Sensitivity Analysis for Equilibrium Problems and Related Problems

Abstract

In this Thesis we consider equilibrium and quasiequilibrium problems and also quasivariational inclusion problems, a more general setting. Beginning with Blum and Oettli (1994), who stated equilibrium problems as a direct generalization of variational inequalities and optimization problems, these general problems have been intensively developed. Although the formulations are simple, this kind of problem settings includes a wide range of optimization-related problems (see particular cases and applications in the Thesis). Furthermore, these formulations allow convenient employments of modern mathematical tools in consideration. During the last decade most efforts have been devoted to the solution existence. When we began this thesis four years ago we observed almost no papers on the stability and sensitivity analysis for the mentioned problems. This motivated our choice of the topic. Among various stability notions for solution sets we committed to investigating semicontinuity and H¨older continuity. It is clear that the stronger stability of the solution set, the better is this set for the use. However, to have a stability level of the solution set we need as usual a similar stability level for the data of the problem. The data of practical problems may be not regular enough. Fortunately, semicontinuity of the solution sets is that sufficient in a number of practical situations as in some mathematical models for competitive economies and others. H¨older continuity is a rather high level of regularity and somehow close to differentiability (due to the Rademacher Theorem for the special Lipschitz case). To justify our choice of the two kinds of stability we note that up to now there have been a number of papers in the literature for the same topics but we still do not observe a study of other kinds of stability.

Date of Award	2007
Original language	English
Supervisor	Phan Quoc Khanh (Supervisor), Jean-Jacques STRODIOT (Jury), Jean-Paul Penot (Jury) & Dinh The Luc (Jury)