Optimality Conditions in Quasidifferentiable Vector Optimization

T Antczak

doi:10.1007/s10957-016-0987-x

Abstract

In the paper, the quasidifferentiable vector optimization problem with the inequality constraints is considered. The Fritz John-type necessary optimality conditions and the Karush---Kuhn---Tucker-type necessary optimality conditions for a weak Pareto solution are derived for such a nonsmooth vector optimization problem. Further, the concept of an F-convex function with respect to a convex compact set is introduced. Then, the sufficient optimality conditions for a (weak) Pareto optimality of a feasible solution are established for the considered nonsmooth multiobjective optimization problem under assumptions that the involved functions are quasidifferentiable F-convex with respect to convex compact sets which are equal to Minkowski sum of their subdifferentials and superdifferentials at this point.

Highlights

Vector optimization problems, known as multiobjective programming problems, have been applied in various fields of science, where optimal decisions need to be PolandJ Optim Theory Appl (2016) 171:708–725 taken in the presence of trade-offs between two or more conflicting objectives
In order to illustrate the result established in Theorem 4.2, we present an example of a nonsmooth vector optimization problem with quasidifferentiable F-convex functions with respect to convex compact sets which are equal to the Minkowski sum of their subdifferentials and superdifferentials
Since all hypotheses of Theorem 4.2 are fulfilled at x, xis a Pareto solution in the considered nonsmooth vector optimization problem with quasidifferentiable convex functions with respect to convex compact sets that are equal to the Minkowski sum of their subdifferentials and superdifferentials at x

Summary

Introduction

Known as multiobjective programming problems, have been applied in various fields of science, where optimal decisions need to be. We prove sufficient optimality conditions for (weak) Pareto optimality of a feasible solution in the considered nonsmooth multiobjective optimization problem under assumptions that the involving functions are quasidifferentiable F-convex with respect to convex compact sets which are equal to the Minkowski sum of their subdifferentials and superdifferentials at this point. The sufficient optimality conditions for (weak) Pareto optimality of a feasible point are proved under assumptions that the functions constituting the considered nonsmooth multiobjective optimization problem are quasidifferentiable F-convex with respect to convex compact sets which are equal to the Minkowski sum of their subdifferentials and superdifferentials at this point. This result established in the paper is illustrated by an example of a nonconvex quasidifferentiable vector optimization problem with F-convex functions with respect to such convex compact sets

Preliminaries

Necessary Optimality Conditions

Sufficient Optimality Conditions

Conclusions