Global optimality test for maximin solution of bilevel linear programming with ambiguous lower-level objective function

A bilevel linear programming problem with ambiguous lower-level objective function is a sequential decision making under uncertainty of rational reaction. The ambiguous lower-level objective function is assumed that the 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. Since an optimal solution exists at a vertex of feasible region, we adopt k-th best method to search an optimal solution. At each iteration of the k-th best method, we check rationality, local optimality and global optimality of the candidate solution. In this study, we propose a global optimality test based on an inner approximation method and compare its computational efficiency to other test methods based on vertex enumeration. We also extensively utilize the history of rationality tests to verify the rationality of the solution in the follower's problem. Numerical experiments show the advantages of the proposed methods.