Some Contributions to Nonlinear Optimization Problems and Variational Inequalities

Abstract

Convex analysis became one of the most beautiful and most developed branch of mathematics newlinedue to the works of Rockafellar, Fenchel and Moreau in the 1960s and 1970s. It has a wide newlinerange of applications including optimization, operations research, engineering, economics, etc. newlineHowever, several practical models involve functions which are not exactly convex, but share newlinecertain nice properties of convex functions. These functions are a modification or generalization newlineof convex functions. newlineThe generalization of convexity inspired a large number of research articles concerned newlinemainly with analysis and application in optimization theory and related areas. In recent years, newlinebesides real-valued generalized convex functions, vector-valued generalized convex functions newlinehave also been investigated intensively. During the last fifty years, a significant increase of newlineresearch activities in this field has been witnessed. See the bibliographical references at the end newlineof this thesis for more details. newlineThe purpose of this thesis is to present an overview (together with some new results) of newlinegeneralized convexity such as, invexity and generalized invexity in optimization problems with newlinesingle valued objective functions and with vector-valued objective functions (Pareto or multiobjective newlineprogramming problems). newlineIn the present thesis, we are concerned with the study of sufficient optimality conditions, newlineduality results for nondifferentiable multiobjective fractional programming problems, mathematical newlineprogramming problems with equilibrium constraints, and to study vector variational newlineinequalities for generalized convex functions. We consider smooth and nonsmooth mathematical newlineprogramming problems with equilibrium constraints and establish duality results relating to newlinethe primal and the dual problems. Further, we deal with nonsmooth vector variational inequalities newlineusing the notion of Mordukhovich subdifferentials and Mordukhovich superdifferentials. newlineThe majority of this thesis focuses on nonsmooth optimization. newline