Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

Qilin Wang,Guolin Yu

doi:10.1155/2012/691018

Abstract

The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.

Highlights

In the last several decades, several notions of derivatives epiderivatives for set-valued maps have been proposed and used for the formulation of optimality conditions in set-valued optimization problems
Throughout this paper, let X, Y, and Z be three real normed spaces, where the spaces Y and Z are partially ordered by nontrivial pointed closed convex cones C ⊂ Y and D ⊂ Z with int C / ∅ and int D / ∅, respectively
Let E be a nonempty subset of X, and F : E → 2Y and G : E → 2Z be two given nonempty set-valued maps

Summary

Introduction

In the last several decades, several notions of derivatives epiderivatives for set-valued maps have been proposed and used for the formulation of optimality conditions in set-valued optimization problems. Li et al studied some properties of higher-order tangent sets and higher-order derivatives introduced in 1 , and obtained higher-order necessary and sufficient optimality conditions for set-valued optimization problems under cone-concavity assumptions By using these higher-order derivatives, they discussed higher-order Mond-Weir duality for constrained set-valued optimization problems based on weak efficiency. Li and Chen proposed higher-order generalized contingent adjacent epiderivatives of setvalued maps and, obtained higher-order Fritz John type necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem. Wang and Li introduced generalized higher-order contingent adjacent epiderivatives of set-valued maps, and investigated both necessary and sufficient conditions for Henig efficient solutions to set-valued optimization problems by employing the generalized higherorder contingent adjacent epiderivatives.

Preliminaries and Notations

Higher-Order Weakly Generalized Epiderivatives

Higher-Order Optimality Conditions