Valid Inequalities for Mixed Integer Bilevel Linear Optimization Problems

Abstract

Despite the success of branch-and-cut methods for solving mixed integer bilevel linear optimization problems (MIBLPs) in practice, there are still gaps in both the theory and practice surrounding these methods. In the first part of this paper, we lay out a basic theory of valid inequalities and cutting-plane methods for MIBLPs that parallels the existing theory for mixed integer linear optimization problems (MILPs). We provide a general scheme for classifying valid inequalities and illustrate how the known classes of valid inequalities fit into this categorization, as well as generalizing several existing classes. In the second part of the paper, we assess the computational effectiveness of these valid inequalities and discuss the myriad challenges that arise in integrating methods of dynamically generating inequalities valid for MIBLPs into a branch-and-cut algorithms originally designed for solving MILPs. Although branch-and-cut methods for solving for MIBLPs are in principle straightforward generalizations of those used for MILP, there are subtle but important differences and there remain many unanswered questions regarding how to suitably modify control mechanisms and other algorithmic details in order to ensure performance in the MIBLP setting. We demonstrate that performance of version 1.2 of the open-source solver MibS was substantially improved over that of version 1.1 through a variety of improvements to the previous implementation.