Disjoint Bilinear Optimization: A Two-Stage Robust Optimization Perspective

Abstract

In this paper, we focus on a subclass of quadratic optimization problems, that is, disjoint bilinear optimization problems. We first show that disjoint bilinear optimization problems can be cast as two-stage robust linear optimization problems with fixed-recourse and right-hand-side uncertainty, which enables us to apply robust optimization techniques to solve the resulting problems. To this end, a solution scheme based on a blending of three popular robust optimization techniques is proposed. For disjoint bilinear optimization problems with a polyhedral feasible region and a general convex feasible region, we show that, under mild regularity conditions, the convex relaxations of the original bilinear formulation and its two-stage robust reformulation obtained from a reformulation-linearization-based technique and linear decision rules, respectively, are equivalent. For generic bilinear optimization problems, the convex relaxations from the reformulation-linearization-based technique are generally tighter than the one from linear decision rules. Numerical experiments on bimatrix games, synthetic disjoint bilinear problem instances, and convex maximization problems demonstrate the efficiency and effectiveness of the proposed solution scheme. Summary of Contribution: Computing solutions for disjoint bilinear optimization problems are of much interest in real-life applications, yet they are, in general, computationally intractable. This paper proposes a computationally tractable approximation as well as a convergent algorithm to the optimal values of such problems. Extensive computational experiments on (i) (constrained) bimatrix games, (ii) synthetic disjoint bilinear problems, and (iii) convex maximization problems are conducted to demonstrate the effectiveness and efficiency of the proposed approach.