A New Infeasible Projection Method for Stochastic Variational Inequality Problem

Abstract

In this paper, we propose a new infeasible stochastic approximation projection method based on the golden ratio for a nonmonotone stochastic variational inequality problem. In the traditional golden ratio methods, the constant phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} is taken as 1+52\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1+\sqrt{5}}{2}$$\end{document}. However, the constant is relaxed to the interval (1,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,\infty )$$\end{document} in our method. A new self-adaptive step size which is admitted to be increasing is generated for dealing with the unknown Lipschitz constant of the mapping. The almost sure convergence and convergence rate of the proposed method are shown. Some numerical examples are given to illustrate the competitiveness of our algorithm compared to the related algorithms in the literature. Finally, we apply our method to solve a network bandwidth allocation problem.