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A Penalization-Gradient Algorithm for Variational Inequalities

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  • Abdellatif Moudafi
  • Eman Al-Shemas

Abstract

This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, find such that for all , where is a single-valued operator, is a closed convex set of a real Hilbert space . Given which acts as a penalization function with respect to the constraint , and a penalization parameter , we consider an algorithm which alternates a proximal step with respect to and a gradient step with respect to and reads as . 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.

Suggested Citation

  • Abdellatif Moudafi & Eman Al-Shemas, 2011. "A Penalization-Gradient Algorithm for Variational Inequalities," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-12, May.
    • Handle: RePEc:hin:jijmms:305856
    DOI: 10.1155/2011/305856
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