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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10898_2025_1523_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月1日

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  • Soodeh Habibi,
  • Michal Kočvara,
  • Michael Stingl

摘要

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 \(\ell _1\) ℓ 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.