Expected Residual Minimization Formulation for Stochastic Absolute Value Equations

In this paper we investigate a class of stochastic absolute value equations (SAVE). After establishing the relationship between the stochastic linear complementarity problem and SAVE, we study the expected residual minimization (ERM) formulation for SAVE and its Monte Carlo sample average approximation. In particular, we show that the ERM problem and its sample average approximation have optimal solutions under the condition of R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document} pair, and the optimal value of the sample average approximation has uniform exponential convergence. Furthermore, we prove that the solutions to the ERM problem are robust for SAVE. For a class of SAVE problems, we use its special structure to construct a smooth residual and further study the convergence of the stationary points. Finally, a smoothing gradient method is proposed by simultaneously considering sample sampling and smooth techniques for solving SAVE. Numerical experiments exhibit the effectiveness of the method.