Solving sparse separable bilinear programs using lifted bilinear cover inequalities

TL;DR

This paper investigates the effectiveness of lifted bilinear cover inequalities in improving the solution of separable bilinear programs, showing they significantly enhance solver performance when added at the root node.

Contribution

It introduces a simple heuristic for separating lifted bilinear cover inequalities and demonstrates their computational benefits over traditional relaxations.

Findings

Lifted bilinear cover inequalities improve gap closure in global optimization.

Semi-definite relaxation offers no advantage over McCormick relaxation for these problems.

Adding inequalities at the root node significantly enhances solver performance.

Abstract

Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semi-definite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick's relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared to when they are not.