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

TL;DR

This paper introduces a dual Riemannian ADMM algorithm for low-rank semidefinite programs with unit diagonal constraints, demonstrating superior performance over existing solvers in accuracy and efficiency.

Contribution

The paper develops a novel dual Riemannian ADMM approach using Burer-Monteiro factorization for low-rank SDPs with unit diagonal, with proven convergence and improved empirical results.

Findings

Outperforms existing SDP solvers in accuracy and speed

Effective on dense and sparse binary quadratic program relaxations

Global convergence established under certain conditions

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.