Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems

Mei-Ju Luo,Yuan Lu

doi:10.1155/2013/497586

Abstract

Expected residual minimization (ERM) model which minimizes an expected residual function defined by an NCP function has been studied in the literature for solving stochastic complementarity problems. In this paper, we first give the definitions of stochasticP-function, stochasticP0-function, and stochastic uniformlyP-function. Furthermore, the conditions such that the function is a stochasticPP0-function are considered. We then study the boundedness of solution set and global error bounds of the expected residual functions defined by the “Fischer-Burmeister” (FB) function and “min” function. The conclusion indicates that solutions of the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in stochastic complementarity problems. On the other hand, we employ quasi-Monte Carlo methods and derivative-free methods to solve ERM model.

Highlights

Given a vector-valued function F : Rn × Ω → Rn, the stochastic complementarity problems, denoted by SCP(F(x, ω)), are to find a vector x∗ such that x∗ ≥ 0, F (x∗, ω) ≥ 0, (1)(x∗)TF (x∗, ω) = 0, ω ∈ Ω a.s., where ω ∈ Ω ⊆ Rm is a random vector with given probability distribution P and “a.s.” means “almost surely” under the given probability measure
Problem (1) is called stochastic nonlinear complementarity problem, denoted by SNCP(F(x, ω)) if F can not be denoted by an affine function of x for any ω
Both expected value (EV) model and expected residual minimization (ERM) model give decisions by a deterministic formulation

Summary

Introduction

Given a vector-valued function F : Rn × Ω → Rn, the stochastic complementarity problems, denoted by SCP(F(x, ω)), are to find a vector x∗ such that x∗ ≥ 0, F (x∗, ω) ≥ 0, (1). By the definition of stochastic P(P0)-function, we have that there exist i ∈ J(x, y), i ∈ ⟨1, n⟩ such that, for every x ≠ y, xi ≠ yi, (19). Suppose on the contrary that F is not a stochastic P(P0)-function, there exist x, ỹ, x ≠ ỹ in Rn for any i ∈ ⟨1, n⟩ satisfying xi ≠ ỹi, (28). Note that there is at most one solution (may not be a solution) for the EV model stochastic complementarity problems if f(x) := E[F(x, ω)] is a P(P0)-function

Boundedness of Solution Set

Robust Solution

Quasi-Monte Carlo and Derivative-Free Methods for Solving ERM Model

Conclusions