Perturbed auxiliary problem methods to solve generalized variational inequalities

Résumé

The first chapter provides some basic definitions and results from the theory of convex analysis and nonlinear mappings related to our work. Some sufficient conditions for the existence of a solution of problem (GVIP) are also recalled. In the second chapter, we first illustrate the scope of the auxiliary problem procedure designed to solve problems like (GVIP) by examining some well-known methods included in that framework. Then, we review the most representative convergence results for that class of methods that can be found in the literature in the case where F is singlevalued as well as in the multivalued case. Finally, we somewhat discuss the particular case of projection methods to solve affine variational inequalities. The third chapter introduces the variational convergence notion of Mosco and combines it with the auxiliary problem principle. Then, we recall the convergence conditions existing for the resulting perturbed scheme before our own contribution and we comment them. Finally, we introduce and illustrate the rate of convergence condition that we impose on the perturbations to obtain better convergence results. Chapter 4 presents global and local convergence results for the family of perturbed methods in the case where F is singlevalued. We also discuss how our results extend or improve the previous ones. Chapter 5 studies the multivalued case. First, we present convergence results generalizing those obtained when there is no perturbations. Then, we relax the scheme by means of a notion of enlargement of an operator and we provide convergence conditions for this inexact scheme. In Chapter 6, we build a bundle algorithm to solve problem (GVIP) and we study its convergence.

la date de réponse	2001
langue originale	Anglais
L'institution diplômante	Universite de Namur
Superviseur	Jean-Jacques STRODIOT (Promoteur), Jean-Pierre Crouzeix (Jury), Van Hien Nguyen (Jury), Jacques HENRARD (Jury) & Michel Willem (Jury)