A Mixed-Integer Linear Programming Reduction of Disjoint Bilinear Programs via Symbolic Variable Elimination

Jihwan Jeong, Scott Sanner, Akshat Kumar

Abstract: A disjointly constrained bilinear program (DBLP) has various practical and industrial applications, e.g., in game theory, facility location, supply chain management, and multi-agent planning problems. Although earlier work has noted the equivalence of DBLP and mixed-integer linear programming (MILP) from an abstract theoretical perspective, a practical and exact closed-form reduction of a DBLP to a MILP has remained elusive. Such explicit reduction would allow us to leverage modern MILP solvers and techniques along with their solution optimality and anytime approximation guarantees. To this end, we provide the first constructive closed-form MILP reduction of a DBLP by extending the technique of symbolic variable elimination (SVE) to constrained optimization problems with bilinear forms. We apply our MILP reduction method to difficult DBLPs including XORs of linear constraints and show that we significantly outperform Gurobi. We also evaluate our method on a variety of synthetic instances to analyze the effects of DBLP problem size and sparsity w.r.t. MILP compilation size and solution efficiency.