Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

Abstract

Consider the variational inequality $\text{V}\text{I}(C,F)$ of finding a point $x^{\mathrm{*}}\in C$ satisfying the property $\langle F{x}^{\mathrm{*}},x-x^{\mathrm{*}}\rangle \ge \mathrm{0}$, for all $x\in C$, where $C$ is the intersection of finite level sets of convex functions defined on a real Hilbert space $H$ and $F:H\to H$ is an $L$-Lipschitzian and $\eta$-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of $\text{V}\text{I}(C,F)$. Since our algorithm avoids calculating the projection $P_C$ (calculating $P_C$ by computing several sequences of projections onto half-spaces containing the original domain $C$) directly and has no need to know any information of the constants $L$ and $\eta$, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.