Stability Analysis of Stochastic Generalized Equation via Brouwer’s Fixed Point Theorem

Received Oct 24, 2017; Accepted Feb 19, 2018

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1. Introduction

In this paper, we focus on the following stochastic generalized equation (SGE): find $x\in {\mathbb{R}}^n$ such that$$\begin{array}{}\text{(1)}& \mathrm{0}\in {\mathbb{E}}_P\left[F\left(x,\xi \left(\omega \right)\right)\right]+\mathcal{G}\left(x\right),\end{array}$$where $F:{\mathbb{R}}^n\times \mathrm{\Xi }\to {\mathbb{R}}^m$ is a continuous function, $\xi :\Omega \to \mathrm{\Xi }$ is a random vector defined on a probability space $(\Omega ,\mathcal{F},P)$ with support set $\mathrm{\Xi }\subseteq {\mathbb{R}}^k$, ${\mathbb{E}}_P[·]$ denotes the expected value with respect to $P$, and $\mathcal{G}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ is an outer semicontinuous set-valued mapping. Throughout the paper, we assume that ${\mathbb{E}}_P[F(x,\xi (\omega ))]$ is well defined for any $x\in {\mathbb{R}}^n.$ To ease notation, we will use $\xi$ to denote either the random vector $\xi (\omega )$ or an element of ${\mathbb{R}}^k$ depending on the context.

Model (1) a natural extension of deterministic parametric generalized equation [1] and the study of stochastic generalized equations can be traced down to King and Rockafellar’s early work [2]. In a particular case when $\mathcal{G}(·)$ is a normal cone operator ${\mathcal{N}}_{\mathcal{K}}(·)$ in which $\mathcal{K}$ is a closed convex cone in ${\mathbb{R}}^m$, (1) reduces to a stochastic variational inequality problem (SVIP) which has been intensively studied over the past few years; see for instance [3–7] and the references therein. The research ranges from numerical schemes such as stochastic approximation method and Monte Carlo method to the fundamental theory and applications.

In this paper, we concentrate our research on the stability of (1); namely, we look into the impact of variation of probability measure $P$ on the solution of the SGE. Like similar existing research in deterministic generalized equation, this kind of stability analysis would address a number of fundamental theoretical issues including robustness, accuracy, and reliability of an optimal solution or an equilibrium against errors arising...

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