Necessary Optimality Conditions for a Class of Nonsmooth Vector Optimization Problems

Abstract

A class of nonsmooth vector optimization problems are considered, where the feasible set defined by cone constraint and the objective and constraint functions are locally Lipschitz. The concept of the Clarke's generalized directional derivative for a locally Lipschitz function is introduced. By using this concept, necessary optimality condition for the unconstrained optimization problem is established. Furthermore, a Slater-type constraint qualification is given in such a way that they generalize the classical one, when the constraint functions are differentiable. Then, Kuhn–Tucker necessary optimality condition in terms of the Clarke subdifferentials is obtained.