On the Leader's Dilemma and a New Idea for Attacking Bilevel Programming Problems

Bilevel programming problems are of growing interest both from a theoretical and a practical points of view. They can be interpreted as certain non-cooperative non-zero sum hierarchical games. Here, one player, the so-called follower, has to solve a parametric optimization problem when computing his decision. If this decision is not uniquely determined, then the other player, the so-called leader, is not able to predict the real outcome of the game since his objective depends on his own choice as well as on the follower's response. In this paper, we treat this case and propose a new idea for generating a satisfactory decision for the leader. In short, we propose the leader to take on an optimistic position combined with a following perturbation of the decision thus obtained. Practicability of a major part of this approach is also shown. 1 Bilevel programming problems Bilevel programming problems are often interpreted within a gametheoretical framework as generalized Stackelberg games 27], i.e. as certain non-cooperative non-zero sum hierarchical games in which a so-called leader has the rst choice and acts rst, while the so-called follower is forced to react optimally on the leader's choice y. Now, assume that the follower chooses his strategy x(y) as an optimal solution of, say, problem x(y) 2 (y) := Argmin x fh 0 (x; y) j h(x; y) 0g: (1.1) Thereby, if this solution is not uniquely determined, the follower's concrete choice is out of the leader's control. In this case, if the leader's objective function value f 0 (x(y); y) depends