CVaR-based formulation and approximation method for stochastic variational inequalities
Xiaojun Chen Guihua Lin
Numerical Algebra, Control & Optimization 2011, 1(1): 35-48 doi: 10.3934/naco.2011.1.35
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: D-gap function Stochastic variational inequalities convergence. conditional value at risk smoothing approximation Monte Carlo sampling approximation
Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks
Mingzheng Wang M. Montaz Ali Guihua Lin
Journal of Industrial & Management Optimization 2011, 7(2): 317-345 doi: 10.3934/jimo.2011.7.317
We consider a class of stochastic nonlinear complementarity problems. We propose a new reformulation of the stochastic complementarity problem, that is, a two-stage stochastic mathematical programming model reformulation. Based on this reformulation, we propose a smoothing-based sample average approximation method for stochastic complementarity problem and prove its convergence. As an application, a supply chain super-network equilibrium is modeled as a stochastic nonlinear complementarity problem and numerical results on the problem are reported.
keywords: Stochastic nonlinear complementarity problem super-network two-stage stochastic mathematical programming convergence. sample average approximation

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