Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Abstract

Consider the variational inequality $\text{V}\text{I}(C,F)$ of finding a point $x^{\mathrm{*}}\in C$ satisfying the property $\langle F{x}^{\mathrm{*}},x-x^{\mathrm{*}}\rangle \ge 0$ for all $x\in C$, where $C$ is a level set of a convex function defined on a real Hilbert space $H$ and $F:H\to H$ is a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets of $H$) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projection $P_C$ (calculating $P_C$ by computing a sequence of projections onto half-spaces containing the original domain $C$) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants of $F$ (i.e., Lipschitz constants on some bounded subsets of $H$)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.