Penalty methods with stochastic approximation for stochastic nonlinear programming

Abstract

In this paper, we propose a class of penalty methods with stochastic approximation for solving stochastic nonlinear programming problems. We assume that only noisy gradients or function values of the objective function are available via calls to a stochastic first-order or zeroth-order oracle. In each iteration of the proposed methods, we minimize an exact penalty function which is nonsmooth and nonconvex with only stochastic first-order or zeroth-order information available. Stochastic approximation algorithms are presented for solving this particular subproblem. The worst-case complexity of calls to the stochastic first-order (or zeroth-order) oracle for the proposed penalty methods for obtaining an ǫ-stochastic critical point is analyzed.

Stochastic Quasi-Newton Methods for Nonconvex Stochastic Optimization

“…In [9], a stochastic block mirror descent method, which incorporates the block-coordinate decomposition scheme into stochastic mirror-descent methodology, was proposed for a nonconvex stochastic optimization problem x * = argmin{f (x) : x ∈ X} with X having a block structure. More recently, Wang et al [44] proposed a penalty method for nonconvex stochastic optimization problems with nonlinear constraints, and also analyzed its SF O-calls complexity.…”

Stochastic optimization over proximally smooth sets

“…Optimization with functional constraints. There have recently appeared a number of papers that develop first-order methods for minimizing smooth (stochastic) objectives over sets cut out by smooth deterministic inequalities [13,26,44]. The convergence guarantees in these works are all stated in terms of a KKT-residual measure, and therefore are not directly comparable to the ones we obtain here.…”

Stochastic Gradient Methods for Unconstrained Optimization

“…On the other hand, given that there are a lot of problems yielding only input-output information, an interesting approach within SA and SAA frameworks is based on zero order information, [17]. Constrained problems are always of great interest and some recent research of penalty methods within SA methodology is presented in [39]. Projection and filter methods with variable sample size might be a valuable topic of future research.…”