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A Dual Riemannian ADMM Algorithm for Low-Rank SDPs with Unit Diagonal

Jie Wang, Liangbing Hu, Bican Xia

TL;DR

This work develops a dual Riemannian ADMM to solve low-rank semidefinite programs with unit-diagonal constraints by applying a Burer-Monteiro factorization and optimizing on the oblique manifold. It proves global convergence under subproblem accuracy and demonstrates dramatic improvements in accuracy, speed, and scalability over state-of-the-art SDP solvers on second-order SOS relaxations for dense and sparse binary quadratic problems. The ManiDSDP algorithm leverages adaptive rank and penalty updates and shows a distinctive residue-diving behavior that yields extremely high-precision solutions in tens of iterations. The results indicate substantial practical impact for large-scale, structured SDPs arising in polynomial optimization and related problems.

Abstract

This paper proposes a dual Riemannian alternating direction method of multipliers (ADMM) for solving low-rank semidefinite programs with unit diagonal constraints. We recast the ADMM subproblem as a Riemannian optimization problem over the oblique manifold by performing the Burer-Monteiro factorization. Global convergence of the algorithm is established assuming that the subproblem is solved to certain optimality. Numerical experiments demonstrate the excellent performance of the algorithm. It outperforms, by a significant margin, a few advanced SDP solvers (MOSEK, COPT, SDPNAL+, ManiSDP) in terms of accuracy, efficiency, and scalability on second-order SDP relaxations of dense and sparse binary quadratic programs.

A Dual Riemannian ADMM Algorithm for Low-Rank SDPs with Unit Diagonal

TL;DR

This work develops a dual Riemannian ADMM to solve low-rank semidefinite programs with unit-diagonal constraints by applying a Burer-Monteiro factorization and optimizing on the oblique manifold. It proves global convergence under subproblem accuracy and demonstrates dramatic improvements in accuracy, speed, and scalability over state-of-the-art SDP solvers on second-order SOS relaxations for dense and sparse binary quadratic problems. The ManiDSDP algorithm leverages adaptive rank and penalty updates and shows a distinctive residue-diving behavior that yields extremely high-precision solutions in tens of iterations. The results indicate substantial practical impact for large-scale, structured SDPs arising in polynomial optimization and related problems.

Abstract

This paper proposes a dual Riemannian alternating direction method of multipliers (ADMM) for solving low-rank semidefinite programs with unit diagonal constraints. We recast the ADMM subproblem as a Riemannian optimization problem over the oblique manifold by performing the Burer-Monteiro factorization. Global convergence of the algorithm is established assuming that the subproblem is solved to certain optimality. Numerical experiments demonstrate the excellent performance of the algorithm. It outperforms, by a significant margin, a few advanced SDP solvers (MOSEK, COPT, SDPNAL+, ManiSDP) in terms of accuracy, efficiency, and scalability on second-order SDP relaxations of dense and sparse binary quadratic programs.

Paper Structure

This paper contains 12 sections, 10 theorems, 52 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The dual ADMM approach
  4. Exploiting the manifold structure
  5. The ManiDSDP algorithm
  6. Numerical experiments
  7. Dense BQPs
  8. Sparse BQPs
  9. ManiDSDP versus ManiSDP
  10. \ref{['dsdp1']} versus \ref{['dsdp']}
  11. The residue diving phenomenon
  12. Conclusions and future work

Key Result

Lemma 3.2

\newlabellm10 For any $k\ge0$, ${\mathcal{A}}(\widetilde{X}^{k+1})=0$. Moreover, for any $k\ge1$, the subproblem subp2 has a closed-form solution $y^{k+1}=({\mathcal{A}}{\mathcal{A}}^*)^{-1}{\mathcal{A}}(S^{k+1}+C)$.

Figures (4)

  • Figure 1: Comparison of maximal factorization sizes reached through outer iterations of ManiDSDP and ManiSDP.
  • Figure 2: Comparison of numbers of outer iterations taken by ManiDSDP and ManiSDP.
  • Figure 3: Comparison of running time in solving \ref{['dsdp1']} and \ref{['dsdp']}.
  • Figure 4: The maximal KKT residue per iteration with ManiDSDP.

Theorems & Definitions (20)

  • Remark 3.1
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Lemma 4.1: wang2025solving, Lemma 4.1
  • Proposition 4.2
  • Proof 3
  • Proposition 4.3
  • Proof 4
  • ...and 10 more