beadded
to any existingsystembyminimizingtheobjective functionconsistingof annualamortizedcostof generation
andtransmissionof thesystem,simultaneouslysatisfyingtheconstraintsof peakandaveragedemands.An
illustrative examplefrom publishedliteratureis workedout.Theeffectof additionalconstraintsfor minimum
generationisalsostudied.Highlightsof theresults,acritical comparisonwith previousstudies,andimportant
conclusionsaregiven.
N. I. M. GOULD & PH.L. TOINT 105
J. R. J. Rao and R. Bradlaw. A unified characterizationof nonunique responseinelastic/perfectly-plasticand elastic/locking structuresusing constraint qualification.TechnicalReport 9, Departmentof MechanicalEngineering,University of Houston,Texas,USA, 1995.
Abstract. Mathematicalprogrammingmethodshave traditionallybeenusedextensively in theanalysisand
designof elastic/plasticstructures.However, someof therecentresultsfrom parametricnonlinearprogram-
minghavenotbeenfully exploitedin mechanicalapplications,particularlythoserelatingto thecausesof sin-
gularitiesdueto parametricvariations.In this paper, we reexaminethephenomenonof nonuniquedisplace-
mentsin elastic/perfectly-plastic structuresunderproportionalloading.Oncetheanalysismodelasderived
from a minimumenergy principleis formulatedasaquadraticprogram,it turnsout thattheresponsecanbe
characterizedasbeinguniqueor nonuniquedependingon thesatisfactionof thelinear independenceor the
Mangasarian-Fromovitz constraintqualification.A completelyanalogousresultis thenalsoderivedfor elas-
tic/locking structures,thusindicatingtheusefulnessof constraintqualificationsin analyzingthenonunique
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Abstract. Comparesthe computationalperformanceof five quadraticprogrammingalgorithms.Thesein-
cludeWolfe’ssimplex method,Lemke’scomplementarypivot method,convex simplex methodandquadratic
differentialalgorithm.Executiontimeanditerationcountareusedasthemajorcriteriafor comparison.Since
Lemke’salgorithmout-performedall othermethodsin thestudy, adetailedstatisticalanalysiswasperformed
to determinetherelative importanceof problemparameterson theefficiency of Lemke’salgorithm.An anal-
ysis of varianceshowed that the numberof variables,the percentof positive linear termsin the objective,
thenumberof constraints,andtheir interactionswerethesignificantfactorsfor bothiterationcountandexe-
cutiontime.Finally, regressionequationsfor iterationcountandexecutiontime arederivedasa functionof
fifteenproblemparameters.
B. S. Razumikhin. Method of physicalmodelling in mathematicalprogrammingand eco-nomics. III. Solution algorithms for problemsof linear and quadraticprogrammingand for equilibrium problemsof exchangemodels. Automationand RemoteControl,33(6), 992–1001,1972.
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rithms for numericalsolutionof linear andquadraticprogrammingproblemsandequilibrium problemsof
linearexchangemodelsaregiven.Thealgorithmsfollow from thephysicalpropertiesof thecorresponding
problemsandthemethodof redundantrelations.
B. D. ReddyandG. P. Mitchell. Theanalysisof elastic-plasticplates:aquadraticprogrammingproblemandits solutionby finite elements.ComputerMethodsin AppliedMechanicsandEngineering, 41(2), 237–248,1983.
Abstract. An extendedkinematicminimumprinciplein classicalplasticityis usedasthebasisfor thefinite
elementformulationof the rateproblemfor elastic-plasticplates.A simplealgorithmis usedto solve the
resultingquadraticprogrammingproblem.Thenumericalsolutionof theproblemis carriedout in two ways:
onemethodinvolvesloadstepsizeswhich arescaledsothatoneor moregausspointsjust becomeplastic,
andthe othermethodinvolves load stepsizeswhich arefixed onceandfor all at the outset.Examplesare
givenanddiscussed.
K. P. Reddy, A. K. Mittal, andS. K. Gupta. Bivalentquadraticprogrammingproblem-acom-putationalstudy. Opsearch, 21(3), 153–166,1984.
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Abstract. Theproblemconsideredis theminimisationof a quadraticfunctionof n 0–1variables.Effective
boundingstrategiesaredevelopedandincorporatedin abranch-and-boundalgorithm.Thealgorithmis used
to identify the relatively difficult problems.A heuristicalgorithmis alsoproposedto solve difficult and/or
large sizeproblems.Computationalresultspertainingto the effectivenessof the proposedalgorithmsare
reported.
A. T. Redpath,B. L. Vickery, andD. H. Wright. A new techniquefor radiotherapy planningusingquadraticprogramming.Physicsin MedicineandBiology, N1(5), 781–791,1976.
Abstract. Previousattemptsat optimisationof radiotherapy planningaredescribedandcriticised,andcon-
siderationof theseattemptshasresultedin thedevelopmentof anew techniqueusingquadraticprogramming.
Uniformity of tumourdoseis selectedasthemostimportantfeatureof any plan,andthis is achievedby min-
imising thevarianceof thedoseto preselectedpointswithin thetumour. Thedoseto vulnerableregionscan
beconstrainednot to exceeda givenpercentageof themeantumourdose.Optimisationof field weightand
field typeis possible.Theoperationof thesystemis describedandsometypical resultsaregiven.
G. F. ReidandL. Hasdorff. Economicdispatchusingquadraticprogramming.IEEE Transac-tionson PowerApparatusandSystems, PAS-92(6), 2015–2023,1973.
Abstract. The economicdispatchproblemis formulatedasa quadraticprogrammingproblemandsolved
usingWolfe’s algorithm.The methodis capableof handlingboth equalityandinequalityconstraintson p,
q, andv andcansolve theloadflow aswell astheeconomicdispatchproblem.Thequadraticprogramming
algorithmdoesnot requirethe useof penaltyfactorsor the determinationof gradientstepsizewhich can
causeconvergencedifficulties.Convergencewasobtainedin threeiterationsfor all testsystemsconsidered
andsolutiontime is small enoughto allow the methodto be usedfor on-line dispatchingat practicaltime
intervals.Resultsarepresentedfor 5, 14,30,57,and118bustestsystems.
R. RhodeandR. Weber. Multiple objectivequadratic-linearprogramming.In J.P. Brans,ed.,‘OperationalResearch’81. Proceedingsof the Ninth IFORSInternationalConference.North-Holland,Amsterdam,Netherlands’,pp.405–420,1981.
Abstract. A vector optimizationproblemis formulatedand discussedwhich containsone quadraticand
several linear objective functions.It is shown that suchproblemsmay be solved easily in differentways.
Algorithmsfor thestrictly concave casearepresented.
F. Riciniello. A survey on themethodsof linearandquadraticprogramming.NoteRecensionie Notizie, 22(2–3),270–315,1973.
Abstract. Somemethodsof linear andquadraticprogrammingare reviewed. Attention is focusedon the
algorithmswhich have beenrecognizedasthemostsuitablefor computerimplementationbecauseof their
limited memoryoccupancy andlow sensitivity to roundoff errors.
N. L. Ricker. Useof quadraticprogrammingfor constrainedinternalmodelcontrol. IndustrialandEngineeringChemistry, ProcessDesignandDevelopment, 24(4), 925–936,1985.
Abstract. Absoluteboundson control actionandothercontrol systemconstraintscanbe handledconve-
niently througha combinationof quadratic-programmingandmultivariableInternalModel Control (IMC).
Threealternative simplex-type quadraticprogramming(QP) algorithmsare considered,and the resulting
IMC algorithmsareappliedto thecontrolof asimulatedmultiple-effect evaporationprocess.Overallcontrol
qualityis excellent;it degradesverylittle whentheprocessis operatednearaconstraintonthecontrolaction.
Calculationsrequiredfor anadjustmentof the controlactionareessentiallyinstantaneouson a PDP11/60
minicomputer. The resultsof the QPcomparisonsuggestthatcertainQPalgorithmscantake advantageof
thespecialstructureof theIMC QPproblem.
K. Rim, R. Chand,and E. J. Haug. Analysis of unbondedcontactproblemsby meansofquadraticprogramming.Journal of OptimizationTheoryandApplications, 20(2), 171–189,1976.
N. I. M. GOULD & PH.L. TOINT 107
Abstract. Two-body, elastic,unbondedcontactproblemsare formulatedasquadraticprogrammingprob-
lems.Uniquenesstheoremsof quadraticprogrammingtheoryareappliedto show thatthesolutionof a con-
tactproblem,if oneexists,is uniqueandcanbereadilyfoundby themodifiedsimplex methodof quadratic
programming.A solution techniquethat is compatiblewith finite-elementmethodsis developed,so that
contactproblemswith complex boundaryconfigurationscanberoutinelysolved.A numberof classicaland
nonclassicalproblemsaresolved.Goodagreementis foundfor problemswith previously known solutions.
K. Ritter. Uber dasmaximum-problemfur nichtkonkavequadratischefuncktionen. Doctoral
dissertation,Albert-LudwigsUniversity, Frieberg, Germany, 1964.
K. Ritter. Stationarypointsof quadraticmaximumproblems.Zeitschrift fur Wahrscheinlichkeit-
stheorieundverwandteGebiete, 4, 149–158,1965.
K. Ritter. A methodfor solving maximumproblemswith a nonconcave quadraticobjective
function. Zeitschrift fur WahrscheinlichkeitstheorieundverwandteGebiete, 4, 340–351,
1966.
K. Ritter. A decompositionmethodfor structuredquadraticprogrammingproblems.Journal
of ComputerandSystemSciences, 1(3), 241–260,1967.
K. Ritter. On parametriclinearandquadraticprogrammingproblems.In R. W. Cottle,M. L.Kelmansonand B. Korte, eds,‘MathematicalProgramming.Proceedingsof the Inter-nationalCongresson MathematicalProgramming.North-Holland,Amsterdam,Nether-lands’,pp.307–335,1984.
Abstract. An algorithmis describedfor determiningtheoptimalsolutionof parametriclinearandquadratic
programmingproblemsas an explicit piecewise linear function of the parameter. Eachlinear function is
uniquelydeterminedby anappropriatesubsetof active constraints.For every critical valueof theparameter
anew subsethasto bedetermined.A simpleruleis givenfor addinganddeletingconstraintsfrom thissubset.
A. G. Robinson,N. Jiang,andC. S. Lerme. On thecontinuousquadraticknapsack-problem.
MathematicalProgramming, 55(1), 99–108,1992.
R.T. Rockafellar. Linear-quadraticprogrammingandoptimalcontrol.SIAMJournalonControlandOptimization, 25(3), 781–814,1987.
Abstract. A generalizedapproachis taken to linear andquadraticprogrammingin which dual aswell as
primal variablesmaybesubjectedto bounds,andconstraintsmayberepresentedthroughpenalties.Corre-
spondingproblemmodelsin optimal control relatedto continuous-timeprogrammingarethensetup and
theoremson duality and the existenceof solutionsarederived. Optimality conditionsareobtainedin the
form of aglobalsaddlepointpropertywhichdecomposesinto aninstantaneoussaddlepointconditiononthe
primalanddualcontrolvectorsat eachtime,alongwith anendpointcondition.
R.T. Rockafellar. Computationalschemesfor large-scaleproblemsin extendedlinear-quadraticprogramming.MathematicalProgramming, SeriesB, 48(3), 447–474,1990.
Abstract. Numericalapproachesaredevelopedfor solvinglarge-scaleproblemsof extendedlinear-quadratic
programmingthat exhibit Lagrangianseparabilityin both primal anddual variablessimultaneously. Such
problemsarekin to large-scalelinear complementaritymodelsasderived from applicationsof variational
inequalities,and they arisefrom generalmodelsin multistagestochasticprogramminganddiscrete-time
optimalcontrol.Becausetheirobjective functionsaremerelypiecewiselinear-quadratic,dueto thepresence
of penaltyterms,they do not fit a conventionalquadraticprogrammingframework. They have potentially
advantageousfeatures,however, whichsofarhavenotbeenexploitedin solutionprocedures.Thesefeatures
arelaid out andanalyzedfor their computationalpotential.In particular, a new classof algorithms,called
108 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
finite-envelopemethods,is describedthat doestake advantageof the structure.Suchmethodsreducethe
solutionof a high-dimensionalextendedlinear-quadraticprogramto thatof a sequenceof low-dimensional
ordinaryquadraticprograms.
R. T. Rockafellar. Large-scaleextendedlinear-quadraticprogrammingand multistageopti-
mization. In ‘Advancesin NumericalPartial Dif ferentialEquationsandOptimization:
Proceedingsof theFifth Mexico-UnitedStatesWorkshop’,Vol. 2, pp.247–261,1991.
R. T. Rockafellar. A simplex-active-setalgorithmfor piecewise quadraticprogramming. In
D.-Z. Du andJ.Sun,eds,‘Advancesin OptimizationandApproximation’,pp.275–292,
Kluwer AcademicPublishers,Dordrecht,TheNetherlands,1994.
R.T. RockafellarandR.J.-B.Wets.A dualsolutionprocedurefor quadraticstochasticprograms
with simplerecourse.TechnicalReportCP-83-017,InternationalInstitute for Applied
SystemsAnalysis,Vienna,Austria,1983.
R. T. RockafellarandR. J. B. Wets. Linear-quadraticprogramming-problemswith stochastic
penalties—thefinite generationalgorithm. Lecture Notesin Control and Information
Sciences, 81, 545–560,1986.
J.E.Rogers,P. T. Boggs,andP. D. Domich.A predictor-corrector-o3dformulationfor quadratic
programming.TechnicalReportNISTIR 5020,NationalInstituteof StandardsandTech-
nology, USA, 1994.
J. S. Rogersand M. Rolko. A quadraticprogrammingmodel for planninggenerationandinter-utility transmission.InternationalJournalof ElectricalPowerandEnergySystems,14(1), 18–22,1992.
Abstract. A new quadraticprogrammingapproachto planningthecoordinatedexpansionof generationand
transmissionbetweentwo electricutilities is described.Earliermethodsusea stepfunctionto approximate
theloaddurationcurve.Theresultinglinearprogrammingmodelswork well if theshapeof theloadcurve is
not changedby the installationof equipment.However, expansionof inter-utility transmissionchangesthe
shapeof theloadcurve whichmustbesuppliedby thegenerationof eachutility, andsoamoreflexible type
of approximationis needed.Theauthorsusea piecewise linearapproximationto theloadcurve to construct
aquadraticprogrammingmodel.Realsystemdataareusedto comparetheresultsfrom thenew modelwith
thosefrom theearliertypeandwith resultsfrom ananalyticalmodelthatgivesanexactsolution.
J. B. Rosenand P. M. Pardalos. Global minimization of large-scaleconstrainedconcave
quadraticproblemsby separableprogramming.MathematicalProgramming, 34(2),163–
174,1986.
V. RuggieroandL. Zanni. A modifiedprojectionalgorithmfor largestrictly-convex quadraticprograms.Journalof OptimizationTheoryandApplications, 104(2), 281–299,2000.
Abstract. In this paper, we proposeamodifiedprojection-typemethodfor solvingstrictly-convex quadratic
programs.This iterative schemerequiresessentiallythe solutionof an easyquadraticprogrammingsub-
problemanda matrix-vector multiplication at eachiteration.The main featureof the methodconsistsin
updatingtheHessianmatrix of thesubproblemsby a convenientscalingparameter. Theconvergenceof the
schemeis obtainedby introducinga correctionformulafor thesolutionof thesubproblemsandvery weak
conditionson the scalingparameter. A practicalnonexpensive updatingrule for the scalingparameteris
suggested.Theresultsof numericalexperimentationenablethisapproachto becomparedwith someclassical
projection-typemethodsandits effectivenessasa solver of largeandvery sparsequadraticprogramsto be
evaluated.
N. I. M. GOULD & PH.L. TOINT 109
M. H. Rusin. A revisedSimplex methodfor quadraticprogramming. SIAM Journal on Ap-plied Mathematics, 20(2), 143–160,1971.Seealso,Bulletin of theOperationsResearchSocietyof America,volume18,pageB17,1970.
Abstract. A computationalmethodfor solvingquadraticprogrammingproblemsis presentedwhichreduces
to therevisedsimplex methodfor linearprogrammingwhentheobjective functionis linear. Thebasismatrix
which is maintainedis symmetricand may vary in size from iteration to iteration.Comparisonson test
problemsindicatethatthemethodis moreefficient thanotheravailablemethods.
R. S.Sacher. A decompositionalgorithmfor quadraticprogramming.MathematicalProgram-ming, 18(1), 16–30,1980.
Abstract. A decompositionalgorithmusing,Lemke’s methodis proposedfor thesolutionof quadraticpro-
grammingproblemshaving possiblyunboundedfeasibleregions.The feasibleregion for eachmasterpro-
gramis ageneralizedsimplex of minimal size.TThispropertyis maintainedby adroppingprocedurewhich
doesnot affect thefinitenessof the convergence.Thedetailsof thematrix transformationsassociatedwith
anefficient implementationof thealgorithmaregiven.Encouragingpreliminarycomputationalexperience
is presented.
H. D. Sahinoglou,P. M. Pardalos,andI. M. Roussos.Characterizationsof globalminima in
nonconvex quadraticprogramming.In ‘Proceedingsof the19thAnnualPittsburh Con-
ferenceon Modelling andSimulationModelingandSimulation’,pp.1823–1828,1988.
T. Satoh,T. Okita,andM. Itoh. Fastcapacitymethodfor solvingconvex quadraticprogrammingproblem. Transactionsof theSocietyof InstrumentandControl Engineers, 32(4), 587–595,1996.
Abstract. The quadraticprogramming(QP) problemis a naturalgeneralizationof well-known linear pro-
gramming(LP) problem.TheQPproblemis formulatedasaproblemto minimizea linearobjective function
subjectto a systemof linear equalitiesand inequalities.The continuationmethodis a powerful tool for
solvinga systemof nonlinearequations.In thecontinuationmethod,thecontinuationparameterbetais in-
troducedinto the original problemP anda parameterizedproblemP( beta) is constructed.Assumingthat
the optimal solutionto P(P),for beta=0, canbe trivially or easily found, andthat the optimal solutionto
P β , for β 1, will bethedesiredsolutionto theoriginalproblemP, a trajectoryof thesolutionto P β is
followedby increasingthevalueof betafrom 0 to 1. In this paper, basedon thecontinuationmethod,a path
following methodcalledthecapacitymethodis constructedfor solvingtheQPproblem.Thebasicideaof the
capacitymethodis to follow a trajectoryof theKuhn-Tucker pointwhenthevalueof betais increasedfrom
0 to 1. It is shown thatthetrajectoryis expressedasapiecewise-linearfunctionof beta. Thereforetheopti-
mal solutioncanbefoundvery efficiently by following thepiecewise-lineartrajectory. In this paper, firstly,
theassumptionthat the right-hand-sidevectorof the constraintequationsmustbe positive, which restricts
theapplicabilityof thecapacitymethod,is droppedanda methodfor transformingany QPproblemto the
standardform of thecapacitymethodis presented.Secondly, thebasismatrix of QP is factorizedby using
anLDLT factorization,andthefactorizationis efficiently updatedby usingmatrix modificationtechniques.
Thentheproposedmethodis appliedto sometestproblems,andnumericalresultsindicatetheeffectiveness
of themethod.
M. A. Saunders.Stablereductionto KKT systemsin barriermethodsfor linearandquadratic
programming.TechnicalReportSOL96-3,Departmentof OperationsResearch,Stanford
University, California,USA, 1996.
C. Schmidand L. T. Biegler. Quadraticprogrammingmethodsfor reducedHessianSQP.ComputersandChemicalEngineering, 18(9), 817–832,1994.
Abstract. ReducedHessiansuccessive quadraticprogramming(SQP)is well suitedfor thesolutionof large-
scaleprocessoptimizationproblemswith many variablesandconstraintsbut only few degreesof freedom.
110 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Thepreprocessingphasedeterminesaninitial consistentpoint, to selecta nonsingularsetof basisvariables
andto identify linear dependency amongthe equality constraints.Fourer’s (1985,1989)piecewise-linear
simplex techniquesallow us to solve a smallerinitialization problemmoreefficiently thanis possiblewith
standardsimplex techniques.Wepresentanew QPsolver, QPKWIK, basedona dualalgorithmwhichonly
requirestheinverseCholesky factorof theHessianmatrixto besupplied.Theresultingsolutiontechniquefor
theQPsubproblemis O n2 with respectto thedegreesof freedomof theproblem.Further, theunconstrained
optimumis dualfeasible,whichprecludestheneedfor phaseI calculations,andmakesthismethodsuperior
even for problemswith few degreesof freedom.QPKWIK hasbeenimplementedso as to enhancethe
efficiency of theactive setidentificationandis alsoableto determinea searchdirectionwheninfeasibleQP
subproblemsareencounteredby relaxingtheequalityconstraintswithoutviolating thesimpleboundsonthe
variables.Finally, numericalresultsareincludedto illustratetheadvantagesof theproposedtechniquesand
to assesstheoverall performanceof thereducedHessianmethod.Thisapproachis especiallywell-suitedfor
processandreal-timeoptimizationproblems.We demonstratethis on several distillation andfractionation
problems.
U. Schmitz,A. Donati,T. L. James,N. B. Ulyanov, andL. J.Yao. Smallstructuralensembles
for a 17-nucleotidemimic of the tRNA TYC-loop via fitting of dipolar relaxationrates
with thequadraticprogrammingalgorithm.Biopolymers, 46, 329–342,1998.
I. E. SchochetmanandR. L. Smith. Solutionexistencein infinite quadraticprogramming.InA. V. Fiacco,ed., ‘Lecture Notesin PureandApplied Mathematics’,Vol. 195.MarcelDekker, 1998.
Abstract. We consideran infinite quadraticprogrammingproblemwith positive semi-definitequadratic
costs,equality constraintsand unboundedvariables.Sufficient conditionsare given for thereto exist an
optimalsolution.Specifically, werequirethat(1) thecostoperatorbestrictly positivedefinitewhenrestricted
to theorthogonalcomplementof its kernel,and(2) theconstraintoperatorhaveclosedrangewhenrestricted
to thekernelof thecostoperator. Condition(1) is shown to beequivalentto thespectrumof therestrictedcost
operatorbeingboundedaway from zero.Similarly, condition(2) is equivalent to theminimummodulusof
therestrictedconstraintoperatorbeingpositive. In thepresenceof separability, wegiveasufficientcondition
for (2) to hold in termsof finite dimensionaltruncationsof the restrictedconstraintoperator. We apply
our resultsto a broadclassof infinite horizonoptimizationproblems.In this setting,thefinite dimensional
truncationscan be consideredto be finite dimensionalapproximationsto our problemwhoselimit, in a
somewhatformalsense,is our infinite dimensionalproblem.Eachof theseapproximationshasproperties(1)
and(2) by virtue of their finite-dimensionality, i.e., eachadmitsanoptimalsolution.However, our infinite
dimensionalproblemmaynot.Thus,we give sufficient conditionsfor our problemto alsoadmitanoptimal
solution.Finally, we illustrate this applicationin the caseof an infinite horizonLQ regulator problem(a
productionplanningproblem).
I. E. Schochetman,R. L. Smith,andS. K. Tsui. Solutionexistencefor time-varying infinite-horizon quadraticprogramming. Journal of MathematicalAnalysisand Applications,195(1), 135–147,1995.
Abstract. Weconsidera general,time-varying, infinite horizon,purequadraticprogrammingproblemwith
positive-definitecostmatricesandunboundeddecisionvariables.Sufficientconditionsareprovidedfor there
to exist anoptimalsolution.Specifically, we show that if theeigenvaluesof the costmatricesarebounded
away from zero, thena (unique)optimal solutionexists. We apply our resultsto the infinite horizonLQ
tracker problemin optimalcontroltheory.
M. Schocm.Uberdie aquivalenzderallgeminenquadratischeroptimierungsaufgabezu einer
parametrischenkomplimentaritarenoptimierungsaufgabe.Mathematische Operations-
forschungundStatistik,SerieOptimization, 15, 211–216,1984.
N. I. M. GOULD & PH.L. TOINT 111
L. Schrage.Integer, andQuadratic Programmingwith LINDO. TheScientificPress,third edn,
1986.
P. Scobey andD. G.Kabe.Directsolutionsto somelinearandquadraticprogrammingproblems.IndustrialMathematics, 29(2), 59–75,1979.
Abstract. Someresultsof univariatenormallinear regressiontheoryareusedto presentdirectsolutionsto
somelinearandquadraticprogrammingproblems.Thelinearprogrammingproblemsincludetheknapsack
problem,theextremepointlinearprogrammingproblem,andtheinterval linearprogrammingproblem.Some
inaccuraciesin interval linearprogrammingproblemsandquadraticprogrammingproblemsarepointedout.
P. D. Scott.A quadraticprogrammingdualalgorithmfor minimaxcontrol. IEEE Transactionson AutomaticControl, AC-20(3), 434–435,1975.
Abstract. Theoptimalcontrolproblemwith peakweightingon trajectoryerror is relatedto quadraticpro-
grammingthroughaduality transformation.A seriesof finite dimensionalquadraticprogramsyieldsfinitely
convergentsolutionsfrom whichtheoptimalcontrolmayberecovered.Eachprogramyieldsupperandlower
boundson theoptimalcost.
J. Semple. Infinite positive-definitequadraticprogrammingin a Hilbert space. Journal ofOptimizationTheoryandApplications, 88(3), 743–749,1996.
Abstract. This note generalizesthe resultsof Benson,Smith, Schochetman,and Bean(1995) regarding
the minimizationof a positive-definitefunctionalover the countableintersectionof closedconvex setsin
a Hilbert space.A finite approximatingsubproblemfor the generalcaseis shown to have the samestrong
convergencepropertiesof the earlier work without any of the specializedstructuresimposedtherein.In
particular, the currentdevelopmentdoesnot rely on any propertiesof L2 anddoesnot requirethe Hilbert
spaceto beseparable.
M. SernaandF. Xhafa. On theparallelapproximabilityof someclassesof quadraticprogram-ming. TechnicalReportR95-58,Departamentde Llenguatgesi SistemesInformatics,UniversitatPolitecnicadeCatalunya,Spain,1995.
Abstract. In thispaperweanalyzetheparallelapproximabilityof two specialclassesof QuadraticProgram-
ming. First, we considerConvex QuadraticProgramming. We show that the problemof Approximating
Convex QuadraticProgrammingis P-complete.We alsoconsidertwo approximationproblemsrelatedto
it, SolutionApproximationandValueApproximationandshow bothof thesecannotbesolved in NC, un-
lessP NC. Secondly, on thepositive side,we show thatwe have anNC ApproximationSchemefor those
instancesof QuadraticProgrammingthatare”smooth” and”positive.” Thenwe show how to extendthere-
sult for positive instancesof boundeddegreeSmoothIntegerProgrammingproblems.Finally, we formulate
severalcombinatorialproblemsaspositive QP(or positive IntegerPrograms)in packing/covering form and
show thatthepresentedtechniquescanbeusedto obtainNC ApproximationSchemesfor ”dense”instances
of suchproblems.
M. SernaandF. Xhafa.Theparallelapproximabilityof asubclassof quadraticprogramming.In‘Proceedings.1997InternationalConferenceon ParallelandDistributedSystems.IEEEComput.Soc,Los Alamitos,CA, USA’, pp.474–481,1997.
Abstract. In this paperwe dealwith theparallelapproximabilityof a specialclassof QuadraticProgram-
ming (QP),calledSmoothPositive QuadraticProgramming.This subclassof QP is obtainedby imposing
restrictionson thecoefficientsof theQPinstance.TheSmoothnessconditionrestrictsthemagnitudesof the
coefficientswhile thepositivenessrequiresthatall thecoefficientsbenon-negative. Interestingly, evenwith
theserestrictionsseveral combinatorialproblemscanbe modeledby SmoothQP. We show NC Approxi-
mationSchemesfor the instancesof SmoothPositive QP. This is doneby reducingthe instanceof QP to
an instanceof Positive Linear Programming,finding in NC an approximatefractionalsolution to the ob-
tainedprogram,andthenroundingthefractionalsolutionto anintegerapproximatesolutionfor theoriginal
112 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
problem.Thenwe show how to extendthe result for positive instancesof boundeddegreeto SmoothIn-
teger Programmingproblems.Finally, we formulateseveral importantcombinatorialproblemsasPositive
QuadraticPrograms(or Positive Integer Programs)in packing/covering form andshow that the techniques
presentedcanbeusedto obtainNC ApproximationSchemesfor ”dense”instancesof suchproblems.
D. G. Shankland.Quadraticprogrammingusinggeneralizedinverses.Technicalreport,AirForceInst.Tech, Wright PattersonAFB, OH, USA, 1975.
Abstract. A methodfor computingtheoptimumvalueof aquadraticfunctionalsubjectto linearinequalities
rapidly ascertainswhich, if any, of theinequalitiesarebindingat theoptimumpoint.Themethodresembles
thatof H. Theil andC. VandePanne,but nocombinatorialanalysisneedbeperformedto isolatethebinding
constraints.All violatedconstraintsareimposedasequalities,andthosewith positiveLagrangianmultipliers
areretained.Contradictoryequalitiesareautomaticallyresolved by theuseof thegeneralizedinverse.The
methodappearsmostusefulin systemswith largenumbersof variablesandconstraints.
D. G. Shankland. A numericallyefficient procedurefor the Theil-Van de Pannequadraticprogrammingmethod. Journal of OptimizationTheoryand Applications, 31(1), 117–123,1980.
Abstract. A procedureis given for implementingthe Theil-Van de Pannealgorithm, which utilizes the
Cholesky decompositionto reducethecomputationsinvolvedin matrix inversions.
J. K. SharmaandS. Kanti. Indefinitequadraticprogrammingand transportationtechnique.Indian Journalof PureandAppliedMathematics, 8(9), 1029–1031,1977.
Abstract. A technique,similar to the transportationtechniquein linearprogramming,is describedto min-
imize a locally indefinitequadraticfunction.Theproblemis attacked directly startingfrom a basicfeasible
solutionandthe conditionsunderwhich the solutioncanbe improved have beenindicated.Conditionsfor
localoptimality have beenobtained.
J.F. SharpandK. M. Suk. A quadraticprogrammingplanningmodel.EngineeringEconomist,20(1), 71–77,1974.
Abstract. Variousauthorshave suggestedusinga quadraticprogrammingmodelof the firm. A quadratic
programmingapproachhasbeenusedto study the optimal useof milk in the Netherlands.Thereseems
to be a lack of otheractualapplicationsreportedin the literature.The authorshave successfullyapplieda
quadraticprogrammingmarketingmodelto anindustrialfirm. It hasseveralfeaturesnot includedin themilk
application.Theseinclude:morethanonemarket for thesameproduct,limits on therangeof applicability,
andlegal restrictions.A simplifiedversionof thismodelis givenbelow.
R. Shen.Reactive power optimizationin power systemquadraticprogrammingmethod.Pro-ceedingsof theChineseSocietyof ElectricalEngineering., 6(5), 40–48,1986.
Abstract. A quadraticprogrammingmethodof comprehensive reactivepoweroptimizationin powersystems
is proposed.With realpower lossastheobjective functionandoperatingvariablesasconstraints,a mathe-
maticalmodelfor comprehensive reactive power optimizationis setupby sensitivity relationsbetweencon-
trol andstatevariablesin thepower system.Theoptimal locationsandcapabilitiesof compensatedreactive
power, optimal terminalvoltagesof generatorsandoptimal ratiosof on-loadtransformertap-changersare
determinedby optimization.Basedoneconomicanalysis,amathematicalmodelof multiobjective program-
ming for comprehensive reactive power optimizationis setup with realpower lossandtotal compensation
capabilityasobjective functions.Threesystemsareoptimizedby thesemethodsandresultsaregiven and
comparedwith thoseof linearprogrammingmethods.
H. D. SheraliandA. Alameddine.Reformulation-linearizationtechniquefor bilinearprogram-
ming problems.Journalof GlobalOptimization, 2(4), 379–410,1992.
N. I. M. GOULD & PH.L. TOINT 113
H. D. SheraliandC. H. Tuncbilek.A reformulation-convexificationapproachfor solvingnon-convex quadraticprogrammingproblems. Journal of Global Optimization, 7(1), 1–31,1995.
Abstract. In this paper, we considerthe classof linearly constrainednonconvex quadraticprogramming
problems,andpresentanew approachbasedonanovel Reformulation-Linearization/Convexification Tech-
nique.In this approach,a tight linear (or convex) programmingrelaxation,or outer- approximationto the
convex envelopeof theobjective functionover theconstrainedregion,is constructedfor theproblemby gen-
eratingnew constraintsthroughtheprocessof employing suitableproductsof constraintsandusingvariable
redefinitions.Varioussuchrelaxationsareconsideredandanalyzed,includingonesthat retainsomeuseful
nonlinearrelationships.Efficientsolutiontechniquesarethenexploredfor solvingtheserelaxationsin order
to derive lower andupperboundson theproblem,andappropriatebranching/partitioning strategiesareused
in concertwith theseboundingtechniquesto derive a convergentalgorithm.Computationalresultsarepre-
sentedon a setof testproblemsfrom the literatureto demonstratethe efficiency of the approach.(Oneof
thesetestproblemshadnot previously beensolved to optimality.) It is shown that for many problems,the
initial relaxationitself producesanoptimalsolution.
W. M. Shi. Comparisonof optimizationwith linearandquadraticprogrammingin 2nd-order
design.In ‘Collectionof 8thInternationalSymposiumonGeodeticComputation’,Vol. 9,
pp.157–163,1991.
J.K. Shim. A survey of quadraticprogrammingapplicationsto businessandeconomics.Inter-nationalJournalof SystemsScience, 14(1), 105–115,1983.
Abstract. Oneof theshortcomingsof linearprogramminglies in its linearity assumption,primarily in the
objective function.This compelsoneto work with a constantmarginal rateof substitutionandconstantre-
turn to scale.However, this assumptionis at variancewith theeconomists’preferencepostulates.Thepaper
presentsa survey of quadraticprogrammingpracticesin corporateandeconomicplanningformulations.It
examinesa variety of applicationsof quadraticprogramming-portfolioselection,monopolists’profit max-
imization, inequalityconstrainedleast-squaresestimation,spatialequilibrium analysis,goal programming
with quadraticpreferences,andoptimaldecisionrules.
Y. Shimazu,M. Fukushima,andT. Ibaraki. A successive over-relaxationmethodfor quadraticprogrammingproblemswith interval constraints. Journal of the OperationsResearchSocietyof Japan, 36(2), 73–89,1993.
Abstract. Hildreth’salgorithm(1957)isaclassicaliterativemethodfor solvingstrictly convex quadraticpro-
grammingproblems,whichusestherowsof constraintmatrixjustoneatatime.Thisalgorithmis particularly
suitedto largeandsparseproblems,becauseit actsuponthegivenproblemdatadirectly andthecoefficient
matrix is never modifiedin thecourseof the iterations.Theoriginal Hildreth’s algorithmis mathematically
equivalentto Gauss-Seidelmethodappliedto thedualof thegivenquadraticprogrammingproblem.In this
paper, we proposea successive overrelaxationmodificationof Hildreth’s algorithmfor solvinginterval con-
strainedquadraticprogrammingproblems.We prove globalconvergenceof thealgorithmandshow thatthe
rateof convergenceis linear. Computationalresultsarealsopresentedto demonstratetheeffectivenessof the
algorithm.
R. I. Shrager. Quadraticprogrammingfor nonlinearregression.Communicationsof theACM,15(1), 41–45,1972.Seealso,CollectedAlgorithmsfromACM, 2395.
Abstract. A quadraticprogrammingalgorithmis describedfor usewith themagnifieddiagonalmethodof
nonlinearregressionwith linearconstraints.Theregressionmethodis publishedin JACM, July 1970.
V. Sima.Algorithm podqp—Solvingpositivedefinitequadraticprogrammingproblems.Stud-iesin InformaticsandControl, 1(1), 1992.
114 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. An efficientandnumericallystabledualalgorithm-PODQP-for solvingpositivedefinitequadratic
programmingproblemsis briefly described.Both inequalityandconstraintsaredealtwiyh. The algorithm
employs anactive setstrategy andcanexploit a priori informationaboutan initial active set.In particular,
the algorithm is strongly recommendedfor solving nonlinearlyconstrainedoptimizationproblemsusing
projectedLagrangiantechniques.Orthogonaltransformationsareusedfor updatingthematricesandmatrix
factorizationsaftereachchangeperformedin theactive set.
S. Simunovic and S. Saigal. Frictionlesscontactwith BEM using quadraticprogramming.Journalof EngineeringMechanics-ASCE, 118(9), 1876–1891,1992.
Abstract. The contactsurface,with its accompanying load transfer, may well constitutethe critical factor
in a structuralmember. Thus it is essentialto be able to performcontactstressanalysisof a component
accuratelyandefficiently. Considerableresearcheffort is representedin the literaturefor contactanalysis
usingfinite elements.To obtain reliable resultsin the contactzone,it is necessaryto provide a very fine
discretizationin thatzone.Many distinctcontactzonesmayexist. which may force theentiredomain,and
not just the contactzones,to be discretizedfinely. This generallyleadsto anexcessive numberof degrees
of freedom(dof), resultingin anuneconomical,andsometimesintractable,analysis.Theboundaryelement
method(BEM), whichdealswith thediscretizationof only theboundaryof thestructurebeinganalyzed,may
beusedto circumventthesedifficultiesandprovide accurate,economicalresults.While afinediscretization
of the contactzoneis still unavoidable,the BEM leadsto a smallernumberof dof’s becausethe rest of
the boundaryneednot have a fine mesh.The problemof frictionlesscontactbetweenan elasticbody and
a rigid surfaceis formulatedasan optimizationproblem.Threedistinct functionsaredefinedin termsof
theunknown variables(displacementsandtractions)correspondingto thecontactsurface,andexpressedas
quadraticobjective functionsthatareto beminimized.Thesolutionis obtainedusingthestandardquadratic
programmingtechniquesof optimization.A numberof exampleproblemswith straightandcurvedcontact
boundariesweresolved.Thepresentformulationswerevalidatedthroughcomparisonof the testproblems
with existing alternative solutions.
S. Simunovic andS. Saigal. Frictionlesscontactwith BEM usingquadraticprogramming—
closure.Journalof EngineeringMechanics-ASCE, 119(12),2540,1993.
S.Simunovic andS.Saigal.Frictionalcontactformulationusingquadraticprogramming.Com-putationalMechanics, 15(2), 173–187,1994.
Abstract. A new solutionprocedurefor contactproblemsin elasticitywith prescribednormaltractionson
contactsurfacehasbeenproposedin thispaper. Theprocedureis basedontheboundaryelementmethodand
quadraticprogramming.It is next usedin a two stepsolutionalgorithmfor theanalysisof contactproblems
with friction. Severalnumericalexamplesarepresentedandcomparedwith resultsobtainedusingalternative
solutionmethods.
S. Simunovic andS. Saigal. Quadratic-programmingcontactformulation for elasticbodiesusingboundary-elementmethod.AIAA Journal, 33(2), 325–331,1995.
Abstract. A methodfor theanalysisof contactof deformablebodiesbasedontheboundaryelementmethod
(BEM) hasbeenpresentedin this paper. The contactproblemis statedin the form of a convex quadratic
programming(QP) problemwritten in termsof the contacttractionson the contactsurface.A strategy for
theincorporationof theBEM contactanalysisinto modelswhosedomainmaybediscretizedusingthefinite
elementmethod(FEM) hasbeeninvestigated.A discussionconcerningthemeritsof theproposedapproach
is providedandseveralexamplesarepresentedto illustratethevalidity of themethod.
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N. I. M. GOULD & PH.L. TOINT 115
Abstract. In thispaperwesurvey someresultsconcerningpolynomialand/orstronglypolynomialsolvability
of someclassesof quadraticprogrammingproblems.Thediscussiononpolynomialsolvability of continuous
convex quadraticprogrammingis followedby acoupleof modelsfor quadraticintegerprogrammingwhich,
dueto their specialstructure,allow polynomial (or even stronglypolynomial)solvability. The theoretical
merit of thoseresultsstemsfrom the fact that a running time (i.e. the numberof elementaryarithmetic
operations)of astronglypolynomialalgorithmis independentof theinputsizeof theproblem.
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Abstract. Variousstatisticaltechniqueshave beenemployed in ’activation analysis’in orderto provide a
bettermeansof estimatingtheamountsof variouspurechemicalelementscontainedin anunknown mixture.
In particular, the methodof leastsquareshasbeenemployed extensively. However, for the mostpart, the
usualleastsquaresapplicationsin activationanalysishaveutilized theordinarymatrixmodelY Xbeta e,
underthe ’error’ assumptions(a) zeromeans,(b) variancesproportionalto Y, and(c) zerocovariances.In
additionto thefact thatassumptions(b) and(c) may leadto erroneousresults,theusualapplicationsallow
only pointestimation,with noprovision for confidenceintervalsandtestsfor modelgoodnessof fit. Further,
theusualapplicationsfail to eliminatethedrawbackthatnegative coefficientsaresometimesobtained.The
presentpapersetsforth aniterative quadraticprogrammingestimationprocedurethatnotonly eliminatesthe
necessityfor assumptions(b) and(c), but alsoalleviatestheotherabove-mentioneddifficulties.
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programming(LCP) format. The LCP model forecastsUnited Statesoil, naturalgas,andcoal pricesand
quantitiesfor boththedemandandthesupplyside.Forecastsarepresentedonaregionalizedlevel. Validation
of theLCPmodelis achievedby forecastinghistoricalpricesandquantities.In addition,themodelexamines
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116 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
Abstract. Theproblemof allocatingthe total systemmargin amongall thegeneratorsin a power systems,
taking into considerationthe maximumpost-disturbancefrequency deviation, is formulatedasa quadratic
programmingproblem.Theoptimizationproblemusesthesensitivity of thefrequency deviation to changes
in eachgenerator’s margin in the allocationof total systemmargin. The quadraticprogrammingproblem
is solved via the Dantzig-Wolfe decompositiontechniqueusing a sequenceof linear programmingsub-
problems.A significant improvementis found in the post-disturbanceperformanceof the power system
in comparisonwith conventionalmargin allocation.Numericalexampleson two testsystemsareincluded.
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constraint.Kuhn-Tucker Conditionsarethenappliedto obtaineitheranunconstrainedoptimizationproblem
or anequalityconstrainedproblem.Thisresultsin theneedto solveonly asystemof linearequationsin order
to obtainthesolutionto thesurrogatedproblem.An iterative procedureis given for obtaininga surrogated
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solveaspecialQPproblemwith anM-matrixastheHessianandsimpleboxconstraints.Theproblemarises
astheresultof anapproximationof a partialDirichlet problemwith obstacleby meansof thefinite element
method.It is proved that the propertiesof the Hessian(beinganM-matrix) imply that if the upperbounds
upon variablesdoesnot appearin the problemand the startingpoint is equal to the lower boundupon
variablesthenthenumberof iterationsof theNewtonprojectionmethoddoesnotexceedn (wheren denotes
the size of the problem); that if the upperboundsdo not exist, and the startingpoint is feasible,and ε0
(parameterusedin thedefinitionof theworking setof active constraints)is forcedto beequalto 0 in step
N. I. M. GOULD & PH.L. TOINT 117
1, thenthenumberof iterationsis not greaterthann 1; andthat if thereexist lower andupperboundson
theproblemvariablesandif someadditionalrequirementsareintroducedinto thealgorithmthentheNewton
projectionmethodgeneratesthesamesequenceof pointsasthatof J. S. Pang(1976).Theanalysiscarried
out indicatessomecorollariesallowing a simplificationof theformulationof theNewton projectionmethod
for theconsideredclassof problems.
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Abstract. Thebehaviour of thegradientprojectionmethodfor thequadraticprogrammingwith M-matrices
and lower boundson variablesis analysed.It is shown that the Chandrasekaranmethodfor solving such
problemsis simply a realizationof theNewton projectionmethodif for the latteronethestartingpoint has
componentsequalto lower boundsonvariables.
V. I. Starostenko. Constantalgorithmsof quadraticprogrammingandsolutionof the inverseproblemsof gravimetry relative to densities.Geofizicheskii-Sbornik., 64, 52–57,1975.
Abstract. Theproblemsof quadraticandlinearprogrammingareformulatedandtheir solutionsby means
of quadraticprogrammingalgorithmsregulatingaccordingto A. N. Tikhonov arepresented.
F. SteinerandL. Zilahysebess.Gravity interpretationwith theaidof quadraticprogramming—
discussion.Geophysics, 48(10),1413–1414,1983.
T. Sugimoto,M. Fukushima,andT. Ibaraki. A parallel relaxationmethodfor quadraticpro-grammingproblemswith interval constraints. Journal of Computationaland AppliedMathematics, 60(1–2),219–236,1995.
Abstract. Optimizationproblemswith interval constraintsareencounteredin variousfieldssuchasnetwork
flowsandcomputertomography. As theseproblemsareusuallyvery large,they arenoteasyto solvewithout
taking their sparsityinto account.Recently”row-actionmethods”,which originatefrom the classicalHil-
dreth’s methodfor quadraticprogrammingproblems,have drawn muchattention,sincethey areparticularly
usefulfor largeandsparseproblems.Variousrow-actionmethodshave alreadybeenproposedfor optimiza-
tion problemswith interval constraints,but they mostlybelongto theclassof sequentialmethodsbasedon
the Gauss-SeidelandSORmethods.In this paper, we proposea highly parallelizablemethodfor solving
thoseproblems,which may be regardedasan applicationof the Jacobimethodto the dual of the original
problems.We prove convergenceof thealgorithmandreportsomecomputationalresultsto demonstrateits
effectiveness.
M. F. Sukhinin. Step-by-stepquadraticpenaltysolutionof a quadratic-programmingproblem.ComputationalMathematicsandMathematicalPhysics, 34(8–9),1125–1131,1994.
Abstract. An numericalalgorithmfor a quadraticprogrammingproblemis considered.Resultsof experi-
mentson thetransportationproblemaregiven.
J. L. Sullivan, J. W. Adams,andR. Roosta. The designof constrainedminimax FIR digitalfilters usingquadraticprogramming.ConferenceRecord of TheTwentyNinth AsilomarConferenceonSignals,SystemsandComputers.IEEEComput.SocPress,LosAlamitos,CA,USA, 2, 826–830,1996.
Abstract. We (seeAdamset al., ibid., pp. 314, 1994)presentedin an earlierarticle a new quadraticpro-
grammingalgorithmthatcandesignconstrainedleast-squaresFIR digital filters. In this articlewe presenta
new quadraticprogrammingalgorithmfor designingconstrainedminimaxFIR filters.Thenew algorithmis
betterthanthe Parks-McClellan(1973)algorithmbecauseit candesigna muchwider variety of filters. In
particular, it candominimaxoptimizationsubjectto arbitraryequalityandinequalityconstraints.
J.Sun.A convergenceproof for anaffine-scalingalgorithmfor convex quadraticprogrammingwithout nondegeneracy assumptions.MathematicalProgramming, 60(1), 69–79,1993.
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Abstract. This paperpresentsa theoreticalresult on convergenceof a primal affine-scalingmethodfor
convex quadraticprograms.It is shown that, as long as the stepsizeis lessthana thresholdvalue which
dependson theinput dataonly, Ye andTse’s interiorellipsoidalgorithmfor convex quadraticprogramming
is globally convergentwithout nondegeneracy assumptions.In addition,its local convergencerateis at least
linearandthedualiterateshave anergodicallyconvergentproperty.
J.Sun.OnpiecewisequadraticNewtonandtrust-regionproblems.MathematicalProgramming,76(3), 451–468,1997.
Abstract. Some recent algorithms for nonsmoothoptimization require solutions to certain piecewise
quadraticprogrammingsubproblems.Two typesof subproblemsareconsideredin this paper. Thefirst type
seekstheminimizationof acontinuouslydifferentiableandstrictly convex piecewisequadraticfunctionsub-
ject to linear equalityconstraints.We prove that a nonsmoothversionof Newton’s methodis globally and
finitely convergentin thiscase.Thesecondtypeinvolvestheminimizationof apossiblynonconvex andnon-
differentiablepiecewisequadraticfunctionover a Euclideanball. Characterizationsof theglobalminimizer
arestudiedundervariousconditions.Theresultsextendaclassicalresulton thetrustregion problem.
J.SunandH. Kuo. Applying a Newton methodto strictly convex separablenetwork quadraticprograms.SIAMJournalon Optimization, 8(3), 728–745,1998.
Abstract. By introducingquadraticpenaltyterms,a strictly convex separablenetwork quadraticprogram
canbe reducedto an unconstrainedoptimizationproblemwhoseobjective is a continuouslydifferentiable
piecewisequadraticfunction.A recentlydevelopednonsmoothversionof Newton’s methodis appliedto the
reducedproblem.The generalizedNewton directionis computedby an iterative procedurewhich exploits
the specialnetwork datastructuresthat originatedfrom the network simplex method.New featuresof the
algorithmincludetheuseof min-maxbasesandadynamicstrategy in computationof theNewtondirections.
Somepreliminarycomputationalresultsarepresented.Theresultssuggesttheuseof ”warmstart” insteadof
”cold start”.
J. SunandJ. Zhu. A predictor-correctormethodfor extendedlinear-quadraticprogramming.ComputersandOperationsResearch, 23(8), 755–767,1996.
Abstract. The saddlepoint form of extendedlinear-quadraticprogramscanbesolved by an interior point
path-following methodin polynomialtime.Thealgorithmmaytake advantageof theblockstructuresof cer-
tainproblemsarisingfrom optimalcontrolandstochasticprogramming.In addition,it needsnoline searches
andtreatsfully or not fully quadraticproblemsequally. Preliminarycomputationalresultsapparentlyshow
thatthealgorithmis effective in solvingaclassof two-stagestochasticprogrammingproblems.
S. M. Sun,H. S. Tzou, andM. C. Natori. Parametericquadraticprogrammingmethodfordynamiccontactproblemswith friction. AIAAJournal, 32(2), 371–378,1994.
Abstract. Basedon theparametricvariationalprinciple, a generalbut effective parametericquadraticpro-
grammingtechniquesatisfyingvariouscontactconditionsif establishedfor dynamicanalysisof contact
problemswith friction anddamping.Thediscretizationwith respectto time andspaceleadsto astaticlinear
complementaryproblem(LCP) for eachtime stepwhich is solved by a quadraticoptimizationalgorithm
suchastheLemkealgorithm,etc.Thus,theconvergenceandnumericalstabilityof thesolutioncanbeguar-
anteed.The substructurecondensationtechniqueis implementedto handlethe unknown contactboundary
conditionsothatthecomputationeffect is considerablyreduced.An anapplicationof themethodpresented,
threedynamiccontactexamplesandnumericalresultsaregiven.
S.SunderandV. Ramachandran.Designof recursive differentiatorsusingquadraticprogram-ming. In ‘Proceedingsof the36thMidwestSymposiumon CircuitsandSystems.IEEE,New York, NY, USA’, Vol. 1, pp.774–777,1993.
Abstract. A methodfor thedesignof first andhigher-degreerecursive differentiatorswith constantgroup-
delaycharacteristicsusinga least-squaresapproachis presented.In this method,a mean-squareerrorbased
on the differencebetweenthe desiredand actual frequency responseis formulatedin a quadraticform.
N. I. M. GOULD & PH.L. TOINT 119
Quadraticprogrammingis employed whereinthe constrainton stability is accommodatedto designstable
differentiators.Our methodis comparedwith the linear programming(LP) approachin termsof the com-
putationalcomplexity and the variation of magnitudeandgroup-delayerrorswith frequency. It is shown
thatthedifferentiatorsdesignedusingourmethodhaveamuchlowercomputationalcomplexity andsmaller
variationof themagnitudeandgroup-delayerrorwith frequency thanthosedesignedusingtheLP approach.
W. R. S.Sutherland,H. Wolkowicz, andV. Zeidan.An explicit linearsolutionfor thequadratic
dynamic- programmingproblem. Journal of OptimizationTheory and Applications,
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K. Swarup. Indefinite quadraticprogramming. Cahiers du Centre d’Etudesde Recherche
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K. Swarup.Programmingwith indefinitequadraticfunctionwith linearconstraints.Cahiersdu
Centre d’EtudesdeRechercheOperationalle, 8, 132–136,1966b.
K. Swarup.Quadraticprogramming.CahiersduCentred’EtudesdeRechercheOperationalle,
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B. A. Szaboand T. T. Chung. The quadraticprogrammingapproachto the finite elementmethod. InternationalJournal for NumericalMethodsin Engineering, 5(3), 375–381,1973.
Abstract. A finite elementapproximationtechniqueis describedin which the potentialenergy function
is minimized subjectto linear constraints.The constraintsrequiresatisfaction of all kinematicboundary
conditionsand interelementcontinuity conditions.An example, indicating the efficiency of the proposed
method,is presented.
K. Tabushi,S. Itoh, M. Sakura,Y. Kutsutaninakamura,T. A. Iinuma,andT. Arai. A methodfor calculatingthe optimum irradiation condition for intracavitary radiotherapy usingquadraticprogramming.Physicsin MedicineandBiology, 33(5), 515–527,1988.
Abstract. A methodof calculatingoptimum irradiation conditionsfor intracavitary radiotherapy using
quadraticprogramminghasbeenformulatedandthenmodifiedfor practicalapplication.Theallowablerange
of obtaineddose,which is usuallyfixed in advance,is automaticallycomputedto be assmall aspossible.
Thevarianceof theproductof theactivity andthe irradiationtime of the tandemsourceis alsominimised
to avoid theoccurrenceof cold and/orhot spots.Optimumirradiationconditionsfor conventionalintracav-
itary radiotherapy of carcinomaof the uterinecervix wereobtainedon the basisof isodosecurvespassed
throughthepointsA of theManchestersystem.Thosefor carcinomaof theotherorgansandspecialcasesof
carcinomaof theuterinecervixcanbedeterminedafterconsiderationof thetumourstate.
M. Takamoto,N. Yamada,Y. Kobayashi,H. Nonaka,and S. Okoshi. 0–1 quadraticpro-grammingalgorithmfor resourceleveling of manufacturingprocessschedules.Systemsand Computers in Japan, 26(10), 68–76,1995. Seealso,Transactionsof the Instituteof Electronics,InformationandCommunicationEngineers,volumeJ77D-II(10)pages2075–2082,1994.
Abstract. In industrialplant constructionscheduling,it is necessaryto minimize thefluctuation,themaxi-
mumpeakof thefluctuationandthemaximumpeakvalueof theamountof daily resources,which is calcu-
latedasthesumof daily resourcesfor eachprocess.Minimization of thefluctuationor themaximumpeak
valuecorrespondsto leveling thepile of resources.To performthis resourceleveling, we needto decideon
anobjective functionwhich is a monotonefunctionthatsimply increaseswith thedegreeof resourcelevel-
ing andthensolve theoptimizationproblemby fixing theprocessstartdateswhile minimizing theobjective
function. Industrialplant constructionis, however, in many casesa large-scaleschedulingwith an entire
120 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
periodof morethan1000daysandmorethan100processes,so it is very difficult to obtaina global opti-
mizationsolution.Wehavedevelopedanalgorithmwhichsolvesa large-scaleoptimizationproblemto level
necessaryresources.This algorithmcanquickly searchfor a goodsuboptimalsolutioncloseto the global
optimalsolutionof a0–1quadraticprogrammingproblem.Thealgorithmsearchesby repeatingapivot oper-
ationusingvariableselectionrulesfor resourceleveling.Weappliedthisalgorithmto large-scalescheduling
for anactualplantconstructionschedule,andsuccessfullyobtaineda practicalsuboptimalsolutionwithin a
few minutes(CPUpower: 28MIPS).Theresultssuggestthatthealgorithmis practicalfor resourceleveling
of large-scaleconstructionscheduling.
T. Takayamaand N. Uri. A note on spatialand temporalprice and allocationmodeling—
quadraticprogrammingor linearcomplementarityprogramming.Regional Scienceand
Urban Economics, 13(4), 455–470,1983.
A. Takeda,Y. Dai, M. Fukuda,andM. Kojima. Towardstheimplementationof successivecon-vex relaxationmethodfor nonconvex quadraticoptimizationproblems.ResearchReporton InformationSciencesB-347,Departmentof MathematicalandComputingSciences,Tokyo Instituteof Technology, Japan,1999.
Abstract. RecentlyKojima and Tuncel proposednew successive convex relaxationmethodsand their
localized-discretized variantsfor generalnonconvex quadraticprograms.Although an upperboundof the
objective function valuewithin a prior precisioncanbe found theoreticallyby solving a finite numberof
linearprograms,severalimportantimplementationproblemsremainunsolved.In thispaperwediscussthese
issues,presentpracticallyimplementablealgorithmsandreportnumericalresults.
N. N. Tam and N. D. Yen. Continuity propertiesof the Karush-Kuhn-Tucker point set inquadraticprogrammingproblems.MathematicalProgramming, 85(1), 193–206,1999.
Abstract. We obtainnecessaryandsufficient conditionsfor thesetof theKarush-Kuhn-Tucker pointsin a
canonicalquadraticprogrammingproblemto beuppersemicontinuousor lowersemicontinuouswith respect
to theproblemparameters.
K. Tammer. Possibilitiesfor theapplicationof theresultsof parametricprogrammingto solvingindefinitequadraticprogrammingproblems. Mathematische OperationsforschungundStatistik, 7(2), 209–222,1976.
Abstract. Somepossibilitiesfor the applicationof the resultsof the convex and nonconvex parametric
quadraticprogrammingto the computationof nonconvex quadraticprogrammingproblemsaregiven. In
thefirst casetheproblemcanbesolvedby solvinga stronglyconvex parametricquadraticproblemwith in
generalmorethanoneparameterandsomesmallerquadraticproblems,in thesecondcaseby solvinga one
parametricnonconvex quadraticproblem.
H. TamuraandY. Kihara. Multistagequadraticprogrammingfor discrete-timeoptimalcontrolwith stateandcontrolconstraints.SystemsandControl, 21(12),702–709,1977.
Abstract. A multistagequadraticprogrammingtechniqueis developedfor discrete-timelinear-quadratic(L-
Q) optimalcontrolproblemwith stateandcontrolconstraints.Theproblemis convertedto a linearprogram
with bilinear constraints.Then, taking advantageof the staircase-structureof equalityconstraints(system
equation),Dantzig-Wolfe decompositionprinciple is appliedrepeatedlyin eachstage,wherethedecompo-
sition techniquefor the bilinear constraintsis newly developedin this paper. The significantadvantageof
themultistagequadraticprogrammingtechniquein this paperis that it canhandlea largenumberof stages
without increasingthecomputationtime andstoragerequirementsenormously, andthat it canhandlestate
andcontrolconstraintswithout difficulty. Therefore,a substantialreductionof computationalburdenis ob-
tainedfrom theexisting discrete-timeoptimalcontrolalgorithms.Numericalexamplesfor comparisonwith
Wolfe’s quadraticprogrammingmethodareincluded.
N. I. M. GOULD & PH.L. TOINT 121
Y. TanandC. Deng.Solvingfor aquadraticprogrammingwith aquadraticconstraintbasedona neuralnetwork frame.Neurocomputing, 30(1–4),117–128,2000.
Abstract. In many applications,a classof optimizationproblemscalledquadraticprogrammingwith aspe-
cial quadraticconstraint(QPQC)oftenoccurs,suchasin thefieldsof maximum-entropy spectralestimation,
FIR filter designwith time-frequency constraints,and the designof an FIR filter bankwith a perfectre-
constructionproperty. In order to deal with this kind of optimizationproblemand to be inspiredby the
computationalvirtueof analogor dynamicneuralnetworks,a feedbackneuralnetwork is proposedfor solv-
ing for thisclassof QPQCcomputationproblemsin realtime.Thestability, convergenceandcomputational
performanceof theproposedneuralnetwork have alsobeenanalyzedandprovedin detail,soasto theoreti-
cally guaranteethecomputationaleffectivenessandcapabilityof thenetwork. Fromthetheoreticalanalyses,
it turnsoutthatthesolutionof aQPQCproblemis just thegeneralizedminimumeigenvectorof theobjective
matrixwith respectto theconstrainedmatrix.A numberof simulationexperimentshavebeengivento further
supportour theoreticalanalysisandto illustratethecomputationalperformanceof theproposednetwork.
H. TanakaandH. Lee. Interval regressionanalysisby quadraticprogrammingapproach.IEEETransactionsonFuzzySystems, 6(4), 473–481,1998.
Abstract. Whenwe uselinear programmingin possibilisticregressionanalysis,somecoefficients tendto
becomecrispbecauseof thecharacteristicof linearprogramming.On theotherhand,a quadraticprogram-
mingapproachgivesmorediversespreadcoefficientsthanalinearprogrammingone.Therefore,to overcome
thecrispcharacteristicof linearprogramming,we proposeinterval regressionanalysisbasedon a quadratic
programmingapproach.Another advantageof adoptinga quadraticprogrammingapproachis to be able
to integrateboth the propertyof centraltendency in leastsquaresand the possibilisticpropertyin fuzzy
regression.By changingthe weightsof the quadraticfunction,we cananalyzethe given datafrom differ-
entviewpoints.For datawith crisp inputsandinterval outputs,thepossibilityandnecessitymodelscanbe
considered.Therefore,theunifiedquadraticprogrammingapproachobtainingthepossibilityandnecessity
regressionmodelssimultaneouslyis proposed.Eventhoughtherealwaysexist possibilityestimationmodels,
theexistenceof necessityestimationmodelsis not guaranteedif we fail to assumea properfunctionfitting
to thegivendataasaregressionmodel.Thus,weconsiderpolynomialsasregressionmodelssinceany curve
canberepresentedby thepolynomialapproximation.Usingpolynomials,we discusshow to obtainapprox-
imationmodelswhich fit well to thegivendatawherethemeasureof fitnessis newly definedto gaugethe
similarity betweenthepossibilityandthenecessitymodels.Furthermore,from theobtainedpossibilityand
necessityregressionmodels,a trapezoidalfuzzyoutputcanbeconstructed.
H. Tanaka,K. Koyama,andH. Lee. Interval regressionanalysisbasedon quadraticprogram-ming. In ‘Proceedingsof the Fifth IEEE InternationalConferenceon FuzzySystems.FUZZ-IEEE’96. IEEE,New York, NY, USA’, Vol. 1, pp.325–329,1996.
Abstract. Weproposeanapproachto interval regressionanalysisbasedonquadraticprogramming.Secondly
weproposea unifiedapproachfor thepossibilityandthenecessityregressionmodelsin interval regression.
Fromtheobtainedtwo models,fuzzy outputcanbedefined.Moreover, a performanceindex for theunified
approachto measurethefitnessof thegivendatato theobtainedmodelsis defined.
J. Tang and D. Wang. An interactive approachbasedon a geneticalgorithm for a type ofquadraticprogrammingproblemswith fuzzy objective andresources.Computers andOperationsResearch, 24(5), 413–422,1997.
Abstract. A type of modelof fuzzy quadraticprogrammingproblems(FQP) is proposed.The modelde-
scribesthefuzzy objective andresourceconstraintswith differenttypesof membershipfunctionsaccording
to differenttypesof fuzzyobjective andfuzzy resourceconstraintsin actualproductionproblems.This arti-
cledevelopsaninexactapproachto solve this typeof modelof quadraticprogrammingproblemswith fuzzy
objective andresourceconstraints.Insteadof finding anexactoptimalsolution,we usea geneticalgorithm
(GA) with mutationalongtheweightedgradientdirectionto find afamily of solutionswith acceptablemem-
bershipdegrees.Thenby meansof thehuman-computerinteraction,thesolutionspreferredby theDM under
differentcriteriacanbeachieved.
122 A QUADRATIC PROGRAMMING BIBLIOGRAPHY
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Abstract. A discrete-timerecurrentneuralnetwork is presentedfor solving convex quadraticprograms.It
is the discrete-timeversionof its continuous-timecounterpartwhich was developedby J. Wang and H.
Li (1994).Sharingthesamecharacteristicwith its continuous-timecounterpart,theproposeddiscrete-time
neuralnetwork couldcomputetheexactoptimalsolutionto a quadraticprogramwithout usingany penalty
parameter. However, thediscrete-timeversionis moredesirablein practicalrealizationin view of theavail-
ability of digital hardwareand the good compatibility to computer. The condition for the neuralnetwork
globallyconverging to theoptimalsolutionof aquadraticprogramis given.Theneuralnetwork is appliedto
FIR filter synthesisfor illustratingits effectiveness.
W. S.TangandJ.Wang.A discrete-timelagrangiannetwork for solvingconstrainedquadraticprograms.InternationalJournalof Neural Systems, 10(4), 261–265,2000.
Abstract. A discrete-timerecurrentneuralnetwork which is calledthediscrete-timeLagrangiannetwork is
proposedfor solvingconvex quadraticprograms.It is developedbasedontheclassicalLagrangeoptimization
methodandsolvesquadraticprogramswithout usingany penaltyparameter. The condition for the neural
network to globally converge to the optimal solutionof the quadraticprogramis given.Simulationresults
arepresentedto illustrateits performance.
Q. TaoandD. Sun.Thesimplificationof neuralnetwork for quadraticprogrammingproblemsandits applicationsin optimal control. In ‘Proceedingsof the 3rd World CongressonIntelligentControlandAutomation,IEEE,Piscataway, NJ,USA’, Vol. 5, pp.3504–3508,2000.
Abstract. In thispaper, akind of neuralnetwork for quadraticprogrammingproblemsis first simplified.The
simplificationis necessaryfor high accuracy solutionsandlow costimplementation;the simplified model
hashigh performance.Theproposedneuralnetwork is usedto solve quadraticcontrolproblemsfor discrete
linearsystemswith constraints.Thesimulationprovedtherationalityof theresultsobtained.
R. M. Teny and A. K. Kochhar. Solution of the aggregateproductionplanning problemin a multi-stage-multi-productmanufacturing systemusing functional spaceanalysisand quadraticprogrammingapproaches. International Journal of SystemsScience,14(3), 325–342,1983.
Abstract. It is shown thatbothof theseapproachescanbeusedto determinetheproductionplanningstrate-
giesfor the numberof periodsunderconsideration.Statisticalinferencetechniquesareusedto determine
whetherthefunctionalspaceanalysistechniqueresultsin aglobalminimum.Testscarriedouton two differ-
entsetsof datashow that thequadraticprogrammingapproachalwaysresultsin a minimumcostsolution,
althoughit requiresa large amountof computingpower. Althoughcomplicatedmathematicsis requiredto
formulatetheproblemandsolve it, theresultingsolutioncanbeeasilyunderstoodandappliedto apractical
manufacturingsystem.
T. Terlaky. A new algorithmfor quadraticprogramming.EuropeanJournalof OperationalRe-search, 32(2), 294–301,1987. Seealso,AlkalmazottMatematikaiLapok.12(3–4):283–293,1986.
Abstract. Presentsanew finite algorithmfor quadraticprogramming.Thealgorithmis basedonthesolution
proceduresof linearprogramming(pivoting,Bland’s rule,HungarianMethods,criss-crossmethod),however
thismethoddoesnotrequiretheenlargementof thebasictableauasFrank-Wolfe methoddoes.It canbecon-
sideredasafeasiblepointactive-setmethod.Thisalgorithmis astraightforwardgeneralizationof Klafszky’s
andTerlaky’s Hungarianmethod.It hasnearlythe samestructureasRitter’s algorithm(which is basedon
conjugatedirections),but it doesnot useconjugatedirections.
N. I. M. GOULD & PH.L. TOINT 123
H. Theil and C. van De Panne. Quadraticprogrammingas an extensionof conventional
quadraticmaximization.ManagementScience, 7, 1–20,1960.
A. L. Tits andJ.L. Zhou.A simple,quadraticallyconvergentinterior-pointalgorithmfor linear-
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Abstract. Duality theoremsof linearandquadraticprogrammingareprovenconstructively in thecombina-
torial settingof orientedmatroids.Oneversionof thealgorithmfor linearprogramminghasthe interesting
featureof maintainingfeasibility. The developmentof the quadraticprogrammingduality result suggests
thestudyof propertiesof squarematricessuchassymmetryandpositive semidefinitenessin thecontext of
orientedmatroids.
M. J.ToddandY. Wang.A projectivealgorithmfor convex quadraticprogramming.Technical
report,Schoolof OperationsResearchand IndustrialEngineering,Cornell University,
Ithaca,New York, USA, 1991.
O. N. Tokareva. Theestimatesof therateof convergenceof algorithmsbasedon barrierfunc-
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1981.
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Center, SanJose,California,USA, 1996.
J. J. Torsti. Inversionof heatcapacityby quadraticprogrammingin thecasesof KCl andCu.AnnalesAcademiaeScientiarumFennicae, SeriesAVI (Physica), 372, 1–20,1971.
Abstract. By usingexperimentalspecificheatdataandby separatingthespecificheatinto two parts,lattice
specificheatandanharmoniccontributions,a methodhasbeenpresentedfor the calculationof the lattice
vibration spectrumand of the anharmoniccontributions to the specificheatwith their confidencelimits.
The calculatedfrequency spectrumof KCl was in qualitative agreementwith the frequency spectrumby
Copley et al. which correspondedto the inelasticneutronscatteringmeasurements.The highestpeakin
the spectrawasat 4 6 & 1012 s' 1 Likewise, the calculatedfrequency spectrumof Cu was found to be in
agreementwith Varshni’s andShukla’s theoreticalspectrum(abstr. A8094 of 1966)which is basedon the
axialsymmetricmodelfor latticedynamics.Thehighestpeakwasat6 3 & 1012 s' 1 If thevolumedependence
of the frequenciesis taken into account,the resultsfor both KCl andCu arein betteragreementwith the
experimentaldatathan when using the harmonicapproximation.The confidencelimits of the frequency
spectrawerecomparatively largeat higherfrequencies.By contrast,thelimits of coefficientsof anharmonic
contributionsto thespecificheatwerenarrow.
J.J.TorstiandA. M. Aurela. A fastquadraticprogrammingmethodfor solvingill-conditioned
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is investigated,usingcomparative judgmentsof all possiblepairsof the N objectsas data.The pairwise
comparisonsfocuson thesimilarity relationsinsteadof therelative importanceof eachobject.A quadratic
programmingmodelis alsoproposed.It processesthesimilarity-basedpairwisecomparisonsanddetermines
the similarity relationsamongthe N objects.The modelhaslinear constraints;thereforeit canbe solved
easilyby transferringit into asystemof linearequations.
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attachedto thebindingconstraintsarenegative. Thebasicalgorithmstartsfrom the freeminimum.At any
stage,aviolatedconstraintis turnedinto adesideratumandtheweightof thisdesideratumis increasedgrad-
ually from 0 to infinity , meanwhileremoving thosebindingconstraints,theLagrangiansattachedto which
becomezero.A modifiedalgorithmis alsoproposedleadingto thesolutionmorerapidly thanthebasical-
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the quadraticprogrammingproblemis introduced,makingit possibleto copewith semidefinitecases.Fi-
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N. I. M. GOULD & PH.L. TOINT 125
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anunconstrainedproblem,theinneriterationsolvesamodifiedequalityconstrainedproblem.Any inequality
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inner iterationfor thesolutionof anequalityconstrainedsubproblem.Theinner iterationsolvesa modified
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N. I. M. GOULD & PH.L. TOINT 127
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andto converge in a finite numberof iterations.The algorithm is easilyprogrammedon a computer. An
example,previously usedby others,is solvedwith this techniquein a fewer numberof iterations.An illus-
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experimentalsystemswith discretedistributionsof lifetimes.Critical analysisof theeffectof signal-to-noise
on the resolvingcapabilityof the algorithmis presented.This techniqueis recommendedfor resolutionof
thedistributionsof quencherconcentrationin heterogeneoussamples,of whichoxygendistributionsin tissue
is animportantexample.Phosphorsof practicalimportancefor biologicaloxygenmeasurements:Pd-meso-
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suchthat the valueof objective function is decreasing.The convergenceof the algorithmis proved under
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satisfy, eachattributeand/ortherelative importanceof theattributesareonly roughlyspecified.In particular,
wewill work with verbalspecificationsof thesequantities,like low, medium,andhigh.Thesespecifications
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it is found that experimentalsimulationscanmatchcomputerpredictionswith accuracieson the orderof
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outliers in time seriesand improves the forecastingresults.New robust estimatesareyieldedby combin-
ing optimally two weight functionssuitablefor innovation andadditive outliers in time series.The tech-
niquewhich is developedhereis basedon an approachof mathematicalprogrammingapplicationto * p-
approximation.The behavior of the estimatorsare illustratednumerically, underthe additive outlier gen-
eratingmodel.Monte Carlo resultsshow that the proposedestimatorscomparedfavorably with respectto
M-estimatorsandboundedinfluenceestimators.Basedon theseresultswe concludethat onecanimprove
therobustpropertiesof AR(p) estimatorsusingquadraticprogramming.