International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 305856, 12 pages
http://dx.doi.org/10.1155/2011/305856
Research Article

A Penalization-Gradient Algorithm for Variational Inequalities

Abdellatif Moudafi1 and Eman Al-Shemas2

1Département Scientifique Interfacultaires, Université des Antilles et de la Guyane, CEREGMIA, 97275 Schoelcher, Martinique, France
2Department of Mathematics, College of Basic Education, PAAET Main Campus-Shamiya, Kuwait

Received 11 February 2011; Accepted 5 April 2011

Academic Editor: Giuseppe Marino

Copyright © 2011 Abdellatif Moudafi and Eman Al-Shemas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, find 𝑥 ∈ 𝐶 such that ⟨ 𝐴 𝑥 , 𝑦 − 𝑥 ⟩ ≥ 0 for all 𝑦 ∈ 𝐶 , where 𝐴 ∶ 𝐻 → 𝐻 is a single-valued operator, 𝐶 is a closed convex set of a real Hilbert space 𝐻 . Given Ψ ∶ 𝐻 → ℝ ∪ { + ∞ } which acts as a penalization function with respect to the constraint 𝑥 ∈ 𝐶 , and a penalization parameter 𝛽 𝑘 , we consider an algorithm which alternates a proximal step with respect to 𝜕 Ψ and a gradient step with respect to 𝐴 and reads as 𝑥 𝑘 = ( 𝐼 + 𝜆 𝑘 𝛽 𝑘 𝜕 Ψ ) − 1 ( 𝑥 𝑘 − 1 − 𝜆 𝑘 𝐴 𝑥 𝑘 − 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.