Efficient Rank Minimization to Tighten Semidefinite Programming for Unconstrained Binary Quadratic Optimization

TL;DR

This paper introduces a computationally efficient method for low-rank semidefinite programming that enhances solutions to unconstrained binary quadratic problems by reducing rank and improving objective values.

Contribution

It presents a novel low-rank semidefinite programming approach that does not require positive-semidefinite projection, improving solution quality for large-scale binary quadratic problems.

Findings

Lower rank solutions achieved in large-scale problems

Objective function improvements demonstrated

Efficient method tested on Gset and Biq Mac collections

Abstract

We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to achieve a lower rank solution. This procedure is computationally efficient as it does not require projecting on the cone of positive-semidefinite matrices. Its performance in terms of objective improvement and rank reduction is tested over multiple graphs of large-scale Gset graph collection and over binary optimization problems from the Biq Mac collection.