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THE OPERATIONS RESEARCH PROGRAM
DEPARTMENT OF INDUSTRIAL ENGINEERING
STATE UNIVERSITY OF NEW YORK AT BUFFALO
MULTILEVEL LINEAR PROGRAMMING
by
Wayne F. Bialas
Mark H. Karwan
Research Report No. 78-1
May 1978
Department of Industrial EngineeringState University of New York at Buffalo
Buffalo, New York 14260
Engineering, StateY. 14260.
IndustrialAmherst, N.
MULTILEVEL LINEAR PROGRAMMING*
Wayne F. Bialast
Mark H. Karwan
Technical Report No. 78-1
*Presented at the Joint National Meeting of the Operations ResearchSociety of America and The Institute of Management Sciences,
May 1-3, 1978.
tAssistant Professor, Department ofUniversity of New York at Buffalo,
+Assistant Professor, Department ofUniversity of New York at Buffalo,
IndustrialAmherst, N.
Engineering, StateY. 14260.
Abstract
The general multilevel programming problem is a set of nested
optimization problems over a single feasible region. Control over the
decision variables is partitioned among the levels, but a decision variable
may impact the objective functions of several, if not all, levels. The
two-level linear resource control problem is a special case where the
level-two planner controls the effective resource space for the level-
one planner. This produces a solution structure with the feasible region
viewed by level two as a nonconvex subset of the overall feasible region.
An algorithm to solve this problem is proposed.
•
Introduction
Many planning situations require the analysis of several objectives.
Multiobjective optimization techniques have been developed to permit a more
faithful analysis of the tradeoffs among competing goals, and assist a planner
in reaching an acceptable compromise. Such methods assume that all objectives
are those of a single planner, impacting directly on his state of well-being.
Multiobjective optimization fails to recognize that many objectives are ordered
within an administrative or other hierarchical structure. For example, the
energy policies set forth by the Federal government affect the objectives and
options, and hence the strategies, of state officials. This process continues
within a hierarchy including local governments, planning agencies, and basic
economic units such as firms and households. Each unit in the hierarchy wishes
to maximize its individual benefits in view of the partial exogenous control
exercised at other levels. In the example, actions at the state level affect
the benefits sought by the Federal government which can, in turn, constitutionally
exercise first, but only partial control over the state.
Multilevel optimization techniques partition control over the decision
variables among the levels, and analyze objectives at all levels of the
planning process. A planner at one level of the hierarchy may have his
objective function determined, in part, by variables controlled at other
levels. However, his control instruments may allow him to influence the
policies at other levels and thereby improve his own objective function.
The problem of multilevel linear programming was offered by Candler
and Norton [2]. Unfortunately, their general problem was imprecisely defined
and they did not realize that the effective feasible regions for higher
1
levels were nonconvex sets. This causes the Candler and Norton algorithm
to fail since it assumes convexity. Although the general problem formulation
and solution technique proposed by them is incorrect, the basic premise of
their problem has far-reaching applications. Some particular problems are
amenable to this framework:
(i) Control of oil imports: With the desire to limit oil imports, theFederal government can impose import quotas and duties. This actionwill affect the consumption of oil by economic units which are maximizingprofits. The decisions made by these units would then impact on theobjectives of the Federal government including a change in consumptionpatterns and levels of oil imports.
(ii) Floodplain planning: Seeking to reduce flood risk, government canimplement floodplain zoning programs and subsidize flood insurance.These policies will influence the land activity patterns based on thebudgets and objectives of land users.
(iii)Job incentive programs: As industry seeks to maximize productivity, itmay implement profit-sharing and other job incentive programs. Suchprograms, in turn, affect the goals of employees whose decisions impacton industrial productivity.
Similar work in this general area has been conducted on Stackelberg
games [1,3,7].Within the broad definition of such games, a static Stackelberg
game with fixed leaders and a continuous decision space could be defined to
encompass multilevel optimization problems. However, current methodology
does not consider the activity space of one player to be a function of the
strategies of other players. Such an extension of Stackelberg games would
require the payoff function of one level to have discontinuities dependent on
the decisions of other players. This formulation is, at best, unwieldly,
and perhaps intractable.
The term "multilevel" has been frequently used to describe approaches
to similar important planning problems. However, in the context of the
multilevel programming problem defined in the next section, these previous
approaches are primarily decomposition techniques applied to single level
problems [4,5,6]. For example, Haimes, Foley and Yu [5] employ lagrangian
duality to decompose and efficiently solve a large model for the control of
2
water quality with a single overall system objective of a central planning
agency. The dual variables are then interpreted, as with Dantzig-Wolfe
decomposition, to determine prices (taxes) to be charged to each subproblem
(polluter) for violating pollution standards.
The flexibility of the multilevel approach can answer questions
regarding the assignment of control over variables to various levels. For
example, in some cases, coalitions of levels can improve the objective functions
of all levels. Furthermore, because of the structure of some problems, a single
level could exercise complete control over all levels even though controlling
only a proper subset of the variables. This methodology can assess the value
of controlling a particular subset of variables, and with this information,
policy makers could determine what control should be relinquished or maintained
over certain variables.
The next section will provide the formal, general definition of a multilevel
programming problem. Then the discussion will focus on a special case of this
general problem, the two-level linear resource control problem, and its
mathematical structure. This will include a characterization of the solution
set and some general algorithmi c approaches to solve the problem. Finally, a
specific algorithm for finding local optimal solutions will be presented.
General Definition of Multilevel Programming Problems
Let the decision variable space (Euclidean n-space) Rn
x = (x1,x
2,...,x
n)
be partitioned among r levels,
nk • k k kR 4 x = (x1
k,1 ' 2 "-- ,xnk=1,...,r ,
k
wherer
nk = n. Denote the maximization of a function f(x) over R
n by
k=1 nk k+1 k+2
nk+2
varying only xk E R given fixed x, x
r in R nk+l
^ R x
by max f(x)
k k+1 k+2
x ix ,x ,...x .
3
The general multilevel programming problem can then be defined as
max fr(x)
xr
st: max fr-1
(x)
r-1 1 rx Ix
st: max fr-2
(x)
xr-2ixr-1
, xr
r-1(P ) 4
r-2(P ) 4
st: max f (x)11 2x ix ,x-
1,...,x
st:xESCR
This establishes a collection of nested mathematical programming problems
{P1 ,.. ,Pr }. The feasible region, S = S l , is defined as the level-one feasible
region. The solutions to P1 in R
1n for each fixed x
2,x
3,..., x
r form a set
S2 = {x E S
1:f
1(x) = max f
1(x) } called the level-two feasible region over
xi 1X2 ,X 3 r
,..., x-
which f2(x) is then maximized by varying x
2 for fixed x
3,...,x
r.
In general, the level-k feasible region is defined recursively as
{ x^
tS = x E Sk-1 :
- fk-1 (x)= max fk-1 (x) }
k-1 ^k ^rx ix ,...,x
rNote that x
k-1 is a function of xlc ,...,x . Furthermore, the problem at
each level can be written as
max fk(x)
k k+1x ix ,...,x
x E Sk
which is a function of xk+1 r
, and (Pr): max f (x) defines the entire
x€Sr r
problem.
( Pk )
4
The Two-Level Linear Resource Control Problem
The two-level linear resource control problem is the multilevel programming
problem, of the form
max c2x
x
st: max c1x
x
A xl
+ A x2 <b
1 2 -x > 0 .
Here, level 2 controls x2 which, in turn, varies the resource space of level
one by restricting A 1xl
< b - A2x2
.
Ill-Defined Problems
Care must be taken when Pk results in alternative optimal solutions for
fixed xk+1r
. Although not affecting the value of the level-k objective
function, fk (x), these solutions can have drastically varying impact on the
objectives of other levels. Therefore, control over the choice among alternative
optimal solutions may have to be delegated to other levels or an incentive
scheme may be required to induce the level-k planner to choose a solution
desirable to level k+1. If no such scheme is employed, the problem may be
ill defined.
Consider the following example of a two-level linear resource control
problem:
( P2
) 4
(P
5
( P2
)1
max
x2st:
(P1
)
1 +
-x1
x1
1 x2 x2
max x2 + x 3xi , x3 1x2
x2+ x
3 = 4
x2> 1
+ 2x2< 2
+ x2< 4
For x2= 2, the unique level-one solution is (x 11 x2, x3 ) = (2,2,2) with value 4.
The corresponding level-two solution value is 3. However, for x 2= 1, there
exists a set of alternate optimal solutions to P1
X = f(x1 ,x2' x3 )
0 < x1 < 3, x 2 = 1, x3
= 3} . The corresponding level-two objective for
1this set of solutions ranges continuously from 1 to 3 2' For a unique solution
to be returned to level two for fixed x 2 and to induce level one to return
x1= 3 for x 2= 1, a side payment to the level-one objective from level two may
be employed. For the example, the level-one objective function,
max x2 + x
3 + E(x
1 + 1 -x
2) with E > 0 sufficiently small, is a perturbation
which accomplishes this. Given this side payment scheme, x 2= 1 is the optimal
decision for level two.
Nonconvexity
In the two-level linear resource control problem,
rS1 = l x > 0 : A1xl + A2 x
2 < b ,
is a convex set. However,
2 1S = Ix f S1 : c
lx = max c1
xi11 2x i x
6
need not be. Therefore, P2, which can be written as
max c2x
x(S 2
involves the optimization of a linear function over a nonconvex region.
Consider the geometric example in Figure 1. The set S2 for the problem in
Figure 1 is a subset of the edges of the boundary of S1. In problems of higher
dimension, S2 is composed of edges and faces of the boundary of S
1. Consider
the following three dimensional example:
max x2 + x
3x2
max x3
xi ,x3 x2
x1 + x
2 - x
3 > 2
x1 - x
2 + x
3 < 2
x1 + x
2 + x
3 < 6
x1 + x
2 + x
3 > 2
x < 1 ,3 -
where S2 is the hatched region shown in Figure 2.
Relationships to Multiobjective Programming
Optimal solutions to the multilevel programming problem may not be
Pareto-optimal. While cooperation might improve the objective functions at
every level, the order and independence with which decisions are made prevent
such cooperation. This rules out any algorithmic approach which seeks only
Pareto-optimal solutions and is one of the main distinguishing characteristics
between multiobjective and multilevel programming.
For an example of this behavior, consider Figure 1. Both levels have
higher objective function values at point (a). However, for x2 fixed at x
2'
level one will choose x l = xl (point (b)), thus point (a) is not in S2This
leads to the best choice of x2
to be x2= 0 with the optimal solution at point (c).
st:
7
A Sufficient Condition for Complete Control
Consider the two-level linear resource control problem. Given any basis
B C A for the set of constraints Ax = b, one can write the equivalent set of
constraints on x:
BxB + Nx
N = b
Or xB = B-1b - B
-1 N xN .
When xN is fixed, xB is uniquely determined. Thus to have complete control of
the solution, the level-two planner need only control the complete set of
nonbasic variables corresponding to any basis.
Further Characterization of S2 and P2
The following theorem and its corollaries help to characterize both S2
and the optimal solution for P2 in the two-level linear resource control
problem.
Theorem I
Suppose S= {x : Ax = b, x > 0} is bounded. Let S 2 = {x =(x 1 ,x2 ) E S i :
11 1 1,c x = max c x /. Then the following hold
x l 1X 2
(i) S2 C S
1
(ii) Let {yd r be any r points of S i , such thatt=1
x = Atyt S
2 with A
t > 0, X
t = 1. Then A
t > 0 implies y
t( S2
-t
Proof: (i) s2
S1 by the definition of S
2.
(ii) (By contradiction) Let y1 , y2"
..' yr
f S1 with
x = (x1,x
2) =
r X A
tyt S
2, and
t=1
> 0 A l > 0,r X A
t = 1.
at - t=1
8
k optimalsolution
Figure 1
Example of NOnconvexity of S 2
Figure 2
EXample of S 2 in Three Dimensions
10
Using (i), Yl E S . Therefore,'
1 1C X
1= c 1A1y1
an x with theand A l > 0, we have established
since S1 is convex.
1 2Given, x = (x ,x )
=AY + XAyc1 1 t=2
t t
- ,Noting that x
2 = x2 and A .
1 >
t=2
S2
1c <t t
y t
1 1cA + cAy1 1
Y 1 ttt
t=2
> 0,t=1
x yt=1
From Theorem 1, this implies .,Yr t S and hence x cannot be an extreme
Suppose y1 2
= (Y1' Y1 ) f SThen there exists y11
such that =-1 2 _2
)Y1 Y1 (I
with c1 171 > c 1
y1l .
following properties:
(a) x= s l
(b) x =2- 2
1-1< c x .(c) c
1x1
This contradicts the definition of S2 since x t S
2 should have maximized c 1 x1
for the fixed value of x2
. Therefore Al > 0 implies y 12S . Since the
choice of yl 'among the y's was arbitrary, we have proven A t > 0 implies yt c S
Any point which positively contributes in any convex combination
forming a point in S2 , must also be in S2. Since this is true of any point,
including y which are extreme points of S1 , the following corollary results:
Corollary 1. If x is an extreme point of S 2 , then x is an
extreme point of S 1 .
Proof: (By contradiction) Let x be an extreme point of S te . Suppose x is
not an extreme point of S1 .
A1
>
Then there exist extreme points y ,...,yr c S1, and
Such that
point of S 2a contradiction.
11
Recalling that P may be formulated as max c2 x and noting the
2x(S
S2correspondence of extreme points in S and S 1 , the following result is derived.
Corollary 2. An optimal solution to the two-level linear resource
control problem (if one exists) occurs at an extreme point of the constraint
set of all variables (S ).
Proof: The two-level linear resource control problem can be written as
2max c x
x(S2
Since c2x is linear,if a solution exists, one must occur at an extreme
2point of'S (alternative optimal solutions at nonextreme points may exist),
By Corollary 1, this must be an extreme point of S 1 .
This result justifies extreme point search procedures as a basis for
algorithmic approaches to solving the two-level linear resource control problem.
Algorithmic Approaches
It has been shown that an optimal solution to the two-level linear
resource control problem occurs at an extreme point of the level-twO feasible
,region, S 2 . Let LS
2 ] denote the convex hull of S 2 . Since the sets of extreme
points for S2 and [S 2] are identical, the problem
(P2 ) max
st:
is an equivalent formulation for the two-level linear resource control problem.
This suggests a search for cutting plane procedures to approximate the convex
hull of S2 as a direction for future research..
2c x
x ( [S 2 ]
12
Any desirable algorithm for the two level linear resource control problem
should exhibit some particular properties.
^1 ^2Consider the solution, x = (x ,x ) to the following problem:
2max c x6; )
st: Ax=b, x>0
In (P), the level two planner is given full control over all variables. Now
^fix x
2 = x
2
max c1x
st:A1 x
1 = (b-A x2
) 1x > 0
then x t S2 is an optimal solution to the overall problem. For
example, note that in Figure 1, the vector c 2 could be changed to produce a
solution to (P) at any extreme point of S 2 ..9
The set $ does not vary with
changes in the second level objective, and, hence quite different choices of c 2
can produce an optimal solution after solving (P). For the example shown in
Figure 1, two particular choices of c 2 which lead to such a condition for the
level-one objective shown are both c2= c 1 and c
2= -c
1. Thus both highly
complementary and highly conflicting objectives (as well as many inbetween)
may lead to solutions after solving the two linear programming problems (P) and
(P). Any reasonable algorithm should have the ability to easily solve any
problems for which x e S 2 .
An Algorithm to Find Local Optimal Solutions
Consider the following portion of abounded simplex tableau to be employed
in the proposed algorithm:
and solve the following problem with solution x to determine if
x f S 2 :
If x = x,
13
2x l
2 2RHS
xB1
xB. 2
xBm
r 21
r2r2
.._
2rk
2
Y 11
Y21..
Yml
Y 1
Y22 —
Y m2
Y lk
—- '''''' " Y2k
Ymk
1--;b2.
_
bm
2The variables, x 2 xk represent the nonbasic level-two variables which are
2 2at nonzero values, and r i ,...,rk represent the reduced costs of these variables
with respect to the level-two objective function. In terms of the present
basisBCA,b=B_ b- y yx2 where (y ,Y = Y, = B
-1(a ,alj 2j mj
,tlj 2jj=
j1
and x denotes the ith basic variable.xB
Assume that S is boUnded with no degeneracy and no alternative optimal
solutions exist for P 1 for any feasible x2 .
The following algorithm guarantees a local optimal solUtion.
^1 ^2Step 1. Solve the following problem with optimal solution x = (x ,x and
optimal tableau T via the simplex method:
Step 2. Set x2= x2
and solve the following problem via bounded simplex
(k=u=x2 ) beginning with tableau T:
max c1x
st: Ax=b2 ^2x =x
x >0
Let the optimal solution be x. If x = x, stop; x is a global optimal
solution. Otherwise, go to step 3a with current tableau T and relax
^2the constraints x
2 = x .
14
Step 3a. If all nonbasic variables are equal to zero, go to step 4 with current
tableau T. Otherwise go to step 3b.
Step 3b. If b. > 0 for all i, go to step 3c. Otherwise, without loss of
generality, consider b t = 0. Choose
Bring x2 into the basis via a degenerate pivot.3
Step 3c. Consider any nonbasic variable which is at a strictly positive value,
2xj2 2
If r. < 0, increase until it enters the basis. If r, %P. 0,3 -
decrease x2 until either it reaches zero or it must enter the basis.
Go to step 3a.
Step 4. Beginning with tableau T solve the following problem via a modified
simplex procedure:
2max c X
st: Ax=b
x>0
The modification is as follows. Given a candidate to enter the basis
(one for which c2 x will increase) only allow it to enter if the
resulting basic solution, x, will be contained in S2.
This is
determined by obtaining the solution x to the following problem:
c1 x
A1 xi < (b-A2 x2
)
1 2 2x > 0, x = x
via dual simplex on repeated applications of step 4. If x = x1
then enter the candidate into the basis. Repeat step 4 until no more
candidates exist which satisfy the above mqdification, then stop,
y 0.3
step 3a.
yQjsuch that 1 < j < k and
Go to
2say x..
max
st:
15
Validation and Convergence
The algorithm begins by finding the maximum of the second level objective
over the entire feasible region, S 1 . A check in step 2 is then made to determine
if the resulting solution is in S 2 . If so, the algorithm terminates with a
global optimal solution and has solved what was previously termed an easy
problem. If termination does not occur in step 2, the resulting solution from
step 2 is by definition contained in S 2 . Since the bounded simplex algorithm
was employed, a number of nonbasic level-two variables may be at nonzero values
2corresponding to appropriate components of x. Degeneracy may also have been
introduced by fixing the components of x2 from step 1.
is indeed contained in 5 2. Since bounded simplex was employed, a number of
nonbasic level-two variables may be at nonzero values corresponding to
appropriate components of x 2 . Degeneracy may also have been introduced by
fixing the components of x 2 from step 1.
The purpose of step 3 is to relax the constraint x = x 2 and to move to an
extreme point x° which satisfies xo S2 and cx
o > cx. If a right hand side,
, from the current tableau is equal to zero then step 3b is entered to
perform a degenerate pivot. Some nonbasic variable, x2 , j=1,2,...,k, is then
brought into the basis at its current positive level and xB becomes nonbasic
at its current value of zero. Thus the number of basic variables at level zero
is reduced by one. This is repeated until no degeneracy is present. Note that
such a pivot is always possible, that is, yk 0 for some j=1,...,k. Suppose
that y2, = 0 for all j=1,...,k. Then repeated applications of step 3c would
result in a degenerate extreme point of the original feasible region, S 1 , since
2xB will remain zero no matter how xl ,...,x are varied.
original nondegeneracy assumption.
This contradicts the
16
If all b.> 0 but there are still nonbasic level-two variables, x2
.. . ,x2k'
xj2
at nonzero values,. then step 3c is entered. Any variable , j=1,...,k, is
chosen to be increased or decreased depending on its reduced level-two cost,
r.. Since there are no explicit upper bounds on2
x2
increase is limited3
by a current basic variable reaching zero. The original problem is bounded,so
this must occur. If x2 is decreased, again a current basic variable may reach
zero or else x2 itself will become zero. In either case, the number of nonzeroJ
nonbasic variables is decreased by one.
The points generated in step 3c can be shown to be contained in S2 which
is assumed when step 4 is entered. Recall that 1;1 0 for all i as a result
of step 3b. Thus there exists two scalars, 0 1 > 0 and 0 2 > 0 such that any
increase or decrease in x, by an amount less than or equal to 01 and 02
respectively results in a feasible solution (i.e., a point in S 1). This implies
that the current solution, which is in S 2 , is a convex combination of two
feasible points resulting from a strict increase and a strict decrease in x 2.3
By Theorem I, such points must also be in S 2 . Thus each point resulting from
step 3c must be contained in S 2 .
Step 4 is entered when an extreme point of S has been obtained. A modified
simplex method is used to take steps in S 2 along which the level two objective
increases. This is accomplished by using the normal simplex rules with
objective c 2x along with a check that no basis change results in leaving S 2 .
The algorithm terminates with an extreme point solution in S 2 which has the
property that all adjacent extreme points either lead to a decrease in c 2 x or
are not in S2. Thus a local optimal solution is obtained.
Convergence of the algorithm is established by noting the following
facts:
17
(i) The feasible region defined by S
Ax = b, x > 0t is bounded
and each basis is nondegenerate.
(ii) Steps 1, 2 and 4 are finite since the simplex, bounded simplex and
dual simplex procedures are finite under fact (i).
(iii)Each application of step 3b strictly decreases the number of basic
variables at level zero and also the number of nonzero nonbasic
variables.
(iv) Each application of step 3c reduces the number of nonzero nonbasic
variables by one.
Conclusions
Multilevel mathematical prograMming problems, if carefully defined, can
serve as useful tools in modelling structured economic units. Such models can
predict the inefficiencies of non-Pareto-optimal decisions and identify the
seats of true control within hierarchical organizations.
This paper has proposed a general mathematical structure for such problems,
and specifically characterized the two-level linear resource control problem.
For this problem, Theorem I illustrates a key property of the nonconvex feasible
region viewed by level two. As a foundation, it justifies extreme point solution
techniques and obviates the need for methods to establish the convex hull of the
level two feasible region. Towards this goal, this paper has offered an adjacent
extreme point method which can find local, and sometimes global, optimal solutions
to the two-level linear resource control problem.
Acknowledgement
The authors wish to expreSs their sincere gratitude to Professbr Daniel P.
Loucks for introducing them to this problem
18
Figure Captions
FigUre 1. Example of Nonconvexity of S
2Figure 2. Example of S in Three Dimensions
2
19
References
1. Basar, T., "On the Relative Leadership Property of StackelbergStrategies," Journal of Optimization Theory and Applications.Volume II, No.6, 1973, pp.655-661.
Candler, W. and R. Norton; Multi-Level Programming, UnpublishedResearch Memorandum, DRC, World Bank, Washington, D.C. August 1976.
3. Cruz, J.B., "Stackelberg Strategies for Multilevel Systems," inDirections in Decentralized Control, Many Person Optimization and Large-Scale Systems, Y.C. Ho and S.K. Mitter, Eds., New York,Plenum Press, 1976, pp.139-147.
4. Goreaux, L.M. and A.S. Manne, Multi-Level Planning: Case Studies in Mexico, North-Holland, Amsterdam, 1973.
5. Haimes, Y.Y., J. Foley and W. Yu, "Computational Results for WaterPollution Taxation Using Multilevel Approach," Water-Resources Bulletin, Volume 8, No.4, August 1972, pp.761-771.
6. Haimes, Y.Y., W.A. Hall and H.T. Freedman, Multiobjective Optimization in Water Resources Systems, Elsevier, Amsterdam, 1975.
7. Simaan, M. and J.B. Craig, "On the Stackelberg Strategy in Nonzero-SumGames," Journal of Optimization Theory and Applications, Volume 11,No,5, 1973, pp.533-555.
20
