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

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.

References in corpus (5)

• Global rates of convergence for nonconvex optimization on manifolds
• No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis
• Matrix Completion has No Spurious Local Minimum
• A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints
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Cited by in corpus (1)

• Stochastic Variance-Reduced Prox-Linear Algorithms for Nonconvex Composite Optimization