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

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0for allx∈C, whereCis a level set of a convex function defined on a real Hilbert spaceHandF:H→His a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets ofH) 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 projectionPC(calculatingPCby computing a sequence of projections onto half-spaces containing the original domainC) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants ofF(i.e., Lipschitz constants on some bounded subsets ofH)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.