Stochastic Linear Complementarity Problems on Extended Second Order Cones

Abstract

In this paper, we study the stochastic linear complementarity problems on extended second order cones (stochastic ESOCLCP). We first convert the problem to a stochastic mixed complementarity problem on the nonegative orthant (SMixCP). Enlightened by the idea of Chen and Lin(2011), we introduce the Conditional Value-at-risk (CVaR) method to measure the loss of complementarity in the stochastic case. A CVaR - based minimisation problem is introduced to achieve a solution which is "good enough" for the complementarity requirement of the original SMixCP. Smoothing function and sample average approximation methods are introduced and the the problem is converted to a form which can be solved by Levenberg-Marquardt smoothing SAA algorithm. At the end of the paper a numerical example illustrates our results.