New SOCP relaxation and branching rule for bipartite bilinear programs

TL;DR

This paper introduces a novel SOCP relaxation and branching rule for bipartite bilinear programs, improving dual bounds in structural engineering applications compared to existing solvers.

Contribution

The paper presents a stronger SOCP relaxation and a new branching rule specifically designed for bipartite bilinear programs, with demonstrated computational advantages.

Findings

The new SOCP relaxation outperforms standard SDP relaxations.

The proposed branching rule improves dual bounds.

Application to structural engineering problems shows enhanced performance.

Abstract

A bipartite bilinear program (BBP) is a quadratically constrained quadratic optimization problem where the variables can be partitioned into two sets such that fixing the variables in any one of the sets results in a linear program. We propose a new second order cone representable (SOCP) relaxation for BBP, which we show is stronger than the standard SDP relaxation intersected with the boolean quadratic polytope. We then propose a new branching rule inspired by the construction of the SOCP relaxation. We describe a new application of BBP called as the finite element model updating problem, which is a fundamental problem in structural engineering. Our computational experiments on this problem class show that the new branching rule together with an polyhedral outer approximation of the SOCP relaxation outperforms a state-of-the-art commercial global solver in obtaining dual bounds.