Projected subgradient methods for paraconvex optimization: Application to robust low-rank matrix recovery

This paper is devoted to the class of paraconvex functions and presents some of its fundamental properties, characterization, and examples that can be used for their recognition and optimization. Next, the convergence analysis of the projected subgradient methods with several step-sizes (i.e., constant, nonsummable, square-summable but not summable, geometrically decaying, and Scaled Polyak's step-sizes) to global minima for this class of functions is studied. In particular, the convergence rate of the proposed methods is investigated under paraconvexity and the H"olderian error bound condition, where the latter is an extension of the classical error bound condition. The preliminary numerical experiments on several robust low-rank matrix recovery problems (i.e., robust matrix completion, image inpainting, robust nonnegative matrix factorization, robust matrix compression, and robust image deblurring) indicate promising behavior for these projected subgradient methods, validating our theoretical foundations.