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

Consider the variational inequality of finding a point satisfying the property for all , where is a level set of a convex function defined on a real Hilbert space and is a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets of ) 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 (calculating by computing a sequence of projections onto half-spaces containing the original domain ) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants of (i.e., Lipschitz constants on some bounded subsets of )), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.