An efficient approach for nonconvex semidefinite optimization via customized alternating direction method of multipliers

TL;DR

This paper introduces a scalable ADMM-based method for solving nonconvex semidefinite optimization problems, including MAX-CUT and community detection, with proven convergence to stationary points and promising empirical results.

Contribution

It proposes two novel reformulations for nonconvex semidefinite problems and proves convergence of ADMM to stationary points under mild conditions.

Findings

ADMM converges to stationary points in nonconvex settings.

The method performs well on MAX-CUT, community detection, and image segmentation.

Local optima are often global when matrix factors are sufficiently wide.

Abstract

We investigate a class of general combinatorial graph problems, including MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints and solved via the alternating direction method of multipliers (ADMM). We propose two reformulations: one using vector variables and a binary constraint, and the other further reformulating the Burer-Monteiro form for simpler subproblems. Despite the nonconvex constraint, we prove the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this latter form, when the matrix factors are wide enough, local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation benchmark and simulated examples.