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Solving Low-Rank Semidefinite Programs via Manifold Optimization

Jie Wang, Liangbing Hu

TL;DR

This work tackles large-scale linear semidefinite programs that admit low-rank solutions, common in moment–SOS relaxations. It blends an inexact augmented Lagrangian method with the Burer–Monteiro factorization and solves ALM subproblems via Riemannian trust-region methods on a fixed-rank manifold, with saddle-point escape mechanics. The authors prove global convergence under milder assumptions and introduce practical boosts: dynamically adjusting the factorization size and self-adaptive penalty parameters, forming the ManiSDP solver. Numerical experiments across diverse SDP relaxations show ManiSDP achieving state-of-the-art efficiency, accuracy, and scalability, often solving problems with millions of constraints that challenge existing solvers. These results suggest a scalable, robust path forward for high-dimensional SDP relaxations in polynomial optimization and related applications.

Abstract

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.

Solving Low-Rank Semidefinite Programs via Manifold Optimization

TL;DR

This work tackles large-scale linear semidefinite programs that admit low-rank solutions, common in moment–SOS relaxations. It blends an inexact augmented Lagrangian method with the Burer–Monteiro factorization and solves ALM subproblems via Riemannian trust-region methods on a fixed-rank manifold, with saddle-point escape mechanics. The authors prove global convergence under milder assumptions and introduce practical boosts: dynamically adjusting the factorization size and self-adaptive penalty parameters, forming the ManiSDP solver. Numerical experiments across diverse SDP relaxations show ManiSDP achieving state-of-the-art efficiency, accuracy, and scalability, often solving problems with millions of constraints that challenge existing solvers. These results suggest a scalable, robust path forward for high-dimensional SDP relaxations in polynomial optimization and related applications.

Abstract

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.
Paper Structure (21 sections, 13 theorems, 74 equations, 8 figures, 10 tables, 2 algorithms)

This paper contains 21 sections, 13 theorems, 74 equations, 8 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. An augmented Lagrangian framework with the Burer-Monteiro factorization
  4. A manifold optimization approach
  5. The Riemannian trust-region method
  6. Escaping from saddle points
  7. Global convergence
  8. The practical algorithm
  9. Dynamically adjusting the factorization size $p$
  10. Self-adaptively updating the penalty parameter $\sigma$
  11. The algorithm
  12. Numerical experiments
  13. The Max-Cut problem
  14. The matrix completion problem
  15. Binary quadratic programs
  16. ...and 6 more sections

Key Result

Lemma 2.1

\newlabellm:optimality0 A matrix $X\in{\mathbb{S}}_n$ is a minimizer of sdp if and only if there exist Lagrange multipliers $y\in{\mathbb{R}}^m$ and $z\in{\mathbb{R}}^l$ such that

Figures (8)

  • Figure 1: The factorization size per iteration in solving \ref{['sec6:bqp']}.
  • Figure 2: The maximal KKT residue per iteration in solving \ref{['sec6:bqp']}.
  • Figure 3: The factorization size per iteration in solving \ref{['sec6:qs']}.
  • Figure 4: The maximal KKT residue per iteration in solving \ref{['sec6:qs']}.
  • Figure 5: The factorization size per iteration in solving \ref{['eq:wahba']}.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Lemma 2.1
  • Proof 1
  • Remark 3.1
  • Remark 3.6
  • Lemma 3.7
  • Proof 2
  • Remark 3.8
  • Lemma 4.1
  • Proof 3
  • Proposition 4.2: cf. wang2023decomposition, Proposition 2.3
  • ...and 23 more