On the numerical solution of Lasserre relaxations of unconstrained binary quadratic optimization problem

TL;DR

This paper develops an efficient interior point method with a specialized preconditioner for solving Lasserre relaxations of unconstrained binary quadratic problems, demonstrating improved performance on MAXCUT and similar problems.

Contribution

It introduces a novel interior point approach with a low-rank structure-based preconditioner and an $ ext{l}_1$-penalty reformulation for solving higher-order Lasserre relaxations efficiently.

Findings

The proposed method outperforms state-of-the-art solvers on benchmark problems.

Second-order relaxations often suffice to find globally optimal solutions.

A hybrid ADMM-interior point method is effective for certain problem classes.

Abstract

The aim of this paper is to solve linear semidefinite programs arising from higher-order Lasserre relaxations of unconstrained binary quadratic optimization problems. For this we use an interior point method with a preconditioned conjugate gradient method solving the linear systems. The preconditioner utilizes the low-rank structure of the solution of the relaxations. In order to fully exploit this, we need to re-write the moment relaxations. To treat the arising linear equality constraints we use an $\ell_1$-penalty approach within the interior-point solver. The efficiency of this approach is demonstrated by numerical experiments with the MAXCUT and other randomly generated problems and a comparison with a state-of-the-art semidefinite solver and the ADMM method. We further propose a hybrid ADMM-interior-point method that proves to be efficient for certain problem classes. As a by-product, we observe that the second-order relaxation is often high enough to deliver a globally optimal solution of the original problem.