Stochastic Successive Convex Approximation for Non-Convex Constrained Stochastic Optimization

This paper proposes a constrained stochastic successive convex approximation (CSSCA) algorithm to find a stationary point for a general non-convex stochastic optimization problem, whose objective and constraint functions are nonconvex and involve expectations over random states. The existing methods for non-convex stochastic optimization, such as the stochastic (average) gradient and stochastic majorizationminimization, only consider minimizing a stochastic non-convex objective over a deterministic convex set. To the best of our knowledge, this paper is the first attempt to handle stochastic non-convex constraints in optimization problems, and it opens the way to solving more challenging optimization problems that occur in many applications. The algorithm is based on solving a sequence of convex objective/feasibility optimization problems obtained by replacing the objective/constraint functions in the original problems with some convex surrogate functions. The CSSCA algorithm allows a wide class of surrogate functions and thus provides many freedoms to design good surrogate functions for specific applications. Moreover, it also facilitates parallel implementation for solving large scale stochastic optimization problems, which arise naturally in today’s signal processing such as machine learning and big data analysis. We establish the almost sure convergence of the CSSCA algorithm and customize the algorithmic framework to solve several important application problems. Simulations show that the CSSCA algorithm can achieve superior performance over existing solutions.