A Decomposition Augmented Lagrangian Method for Low-rank Semidefinite Programming

TL;DR

This paper introduces a decomposition augmented Lagrangian method for efficiently solving complex semidefinite programming problems, leveraging matrix factorization and manifold optimization techniques.

Contribution

It presents a novel framework combining decomposition, augmented Lagrangian, and manifold methods to handle broad SDP problems with nonlinear and nonsmooth features.

Findings

Effective on large-scale problems like max-cut and PCA

Outperforms existing state-of-the-art methods

Provides convergence guarantees under certain conditions

Abstract

We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems, possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints. In particular, the positive semidefinite variable along with a group of linear constraints can be transformed into a variable on a smooth manifold via matrix factorization. The nonsmooth regularization and other general linear constraints are handled by the augmented Lagrangian method. Therefore, each subproblem can be solved by a semismooth Newton method on a manifold. Theoretically, we show that the first and second-order necessary optimality conditions for the factorized subproblem are also sufficient for the original subproblem under certain conditions. Convergence analysis is established for the Riemannian subproblem and the augmented Lagrangian method. Extensive numerical experiments on large-scale semidefinite programming problems such as max-cut, nearest correlation estimation, clustering, and sparse principal component analysis demonstrate the strength of our proposed method compared to other state-of-the-art methods.