Neural network smoothing approximation method for stochastic variational inequality problems

Hui-Qiang Ma; Nan-Jing Huang

doi:10.3934/jimo.2015.11.645

Article Contents

2015, Volume 11, Issue 2: 645-660. Doi: 10.3934/jimo.2015.11.645

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Neural network smoothing approximation method for stochastic variational inequality problems

Hui-Qiang Ma^1, and
Nan-Jing Huang^2,

1.
School of Economics, Southwest University for Nationalities, Chengdu, Sichuan 610041
2.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received: May 31, 2012

Revised: April 30, 2014

Published: March 2015

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Abstract

This paper is concerned with solving a stochastic variational inequality problem (for short, SVIP) from a viewpoint of minimization of mixed conditional value-at-risk (CVaR). The regularized gap function for SVIP is used to define a loss function for the SVIP and mixed CVaR to measure the loss. In this setting, SVIP can be reformulated as a deterministic minimization problem. We show that the reformulation is a convex program for a huge class of SVIP under suitable conditions. Since mixed CVaR involves the plus function and mathematical expectation, the neural network smoothing function and Monte Carlo method are employed to get an approximation problem of the minimization reformulation. Finally, we consider the convergence of optimal solutions and stationary points of the approximation.

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Mathematics Subject Classification: Primary: 90C33, 90C15; Secondary: 91B30.

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