Optimality and Duality in Nonsmooth Conic Vector Optimization

Abstract

This article is concerned with a nonsmooth vector optimization problem involving conic constraints. We employ some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of the conic vector optimization problem, where the fuzzy necessary condition and sequential necessary condition are expressed in terms of the Frechet subdifferential and the exact necessary condition is in terms of the limiting/Mordukhovich subdifferential of the related functions. Sufficient conditions for (weakly) efficient solutions of the underlying problem are also provided by means of introducing the concepts of (strictly) generalized convex vector functions with respect to a cone. In addition, we propose a dual problem to the conic vector optimization problem and explore weak, strong, and converse duality relations between these two problems.