A note on the definition of a linear bilevel programming solution

Transcription

1 A note on the definition of a linear bilevel programg solution Charles Audet 1,2, Jean Haddad 2, Gilles Savard 1,2 Abstract An alternative definition of the linear bilevel programg problem BLP has recently been proposed by Lu, Shi, and Zhang. This note shos that the proposed definition is a restriction of BLP. Indeed, the ne definition is equivalent to transferring the first-level constraints involving second-level variables into the second level, resulting in a special case of BLP in hich there are no first-level constraint involving second-level variables. Thus, contrary to hat is stated by the authors ho suggested the ne definition, this does not allo to solve a ider class of problems, but rather relaes the feasible region, alloing for infeasible points to be considered as feasible. Key ords: Optimization, Linear bilevel programg 1 Introduction The aim of this note is to sho that the deficiency pointed out in [8,9,10] is not really a deficiency and that the ne definition proposed is equivalent to moving the first-level constraints involving the second-level variables into the second level, hich changes the nature of the problem. A bilevel program is a program in hich a subset of the variables is required to be an optimal solution of a second mathematical program [1,2,3,5,6,7,12]. The linear bilevel program is a special case in hich all the constraints and the addresses: Charles.Audet@gerad.ca (Charles Audet ), Jean.Haddad@polymtl.ca (Jean Haddad), Gilles.Savard@gerad.ca (Gilles Savard). URL: (Charles Audet ). 1 GERAD 2 Département de Mathématique et de Génie Industriel, École Polytechnique de Montréal, C.P. 6079, succ. Centre-ville, Montréal (Québec), H3C 3A7 Canada Preprint submitted to Elsevier Science 3rd October 2005

2 objective functions of both programs are linear. An overvie of linear bilevel programg can be found in the tetbook [4]. The linear bilevel program can be formulated as: (BLP ) ct + d t y X (, y) P 1 y arg a t S BLP (), here, c R n, y, d,, a R ny. X R n and P 1 R n+ny are polyhedral sets, and S BLP () = { : B b A} R ny ith A R m n, B R m ny, b R m. Let P 2 = {(, y) : A + By b} R n+ny be the polyhedra defined by the second-level constraints. Thus, S BLP () corresponds to the projection of P 2 on the y-space for a given. Let us define the folloing sets: S = { (, y) : X, (, y) P 1 P 2}, M BLP () = arg { a t : S BLP () }, (1) IR BLP = {(, y) S : y M BLP ()}. S is the polyhedra defined by the intersection of both the first-level and the second-level constraints. M BLP () corresponds to the set of optimal solutions for the second-level program for a given. Finally, IR BLP, called the induced (or inducible) region, represents the feasible region of BLP. With these definitions, BLP is equivalent to the problem. {ct + d t y : (, y) IR BLP } In the presence of first-level constraints involving the second-level y variables, IR BLP is not necessarily a connected set, it may even be discrete [12,1]. Hoever, in the case here there are no first-level constraint involving y, then IR BLP is alays connected [12]. Many algorithms have been developed to solve BLP. The special case here there are no constraints involving the y variables in the first-level has also been studied, and it is important to note that transferring a first-level constraints 2

3 into the second-level is not equivalent to the original problem, as illustrated by the folloing eample (see figure 1): (BLP 1 ) ma 0 1 y = 0 y arg ma, (BLP 2 ) ma 0 1 y arg ma = 0. Both problems contain the same constraints, ecept that the first-level con- y BLP1 y BLP2 (0,1) (0,1) y = =1 y = =1 Second level objective Second level objective S IR First level objective (1,0) IR = S First level objective (1,0) Figure 1. The effects of moving the constraint y = 0 from the first level in BLP 1 to the second level in BLP 2. straint y = 0 in BLP 1 is moved to the second level in BLP 2. It follos that S = {(, 0) : 0 1} for both problems, and for any R e have {0} if 0 M BLP1 () = {}, M BLP2 () = else, IR BLP1 = {(0, 0)}, IR BLP2 = {(, 0) : 0 1}. For BLP 1, the only feasible point and optimal solution is (0, 0) ith objective function value 0. For BLP 2, the optimal solution is (1, 0) ith objective function value 1. 3

4 2 A restrictive definition for BLP Recently, [8,9,10] have proposed an alternative definition for the solution of BLP. They motivate their ne definition by giving an instance of BLP for hich no solution can be found even if S is not empty. They then sho that this instance must have a Pareto optimal solution [10]. They claim that the fact that a Pareto optimal solution eists but could not be found using the actual definitions is a deficiency of the theory. In order to obtain this Pareto optimal solution, they propose a ne definition for the solution of BLP. They state that this ne definition can solve a ider class of problems than current capabilities permit. Let us first state the definitions of [8,9,10]. Let S be as above and define: S ne BLP () = {y : (, y) S}, { BLP () = arg a t : SBLP ne () }, (2) M ne IR ne BLP = {(, y) S : y M ne BLP ()}. The difference beteen definitions (1) and (2) is that (2) implies that the second level is no responsible for respecting the first-level constraints involving the y variables. Let us first note that the solution of the original BLP, as it is strictly defined, does not have to be Pareto optimal [11], and this is due to the intrinsic noncooperative nature of the model. So the fact that no Pareto optimal solution could be found for the eample presented in [10] is not really a deficiency: no solution as found because no solution eists, and this may happen even if S is not empty. Let us consider the instance BLP 1. According to the ne definition, the optimal solution is (1, 0) and this does not solve BLP 1 as it is stated. Indeed, if = 1, then the constraint y arg ma implies y = 1, but this contradicts the first-level constraint y = 0, so that this solution is not feasible. Using definition (2) implies that the second-level also takes responsibility for satisfying the first-level constraints involving the y variables. This is equivalent to simply moving the constraint (, y) P 1 from the first to the second level, and this is a relaation of problem BLP but still NP-hard [12]. 4

5 3 An equivalence This section formally shos that the ne BLP definition is equivalent to moving the first-level constraints to the second level. Define: ct + d t y X (LSZ) (, y) P 1 y arg a t S ne LSZ(), to be the bilevel formulation of BLP proposed in [8,9,10], and ct + d t y (BLP ) X y arg a t S BLP (), to be the bilevel problem in hich the first-level constraints involving y variables (P 1 ) are moved to the second level, here S ne LSZ() = {y : (, y) S}, S BLP () = {y : (, y) P 1 P 2 }. Define IR LSZ to be the induced region of LSZ, and IR BLP to be the induced region of BLP. Theorem 1 IR LSZ = IR BLP. Proof: By definition, IR BLP = {(, y) S : y M BLP ()}, IR LSZ = {(, y) S : y M ne LSZ()}, here S is the same in both induced regions since it is defined to be the intersection of the first and second-level constraints. We need to sho that 5

6 the second-level optimal sets are identical for any R n. M ne LSZ() = arg {a t : SLSZ()} ne = arg {a t : (, ) S} = arg {a t : S BLP ()} = M BLP (). Define IR to be the induced region of these to problems, that is, IR = IR LSZ = IR BLP. The net corollaries follo trivially: Corollary 2 (, y ) is optimal for LSZ (, y ) is optimal for BLP. Corollary 3 The induced region IR is connected. Corollary 4 If S is nonempty and bounded, IR is nonempty and bounded and an optimal solution (, y ) is attained at an etreme point of IR. Thus, solving LSZ is equivalent to solving BLP, hich is not equivalent to the original BLP since BLP is a relaation of BLP. References [1] C. Audet, P. Hansen, B. Jaumard, and G. Savard. Links beteen linear bilevel and mied 0-1 programg problems. Journal of Optimization Theory and Applications, 93(2): , [2] C. Audet, G. Savard, and W. Zghal. Ne branch-and-cut algorithm for bilevel linear programg. Technical Report G , Les Cahiers du GERAD, Submitted to Journal of Optimization Theory and Applications. [3] J. F. Bard and J. T. Moore. A branch and bound algorithm for the bilevel programg problem. SIAM Journal on Scientific and Statistical Computing, 11(2): , [4] S. Dempe. Foundations of Bilevel Programg. Kluer Academic Publishers, Dordrecht, [5] J. Fortuny-Amat and B. McCarl. A representation and economic interpretation of a to-level programg problem. Journal of the Operations Research Society, 32: , [6] P. Hansen, B. Jaumard, and G. Savard. Ne branch and bound rules for linear bilevel programg. SIAM Journal on Scientific and Statistical Computing, 13: ,

7 [7] B. Jaumard, G. Savard, and X. Xiong. An eact algorithm for conve bilevel programg. Technical Report G 95 33, Les Cahiers du GERAD, [8] J. Lu, C. Shi, and G. Zhang. An etended kth-best approach for linear bilevel programg. Applied Mathematics and Computation, Article in press. [9] J. Lu, C. Shi, and G. Zhang. An etended kuhn-tucker approach for linear bilevel programg. Applied Mathematics and Computation, 162:51 63, [10] J. Lu, C. Shi, and G. Zhang. On the definition of linear bilevel programg solution. Applied Mathematics and Computation, 160: , [11] P. Marcotte and G. Savard. A note on the pareto-optimality of solutions to the linear bilevel programg problem. Computers and Operations Research, 18(4): , [12] G. Savard. Contribution à la programmation mathématique à deu niveau, Ph.D. Thesis. 7