Linear Bilevel Programming With Upper Level Constraints Depending on the Lower Level Solution

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1 Linear Bilevel Programming With Upper Level Constraints Depending on the Lower Level Solution Ayalew Getachew Mersha and Stephan Dempe October 17, 2005 Abstract Focus in the paper is on the definition of linear bilevel programming problems, the existence of optimal solutions and necessary as well as sufficient optimality conditions. In the papers [9] and [10] the authors claim to suggest a refined definition of linear bilevel programming problems and related optimality conditions. Mainly their attempt reduces to shifting upper level constraints involving both the upper and the lower level variables into the lower level. We investigate such a shift in more details and show that it is not allowed in general. We show that an optimal solution of the bilevel programm exists under the conditions in [10] if we add the assumption that the inducible region is not empty. The necessary optimality condition reduces to check optimality in one linear programming problem. Optimality of one feasible point for a certain number of linear programs implies optimality for the bilevel problem. 1 Introduction The bilevel programming problem is an optimization problem whose constraints are (in part) determined by an other optimization problem. In other words it is an hierarchical optimization problem consisting of two levels, the Department of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Germany Department of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Germany 1

2 first of which (the leader s level) is dominant over the other (the follower s one). The order of the play is very important. The choice of the dominant level limits or highly affects the choice or strategy of the lower level. Knowing the selection of the leader, the follower solves an ordinary parametric optimization problem. Anticipating the reaction of the follower the leader intends to find such values for his variables which together with the follower s reaction minimize his objective function. Hence, bilevel programming can be used to describe models for decision making situations where an hierarchy exists. Due to the hierarchical structure the bilevel programming problem is not convex and not differentiable. These problems have attainted large interest both from the theoretical as well as from the practical points of view, see [1], [2], [3] and the references therein. Bilevel programming problems are sensitive with respect to the addition of constraints to the lower level. Even the addition of a constraint which is not active at an optimal solution can change the problem drastically [7]. In this paper focus is on the shift of constraints between the upper and the lower level problems. Usually the position of constraints in the modelling phase depends strongly on the application under consideration. Problems with upper level constraints involving both the upper and the lower level variables (called connecting constraints in what follows) can arise e.g. if the leader is not willing (or able) to accept certain optimal solutions of the follower. This may be strongly related to the existence of multiple optimal lower level solutions or not. We will show, that the shift of such connecting constraints can change the problem essentially. This shows that the ideas in the papers [9] and [10] being essentially based on the shift of constraints violate the modelling procedure. Next we give conditions guaranteeing the existence of optimal solutions of the bilevel programming problem with connecting constraints (which is the main aim in [10]). The paper is then closed with optimality conditions for such problems thus generalizing (or correcting) the results in [9]. 2 Formulation of bilevel programming problems In this paper we follow mainly the definition given by Bard [1]: 2

3 min c 1 x + d 1 y x subject to (1) where y solves subject A 1 x + B 1 y b 1 to min y c 2 x + d 2 y A 2 x + B 2 y b 2, (2) where x R n, y R m, c 1, c 2 R n, d 1, d 2 R m, b 1 R p, b 2 R q. The only difference is the appearance of the quotation marks around the minimum operator in the upper level objective function. These are used to express certain ambiguity of the problem formulation in the case of nonunique lower level solutions. We will come back to this problem later. Based on these we get the following definitions (see [1]): 1. Constraint set of the problem M = {(x, y) : A 1 x + B 1 y b 1, A 2 x + B 2 y b 2 } 2. Feasible set for the follower for each x M(x) = {y : B 2 y b 2 A 2 x} 3. Projection of M onto the leader s decision space M(X) = {x : y, A 1 x + B 1 y b 1, A 2 x + B 2 y b 2 } 4. Follower rational reaction set for x M(X), Ψ L (x) = Argmin y 5. Inducible region {c 2 x + d 2 y : y M(x)} IR = {(x, y) : (x, y) M, y Ψ L (x)}. The inducible region is the most important object since it defines the feasible set at least in the case when the lower level problem has unique optimal solutions for all values of x. Then the quotation marks can be dropped. Note that the constraints A 2 x + B 2 y b 2 can be dropped in the set M since they are satisfied necessarily by an optimal solution y Ψ L (x) of problem (2). 3

4 Definition 1 Consider problem (1), (2) and assume that the rational reaction set of the follower consists of at most one point for all x M(X): Ψ L (x) 1 for all x M(X). Then, a point (x, y) IR is an optimal solution of problem (1), (2) if c 1 x + d 1 y c 1 x + d 1 y (x, y) IR. The situation is a little bit different in the case when the lower level problem can have multiple optimal solutions. The difficulty is due to an unclear value of the leader s objective function (and feasibility of the point (x, y)) prior to the publication of the optimal solution taken by the follower. If the leader has control over x only he is then not able to predict this value and hence he cannot find a best decision. There are mainly two ways out of this situation discussed in the literature: the optimistic (or weak) and the pessimistic (or strong) formulation of the bilevel programming problem, see e.g. [2],[4],[5]. The pessimistic formulation needs to be selected if the leader is not able to influence the follower in the sense that he can ask the follower to select one preferable solution out of Ψ L (x). Then, the leader is forced to protect himself against bad selections of the follower. But, if the leader can influence the follower s selection he is able to motivate the follower to select a point y Argmin {c 1 x + d 1 y : y Ψ L (x)}, y i.e. a best solution in Ψ L (x) from the leader s point of view. The resulting problem is equivalent to the following one provided this has an optimal solution [6]: min {c 1x + d 1 y : (x, y) IR}. (3) x,y Hence, we get the following definition of an optimal solution: Definition 2 Consider problem (1), (2) in the optimistic formulation, i.e. consider problem (3). Then, a point (x, y) IR is an optimal solution if c 1 x + d 1 y c 1 x + d 1 y (x, y) IR. The point (x, y) IR is a local optimal solution for (3) provided that this inequality is valid for all (x, y) IR sufficiently close to (x, y). This implies now that the problem with uniquely solvable lower level problems is a special case of the optimistic formulation and we need only to consider the last one when searching for sufficient conditions for the existence of an optimal solution. 4

5 3 Existence of optimal solution Consider the assumption A1: M is a nonempty compact set. Then we have the following lemma. Lemma 1 If A1 holds then the set IR is closed. Proof: Since M is not empty there is at least one parameter value x M(X) with M(x). By compactness, Ψ L (x). Consider a sequence {(x k, y k )} k=1 IR converging to (x, y). Then, by well-known results of linear parametric optimization [8], y Ψ L (x). Hence, (x, y) IR showing the proof. q.e.d. The same result has been shown e.g. in [1] under the additional assumption that the lower level problem has at most one optimal solution for all parameter values. Corollary 2 If assumption A1 is satisfied and there do not exist connecting constraints, then problem (1),(2) has an optimal solution. The proof follows from the Weierstraß Theorem since IR is a nonempty compact set in this case. The main idea in the papers [10] reduces to shifting connecting constraints to the lower level to get the existence of an optimal solution. They use the following example with the intention to motivate that the problem formulation (1),(2) is not correct. Example 1 Consider the following example subject to where y solves min x 0 x 8y 5x + 2y 33 x + 2y 9 subject to min y 0 y 7x + 3y 5 x + y 15. 5

6 Clearly M is a non-empty, compact set. But the bilevel problem has no solution since IR =. If we shift the upper level constraints to the lower level then M remains closed and bounded. Moreover, the resulting problem has solution. Therefore, from this example one can see that one needs more assumptions besides non-emptiness and compactness of M to guarantee the existence of a solution. Namely, Lemma 1 does not imply that the set IR =. A2: IR is non-empty. Lemma 3 If A1 and A2 are satisfied then IR is non-empty and is the finite union of compact polyhedral sets. The proof follows since the graph grph Ψ L ( ) of Ψ L ( ) is polyhedral [8] and IR is equal to the intersection of two polyhedral sets. Corollary 4 If A1 and A2 are satisfied and Ψ L (x) 1 for all x M(X), then IR describes a piecewise linear function. Corollary 5 If A1 and A2 are satisfied then IR is equal to the union of faces of the polyhedral set M. Theorem 6 If A1 and A2 are satisfied problem (3) has a solution. Proof: By Lemma 1, IR is closed and due to IR M it is also bounded. By A2 it is also not empty. Hence, problem (3) consists in minimizing a continuous function over a compact non-empty set, which implies the proof by Weierstraß s Theorem. q.e.d. However, unlike in the optimistic case the existence of a solution in the pessimistic case is generally not guaranteed even under the above assumptions [2]. 4 The effect of shifting constraints from one level to the other on the solution of BLP As we tried to indicate in the previous section, it is easy to see that the model may completely change if constraints are shifted from one level to the other. Different authors have already noticed this. See [1],[2] and others. 6

7 Figure 1: The problem with upper level connecting constraints. The feasible set IR is depicted in bold lines. Global optimal solution at point C, local optimal solution at point A. However, as we mentioned above recently [10] reported a result based on shifted constraints. The following simple example shows that the optimal solution of bilevel linear optimization problem may be not longer optimal if the constraints are shifted from one level to the other. Example 2 Consider the problem subject to where y solves subject to min x 2y x 2x 3y 12 x + y 14 min y y 3x + y 3 3x + y 30 The optimal solution for this problem is (x, y) = (8, 6) (see fig. 1). But if we shift the two upper level constraints to the lower level we get ( x, ỹ) = (6, 8) as an optimal solution (see fig. 2). So it is easy to see from this example that if we shift constraints from the upper level to the lower one, the optimal solution obtained prior to shifting is no more optimal. Hence ideas based on shifting constraints from one level to another will lead to a solution which may not be a solution prior to shifting constraints. 7

8 Figure 2: The problem without upper level connecting constraints. The feasible set IR is depicted in bold lines. Global optimal solution at point B. 5 Optimality Conditions The following lemma gives the necessary condition: Lemma 7 [1] [2] A necessary condition that (x, y ) solves the optimistic linear bilevel programming problem (3) locally is that there exists a vector λ such that (x, y, λ ) locally solves min c 1 x + d 1 y subject to A 1 x + B 1 y b 1 A 2 x + B 2 y b 2 B 2 λ + d 2 = 0 λ (b 2 A 2 x B 2 y) = 0 λ 0 (4) This result is a simple consequence from duality of linear programming (i.e. that the Karush-Kuhn-Tucker conditions are sufficient and necessary optimality conditions in linear programming). Moreover, this result is valid for all optimal solutions of the dual linear programming problem to (2). However, the opposite implication is not true in general if the optimal solution of the dual to (2) is not uniquely determined, cf. fig. 3. The reason for this is that new variables have been added to the optimistic bilevel programming problem (3). 8

9 The above problem can be investigated by decomposing it into a system which can be handled by linear programming. For this take any index set Ĩ with {i : λ i > 0} Ĩ {i : (A 2x + B 2 y b 2 ) i = 0} and consider min c 1 x + d 1 y subject to A 1 x + B 1 y b 1 A 2 x + B 2 y b 2 B 2 λ + d 2 = 0 (A 2 x + B 2 y b 2 ) i = 0, i Ĩ λ i = 0, i / Ĩ λ 0 (5) Here, the matrix (B 2 ) i Ĩ has full row rank. Then, if (x, y ) a local optimal solution of (3), (x, y, λ ) is a global optimal solution of (5). Now consider min c 1 x + d 1 y subject to A 1 x + B 1 y b 1 A 2 x + B 2 y b 2 (A 2 x + B 2 y b 2 ) i = 0, i Ĩ where the dual variable λ is a solution of (6) B 2 λ + d 2 = 0 λ i = 0, i / Ĩ (7) λ 0 with sets Ĩ D(x, y) := {I : λ 0, B T 2 λ + d 2 = 0, λ i = 0, i / I, (A 2 x + B 2 y b 2 ) i = 0, i I, (B 2 ) i Ĩ has full rank}. Then, if (x, y ) is a local optimal solution of (3), (x, y ) is an optimal solution of problem (6) for all Ĩ D(x, y ). Moreover, if (x, y ) is an optimal solution of (6) for all Ĩ D(x, y ) then (x, y ) is a local optimal solution of (3). Fig. 3 illustrates this. If (x, y, λ ) is a local optimal solution of problem (4) then each point (x, y, λ) in a sufficient small neighborhood of (x, y, λ ) cannot have a smaller objective function value. The neighborhood is shown in the left picture of fig. 3. By duality this neighborhood is associated to a facet of M corresponding to some part of IR (the part on the right-hand side of the point (x 0, y 0 )). The point (x 0, y 0 ) is the optimal solution of the problem (6) for the set I corresponding to λ 0. But, the point (x 0, y 0 ) is not optimal for (3). 9

10 Figure 3: Necessary optimality condition for MPEC is not sufficient for bilevel problem Theorem 8 The point (x, y ) is an optimal solution of the problem (3) if and only if it is an optimal solution of the problem (6) for all Ĩ D(x, y ). Proof: Since the Karush-Kuhn-Tucker conditions are necessary and sufficient optimality conditions for the lower level problem, local optimality of (x, y ) for the bilevel programming problem implies that (x, y, λ ) is an optimal solution for the problems (6) for each solution λ for (7). To show the if part of the theorem assume that (x, y ) IR is not an optimal solution of (3). Then there exists a sequence {(x n, y n )} converging to (x, y ) with (x n, y n ) IR, c 1 x n + d 1 y n < c 1 x + d 1 y for all n. Since (x n, y n ) IR, A 1 x n + B 1 y n b 1, y n Ψ(x n ), i.e λ n 0, B 2 λ n + d 2 = 0, λ n (b 2 A 2 x n B 2 y n ) = 0. Here, λ n can be taken as vertex of {λ 0 : B 2 λ + d 2 = 0, λ T (b 2 A 2 x n B 2 y n ) = 0.} Hence also as a vertex of {λ 0 : B 2 λ + d 2 = 0, λ i = 0, for i / I n } 10

11 for some set I n satisfying (b 2 A 2 x n B 2 y n ) i = 0, i I n. Since the number of different sets I {1, 2,...q} is finite there exists an infinite subsequence of {(x n, y n, λ n )} with λ n is a vertex of {λ 0 : B 2 λ + d 2 = 0, λ i = 0, for i / Î} for a fixed set Î satisfying (b 2 A 2 x B 2 y ) i = 0, i I for all n. This means that Î D(x, y ). This shows that (x, y ) can not an optimal solution of (6) for this set Î. q.e.d. 6 Conclusion In this paper we have discussed conditions under which the existence of an optimal solution of the linear bilevel programming problem is guaranteed. If the optimistic case is considered and the inducible region IR is not empty and compact an optimal solution of the bilevel programming problem exists. Ideas being based on shifting constraints between the upper and the lower level problems violate the modelling procedure. One example illustrates this fact is given in this paper. Necessary and sufficient optimality condition are derived using the Karush-Kuhn-Tucker reformulation of the optimistic bilevel programming problem. References [1] J.F. Bard. Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Dordrecht, [2] S. Dempe. Foundations of Bilevel Programming. Kluwer Academie Publishers, Dordrecht, [3] S. Dempe. Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization, 52: , [4] P. Loridan and J. Morgan. Least-norm regularization for weak two-level optimization problems. In Optimization, Optimal Control and Partial Differential Equations, volume 107 of International Series of Numerical Mathematics, pages Birkhäuser Verlag, Basel, [5] P. Loridan and J. Morgan. Weak via strong Stackelberg problem: New results. Journal of Global Optimization, 8: ,

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