Optimality conditions for directionally differentiable multi-objective programming problems

This paper is concerned with the optimality for multi-objective programming problems with nonsmooth and nonconvex (but directionally differentiable) objective and constraint functions. The main results are Kuhn-Tucker type necessary conditions for properly efficient solutions and weakly efficient solutions. Our proper efficiency is a natural extension of the Kuhn-Tucker one to the nonsmooth case. Some sufficient conditions for an efficient solution to be proper are also given. As an application, we derive optimality conditions for multi-objective programming problems including extremal-value functions.This work was done while the author was visiting George Washington University, Washington, DC.