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<p>We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions, with an emphasis on SDP relaxations for polynomial optimization problems. This approach incorporates the inexact augmented Lagrangian method (ALM) and the Burer-Monteiro factorization, and features the self-adaptive strategies for updating the factorization size and the penalty parameter. We establish global convergence of the inexact ALM, despite the non-convexity brought by the Burer-Monteiro factorization. We further provide a practical algorithm building on the inexact ALM, and along with the algorithm we release an open-source SDP solver <Emphasis FontCategory="NonProportional">ManiSDP</Emphasis>. Comprehensive numerical experiments demonstrate that <Emphasis FontCategory="NonProportional">ManiSDP</Emphasis> achieves state-of-the-art in terms of efficiency, accuracy, and scalability, and is faster than several advanced SDP solvers (<Emphasis FontCategory="NonProportional">MOSEK</Emphasis>, <Emphasis FontCategory="NonProportional">SDPLR</Emphasis>, <Emphasis FontCategory="NonProportional">SDPNAL+</Emphasis>, <Emphasis FontCategory="NonProportional">STRIDE</Emphasis>) by up to orders of magnitudes on a variety of linear SDPs. The largest SDP solved by <Emphasis FontCategory="NonProportional">ManiSDP</Emphasis> (in about 8.5 hours with maximal KKT residue 3.5e-13) is the second-order moment relaxation of a binary quadratic program with 120 variables, which has matrix dimension 7261 and contains 17,&#xa0;869,&#xa0;161 affine constraints.</p>

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Solving Low-Rank Semidefinite Programs via Manifold Optimization

  • 期刊论文

  • 发表日期：

    2025年5月31日

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  • Jie Wang,
  • Liangbing Hu

摘要

We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions, with an emphasis on SDP relaxations for polynomial optimization problems. This approach incorporates the inexact augmented Lagrangian method (ALM) and the Burer-Monteiro factorization, and features the self-adaptive strategies for updating the factorization size and the penalty parameter. We establish global convergence of the inexact ALM, despite the non-convexity brought by the Burer-Monteiro factorization. We further provide a practical algorithm building on the inexact ALM, and along with the algorithm we release an open-source SDP solver ManiSDP. Comprehensive numerical experiments demonstrate that ManiSDP achieves state-of-the-art in terms of efficiency, accuracy, and scalability, and is faster than several advanced SDP solvers (MOSEK, SDPLR, SDPNAL+, STRIDE) by up to orders of magnitudes on a variety of linear SDPs. The largest SDP solved by ManiSDP (in about 8.5 hours with maximal KKT residue 3.5e-13) is the second-order moment relaxation of a binary quadratic program with 120 variables, which has matrix dimension 7261 and contains 17, 869, 161 affine constraints.