An Effective Method for Optimality Test over Possible Reaction Set for Maximin Solution of Bilevel Linear Programming with Ambiguous Lower-Level Objective Function

Abstract

A bilevel linear optimization problem with ambiguous lower-level objective requires a decision making under uncertainty of rational reaction. With the assumption that the ambiguous coefficient vector of the follower lies in a convex polytope, we apply the maximin solution approach and formulate it as a special kind of three-level programming problem. According to its property that the optimal solution locates on an extreme point, we adopt k-th best method to search the optimal solution equipped with tests for possible optimality, local optimality and global optimality of a solution. In this study, we propose an effective method to verify the rational reaction of the follower which is essential to all steps of optimality test. Our approach uses a relatively small memory to avoid repetition of possible optimality tests. The numerical experiments demonstrate our proposed method significantly accelerates the optimality verification process and eventually computes an optimal solution more efficiently.