CVaR-based formulation and approximation
method for stochastic variational inequalities
Xiaojun Chen - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, 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.
Received: August 2010; Revised: November 2010; Available Online: February 2011.