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.