Convergence Analysis of Sample Average Approximation of Two-stage Stochastic Generalized Equations

TL;DR

This paper analyzes the convergence properties of the sample average approximation method for two-stage stochastic generalized equations, establishing exponential convergence rates and conditions for solution existence and uniqueness.

Contribution

It provides new theoretical results on convergence rates and solution properties for two-stage stochastic generalized equations without requiring relatively complete recourse.

Findings

Exponential convergence rate of sample average approximation established.

Sufficient conditions for existence, uniqueness, and regularity of solutions provided.

Applications demonstrated through two-stage stochastic non-cooperative games.

Abstract

A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables.This paper studies convergence of the sample average approximation of two-stage stochastic nonlinear generalized equations. In particular an exponential rate of the convergence is shown by using the perturbed partial linearization of functions. Moreover, sufficient conditions for the existence, uniqueness, continuity and regularity of solutions of two-stage stochastic generalized equations are presented under an assumption of monotonicity of the involved functions. These theoretical results are given without assuming relatively complete recourse, and are illustrated by two-stage stochastic non-cooperative games of two players.