Penalty Methods with Stochastic Approximation for Stochastic Nonlinear Programming

TL;DR

This paper introduces penalty methods combined with stochastic approximation techniques to solve stochastic nonlinear programming problems using noisy gradient or function evaluations, providing complexity analysis for reaching approximate critical points.

Contribution

It develops a novel class of penalty methods with stochastic approximation for stochastic nonlinear programming, including complexity bounds for oracle calls to achieve approximate solutions.

Findings

Complexity bounds for stochastic first-order oracle calls.

Extension to stochastic zeroth-order oracle scenarios.

Convergence to epsilon-stochastic critical points.

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 $\epsilon$-stochastic critical point is analyzed.