A Penalization-Gradient Algorithm for Variational Inequalities

This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, findx̅∈Csuch that〈Ax̅,y-x̅〉≥0for ally∈C, whereA:H→His a single-valued operator,Cis a closed convex set of a real Hilbert spaceH. GivenΨ:H→R ∪ {+∞}which acts as a penalization function with respect to the constraintx̅∈C, and a penalization parameterβk, we consider an algorithm which alternates a proximal step with respect to∂Ψand a gradient step with respect toAand reads asxk=(I+λkβk∂Ψ)-1(xk-1-λkAxk-1). Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing the multivalued operator by its Yosida approximate which is always Lipschitz continuous.

The Shrinking Projection Method for Solving Variational Inequality Problems and Fixed Point Problems in Banach Spaces

We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-ϕ-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.