A Primal-Dual Smoothing Framework for Max-Structured Non-Convex Optimization

We propose a primal-dual smoothing framework for finding a near-stationary point of a class of nonsmooth nonconvex optimization problems with max-structure. We analyze the primal and dual gradient complexities of the framework via two approaches, that is, the dual-then-primal and primal-the-dual smoothing approaches. Our framework improves the best-known oracle complexities of the existing method, even in the restricted problem setting. As an important part of our framework, we propose a first-order method for solving a class of (strongly) convex-concave saddle-point problems, which is based on a newly developed non-Hilbertian inexact accelerated proximal gradient algorithm for strongly convex composite minimization that enjoys duality-gap convergence guarantees. Some variants and extensions of our framework are also discussed. Funding: R. Zhao’s research is partially supported by AFOSR Grant FA9550-22-1-0356.