Stochastic Methods Based onVU-Decomposition Methods for Stochastic Convex Minimax Problems

Yuan Lu,Ming Huang,Wei Wang,Shuang Chen

doi:10.1155/2014/894248

Abstract

This paper applies sample average approximation (SAA) method based onVU-space decomposition theory to solve stochastic convex minimax problems. Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one and convergence is exponentially fast with the increase of sample size. Based on theVU-theory, a superlinear convergentVU-algorithm frame is designed to solve the SAA problem.

Highlights

In this paper, the following stochastic convex minimax problem (SCMP) is considered: minf (x), x∈Rn (1)where f (x) = max {E [fi (x, ξ)] : i = 0, . . . , m}, (2)and the functions fi(x, ξ) : Rn → R, i = 0, . . . , m, are convex and C2, ξ : Ω → Ξ ⊂ Rn is a random vector defined on probability space (Ω, Υ, P); E denotes the mathematical expectation with respect to the distribution of ξ.SCMP is a natural extension of deterministic convex minimax problems (CMP for short)
The CMP has a number of important applications in operations research, engineering problems, and economic problems
A well-known approach based on the sampling is the so-called sample average approximation (SAA) method, that is, using sample average value of fi(x, ξ) to approximate its expected value because the classical law of large number for random functions ensures that the sample average value of fi(x, ξ) converges with probability 1 to E[fi(x, ξ)] when the sampling is independent and identically distributed

Summary

Introduction

The following stochastic convex minimax problem (SCMP) is considered: minf (x) , x∈Rn (1). A well-known approach based on the sampling is the so-called SAA method, that is, using sample average value of fi(x, ξ) to approximate its expected value because the classical law of large number for random functions ensures that the sample average value of fi(x, ξ) converges with probability 1 to E[fi(x, ξ)] when the sampling is independent and identically distributed (idd for short). A VU-space decomposition method for solving a constrained nonsmooth convex program is presented in [12]. A decomposition algorithm based on proximal bundle-type method with inexact data is presented for minimizing an unconstrained nonsmooth convex function in [13]. Based on the VU-theory, a superlinear convergent VU-algorithm frame is designed to solve the SAA problem. The VU-decomposition algorithm frame of the SAA problem is designed

Convergence Analysis of SAA Problem

The VU-Theory of the SAA Problem

Algorithm and Convergence Analysis