Necessary optimality conditions in terms of convexificators for nonsmooth optimization problems

Abstract

The aim of this work is to study constraint qualifications and Kuhn-Tucker type necessary optimality conditions for nonsmooth optimization problems involving locally Lipschitz functions by means of the idea of convexificators that has recently been used to extend and strengthen various results in nonsmooth analysis and optimization. First, for minimization problems with an arbitrary set inclusion constraint or inequality constraints and a maximization problem with inequality constraints, several constraint qualifications are introduced and Kuhn-Tucker type necessary optimality conditions are derived under the qualifications. Secondly, for an inequality constrained multiobjective optimization problem, several constraint qualification are proposed and under the qualifications, stronger Kuhn-Tucker type necessary optimality conditions are developed in which all multipliers associated with the objective functions are positive. Finally, for a multiobjective optimization problem with inequality and an arbitrary set inclusion constraints, two constraint qualifications are given and shown to be necessary and sufficient conditions for the sets of Kuhn-Tucker multipliers to be nonempty and bounded. The constraint qualification and Kuhn-Tucker type necessary optimality conditions discussed in this thesis are expressed in terms of upper or lower convexificators. The work of this thesis makes a contribution to the unified approach to the study of constraint qualifications and optimality conditions for nonsmooth optimization problems, and the results obtained extend some already obtained ones concerning Kuhn-Tucker type necessary optimality conditions that use the convex-valued subdifferentials such as the Clarke subdifferentials. Key Words. Nonsmooth optimization, necessary optimality conditions, stronger necessary optimality conditions, qualifications, convexificators, locally Lipschitz functions.