Solution Methods for Pseudomonotone Variational Inequalities

Abstract

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators.