Numerical Algebra, Control and Optimization (NACO)

CVaR-based formulation and approximation method for stochastic variational inequalities

Pages: 35 - 48, Volume 1, Issue 1, March 2011      doi:10.3934/naco.2011.1.35

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Xiaojun Chen - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China (email)
Guihua Lin - School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China (email)

Abstract: In this paper, we study the stochastic variational inequality problem (SVIP) from a viewpoint of minimization of conditional value-at-risk. We employ the D-gap residual function for VIPs to define a loss function for SVIPs. In order to reduce the risk of high losses in applications of SVIPs, we use the D-gap function and conditional value-at-risk to present a deterministic minimization reformulation for SVIPs. We show that the new reformulation is a convex program under suitable conditions. Furthermore, by using the smoothing techniques and the Monte Carlo methods, we propose a smoothing approximation method for finding a solution of the new reformulation and show that this method is globally convergent with probability one.

Keywords:  Stochastic variational inequalities, conditional value at risk, D-gap function, Monte Carlo sampling approximation, smoothing approximation, convergence.
Mathematics Subject Classification:  Primary: 90C33; Secondary: 90C15.

Received: August 2010;      Revised: November 2010;      Available Online: February 2011.