  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.
