Proximal algorithms for constrained composite optimization, with applications to solving low-rank SDPs
This work addresses optimization challenges in low-rank semidefinite programming, which is incremental as it builds on existing proximal methods and factorization techniques.
The paper tackles constrained composite optimization problems by proving that proximal algorithms applied to exact penalty formulations achieve local linear convergence under a quadratic growth condition, with concrete results applied to low-rank semidefinite optimization using Burer-Monteiro factorizations.
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.