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

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 $$\ell _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -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.