Bilevel Mixed-Integer Linear Program with Binary Tender

Abstract: Bilevel programs model sequential decision interactions between two sets of players and find wide applications in real-world complex systems. In this paper, we consider a bilevel mixed-integer linear program with binary tender, wherein the upper and lower levels are linked via binary decision variables and both levels may involve additional mixed-integer decisions. We recast this bilevel program as a single-level formulation through a value function for the lower-level problem and then propose valid inequalities to replace and iteratively approximate the value function. We first derive a family of Lagrangian-based valid inequalities that give a complete description of the value function, providing a baseline method to obtain exact solutions for the considered class of bilevel programs. To enhance the strength of this approach, we further investigate another two types of valid inequalities. First, when the lower-level value function has intrinsic special properties such as supermodularity or submodularity, we exploit such properties to separate the Lagrangian-based inequalities quickly. Second, we derive decision rule-based valid inequalities, where linear decision rules and learning techniques are explored respectively. We demonstrate the effectiveness and efficiency of the proposed methods in extensive numerical experiments, including instances of general bilevel mixed-integer programs and those of a facility location interdiction problem.