New Understandings and Computation on Augmented Lagrangian Methods for Low-Rank Semidefinite Programming

TL;DR

This paper provides a rigorous theoretical analysis of augmented Lagrangian methods combined with Burer-Monteiro factorization for low-rank semidefinite programming, establishing conditions for global optimality and proposing a practical, rank-adaptive algorithm with strong numerical results.

Contribution

It offers the first comprehensive theoretical understanding of ALM-BM subproblems, including their structural properties and solvability by first-order methods, and introduces ALORA, a novel rank-adaptive algorithm.

Findings

ALM subproblems inherit low-rankness and strict complementarity under primal simplicity.

Non-convex ALM-BM subproblems can be solved to global optimality via gradient descent.

ALORA efficiently solves large-scale SDPs, demonstrating practical scalability.

Abstract

Augmented Lagrangian Method (ALM) combined with Burer-Monteiro (BM) factorization, dubbed ALM-BM, offers a powerful approach for solving large-scale low-rank semidefinite programs (SDPs). Despite its empirical success, the theoretical understandings of the resulting non-convex ALM-BM subproblems, particularly concerning their structural properties and efficient subproblem solvability by first-order methods, still remain limited. This work addresses these notable gaps by providing a rigorous theoretical analysis. We demonstrate that, under appropriate regularity of the original SDP, termed as primal simplicity, ALM subproblems inherit crucial properties such as low-rankness and strict complementarity when the dual variable is localized. Furthermore, ALM subproblems are shown to enjoy a quadratic growth condition, building on which we prove that the non-convex ALM-BM subproblems can be solved to global optimality by gradient descent, achieving linear convergence under conditions of local initialization and dual variable proximity. Through illustrative examples, we further establish the necessity of these local assumptions, revealing them as inherent characteristics of the problem structure. Motivated by these theoretical insights, we propose ALORA, a rank-adaptive augmented Lagrangian method that builds upon the ALM-BM framework, which dynamically adjusts the rank using spectral information and explores negative curvature directions to navigate the nonconvex landscape. Exploiting modern GPU computing architectures, ALORA exhibits strong numerical performance, solving SDPs with tens of millions of dimensions in hundreds of seconds.