Perturbation Techniques for Convergence Analysis of Proximal Gradient Method and Other First-Order Algorithms via Variational Analysis

We develop new perturbation techniques for conducting convergence analysis of various first-order algorithms a class nonsmooth optimization problems. consider the iteration scheme an algorithm to construct perturbed stationary point set-valued map, and define perturbing parameter by difference two consecutive iterates. Then, we show that calmness condition induced together with local version proper separation value condition, is sufficient ensure linear algorithm. The equivalence one canonically map proved, this allows us derive some conditions using recent developments in variational analysis. These are different from existing results (especially, those error-bound-based ones) they can be easily verified many concrete application models. Our focused on fundamental proximal gradient (PG) method, it enables any accumulation sequence generated PG method must terms subdifferential, instead limiting subdifferential. This result finds surprising fact solution quality found general superior. also leads improvement convex case. technique conveniently used rate number other methods including well-known alternating direction multipliers primal-dual hybrid under mild assumptions.