Proximal algorithms for constrained composite optimization, with applications to solving low-rank SDPs

TL;DR

This paper introduces proximal algorithms for constrained composite optimization problems, demonstrating local linear convergence under certain conditions, with applications to low-rank semidefinite programming and matrix factorization.

Contribution

It provides a novel analysis of non-smooth geometry and establishes convergence results for proximal algorithms applied to non-convex problems with applications to low-rank SDPs.

Findings

Proximal algorithms achieve local linear convergence under quadratic growth conditions.

Identifies conditions for quadratic growth in low-rank semidefinite problems.

Offers insights into matrix factorization structures for optimization.

Abstract

We study a family of (potentially non-convex) constrained optimization problems with convex composite structure. Through a novel analysis of non-smooth geometry, we show that proximal-type algorithms applied to exact penalty formulations of such problems exhibit local linear convergence under a quadratic growth condition, which the compositional structure we consider ensures. The main application of our results is to low-rank semidefinite optimization with Burer-Monteiro factorizations. We precisely identify the conditions for quadratic growth in the factorized problem via structures in the semidefinite problem, which could be of independent interest for understanding matrix factorization.