Some properties of the bilevel programming problem - Springer Link

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 68, No. 2, FEBRUARY 1991

TECHNICAL NOTE Some Properties of the Bilevel Programming Problem' J. F.

BARD 2

Communicated by M. Avriel

Abstract. The purpose of this paper is to elaborate on the difficulties accompanying the development of efficient algorithms for solving the bilevel programming problem (BLPP). We begin with a pair of examples showing that, even under the best of circumstances, solutions may not exist. This is followed by a proof that the BLPP is NP-hard. Key Words. complexity.

Bilevel programming, Stackelberg games, computational

1. Introduction A s e q u e n t i a l o p t i m i z a t i o n p r o b l e m in which i n d e p e n d e n t d e c i s i o n m a k e r s act in a n o n c o o p e r a t i v e m a n n e r to m a x i m i z e their i n d i v i d u a l benefits m a y be c a t e g o r i z e d as a S t a c k e l b e r g game. The bilevel p r o g r a m m i n g p r o b l e m is a static, o p e n - l o o p v e r s i o n o f this g a m e w h e r e the l e a d e r c o n t r o l s the d e c i s i o n v a r i a b l e s x~XC_R ~,, while the f o l l o w e r s e p a r a t e l y controls the d e c i s i o n v a r i a b l e s y ~ Y _ R n~ (Refs. 1-3). In the sequel, it will be a s s u m e d that the l e a d e r goes first a n d c h o o s e s x to m a x i m i z e his o b j e c t i v e f u n c t i o n F(x, y). The f o l l o w e r t h e n reacts by selecting y to m a x i m i z e his i n d i v i d u a l o b j e c t i v e f u n c t i o n f(x, y) w i t h o u t r e g a r d to the i m p a c t this c h o i c e has on the first player. Here, F : X x Y ~ R 1 a n d f : X x Y ~ R i. The focus o f this p a p e r is on the l i n e a r - q u a d r a t i c case,

1This work was partially supported by a grant from the Advanced Research Program of the Texas Higher Education Coordinating Board. 2 Associate Professor, Operations Research Group, Department of Mechanical Engineering, University of Texas, Austin, Texas. 371 0022-3239/91 /0200-037 IS06.50/0 ~) 1991 Plenum Publishing Corporation

372

JOTA: VOL. 68, NO. 2, FEBRUARY 1991

given by max F(x, y) = cl x + c2y,

(la)

x

s.t.

xcX={x:Dx>-d},

max f ( x , y) = c3y + xrQo, + (1/2)y ~Q2y,

(lb) (lc)

Y

s.t.

g(x,y)=Ax+By>--b,

(ld)

y ~ Y = { y : Ey>-e},

(le)

where A is m l × n l , B is ml× n2, D is m2×nl, E is m3×n2, Q~ is njxn2, Q2 is n2x n2 symmetric, negative semidefinite, and c ~, c 2, c 3, b, d, e are vectors of conformal dimension. Note that it is always possible to drop components separable in x from the follower's objective function without altering the results. Hence, (lc) does not contain linear and quadratic terms in x.

2. Terminology and Examples

As is customary in the formulation of Stackelberg games, it will be assumed that full information is available to the players, and that cooperation is prohibited. This precludes the use of correlated strategies and side payments. The following notation is used in the development. Follower's Rational Reaction Set:

M(x) = {y: y = argmax[f(x, y):y ~ Y, g(x, y) >-b]}. Inducible Region: IR = {(x, y): x ~ X, y c M ( x ) } . Out of practical considerations, we further suppose that the feasible region (lb), (ld), (le) is nonempty and compact and that, for each decision taken by the leader, the follower has some room to respond. The rational reaction set M(x) defines this response while the inducible region IR represents the set over which the leader may optimize. In the play, y is restricted to M(x). Given these assumptions, problem (1) may still not have a well-defined solution. In particular, difficulties may arise when M(x) is multi-valued and discontinuous. This is shown by way of example making use of the following definitions from Ref. 4.

JOTA: VOL. 68, NO. 2, FEBRUARY 1991

373

Definition 2.1. A point-to-set m a p M : X ~ Y is open at a point ff in X if the sequence {x k} C X, x k ~ if, and 37c M ( f f ) imply the existence o f an integer m a n d a sequence {yk}C Y such that ygc M ( x k) for k-> m a n d yk -~ 35. Definition 2.2. A point-to-set m a p M is closed at a point ~ in X if {x k} C X, x k ~ ~, yk e M(xk), and yk _~ 25 imply that )5 ~ M ( ~ ) .

E x a m p l e 2.1.

Consider

max F = -x

-

Y2,

x~O

s.t.

2- 0}. Now, given the sequences x k ~ f f = 2 , y ~ - ~ 4 + Y , y ~ = 0 , and the point 25= (0, 6), we note that 37 ~ M(22), but that there does not exist m such that, for k-> m, yk _~ ~. Thus, M ( x ) is not open at x = 2, although it is closed for all x c [2, 4]. Proposition 2.1. I f M ( x ) is not single-valued for all permissible x, the leader m a y not achieve his m a x i m u m objective. In order to deal with this situation, two possibilities present themselves. The first would require replacing " m a x " with " s u p " in ( l a ) and define e-optimal solutions. This would work for E x a m p l e 2.1 as currently fbrmulated, but a slight change in the follower's objective function t o f = -2y~ - 2y2 would reintroduce the multi-valued condition. The second a p p r o a c h argues for a conservative strategy" that redefines p r o b l e m (1) as

m a x rain xeX

F(x,y).

3'~ M ( x )

I f M ( x ) is single-valued, however, we have the following proposition.

374

JOTA: VOL 68, NO. 2, FEBRUARY 1991

Proposition 2.2. In general, if all the functions in (1) are twice continuously differentiable, and if all the solutions to the subproblems (1 c ) - ( l e ) are unique, then the inducible region IR is continuous. The basis for the p r o o f can be found in Ref. 4, Corollary 8.1; the same result was established for the linear BLPP by Bard (Ref. 1) using duality arguments. Finally, we note the following proposition. Proposition 2.3. The rational reaction set M ( x ) for the pure linear BLPP (QI = Q2 = 0 ) is closed. The p r o o f follows from the observations that the subproblem ( l c ) - ( l e ) is a right-hand-side perturbed linear program and that the accompanying optimal-value function ¢b(x) = max{c3y: B y >- b - A x , y ~ Y } is continuous. In fact, ~b(x) is piecewise linear and concave. The closedness of the rational reaction set for the pure linear BLPP does not guarantee that it is always single-valued. The next example 3 further illustrates the complications that may arise in such a case. Example 2.2.

Consider

max F = x - 10yl + Y 2 , X~O

max f --- y~ + Y2, y>--O

s.t.

x + y~