# 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.

## Key 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.