Exact and Heuristic Methods for $Γ$-Robust Min-Max Problems

Abstract

Bilevel optimization is a powerful tool for modeling hierarchical decision-making processes, which arise in various real-world applications. Due to their nested structure, however, bilevel problems are intrinsically hard to solve, even if all variables are continuous and all parameters of the problem are exactly known. Further challenges arise if mixed-integer aspects and problems under uncertainty are considered. In this article, we summarize selected results from the author's dissertation. We study mixed-integer linear min-max problems with a $Γ$-robust treatment of uncertain data, for which we present exact and heuristic solution approaches. The performance of the methods is assessed in a computational study on 560 instances of the knapsack interdiction problem. Our results show that the heuristic closes the optimality gap for a significant portion of the considered instances and often practically outperforms both heuristic and exact benchmark approaches.