Variance-based stochastic projection gradient method for two-stage co-coercive stochastic variational inequalities

The existing stochastic approximation (SA)-type algorithms for two-stage stochastic variational inequalities (SVIs) are based on the uniqueness of the second-stage solution, which restricts the use of those algorithms. In this paper, we propose a dynamic sampling stochastic projection gradient method (DS-SPGM) for solving a class of two-stage SVIs satisfying the co-coercive property. With the co-coercive property and the dynamic sampling technique, we can handle the two-stage SVIs when the second-stage problem has multiple solutions and achieve the rate of convergence with O(1/K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{O}(\varvec{1/\sqrt{K}})$$\end{document}. Moreover, numerical experiments show the efficiency of the DS-SPGM.