Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives

This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.


Introduction
The study on the optimality conditions for non-smooth and generalized convex vector optimization problem in abstract spaces is a lively subject.Recently, there is a growing interest on this topic by using Dini set-valued directional derivatives.For example: Yang [1] introduced Dini set-valued directional derivatives for a vector valued function in infinite dimensional vector spaces and used this concept to establish the optimality conditions for weakly efficient solution in vector optimization problem under cone-convexity assumption; Ginchev [2] obtained first-order necessary and sufficient optimality conditions in terms of Dini set-valued directional derivatives in finite dimensional linear spaces for locally Lipschitz vector optimization.
It is well known that the concept of convexity and its various generalizations play an important role in operations research and applied mathematics.A meaningful generalized convex function was called the preinvex functions, which was introduced by Weir and Mond [3] and by Weir and Jeyakumer [4] in -dimensional Euclidean space.Nowadays, this class of functions has been extended to the abstract spaces and applied to establish optimality criteria and duality in vector optimization [5][6][7][8].Recently, Qiu [9] in normed linear spaces considered a class of functions called generalized prein-vex and established the unified optimality conditions for set-valued vector optimization problems.n On the other hand, the (weakly) efficient solution is a kind of extremely efficient solutions in vector optimization.Since the range of the set of (weak) efficient solutions is often too large, contracting the solution range is a basic topic in vector optimization.For this purpose, many kinds of proper efficiency have been presented.Among them, an important proper efficiency is called Henig proper efficiency, which was introduced by Henig [10].It is worthy to notice that the super efficiency, introduced by Borwein [11], equals to the Henig efficiency when the convex cone has a bounded base.
The aim of this paper is to deal with the optimality of Henig proper efficient solutions for vector optimization problems in terms of Dini set-valued directional derivatives under the generalized preinvex assumptions.

Preliminaries
In this note, it is assumed that X and Y are two locally convex spaces with topological duals X and Y , respectively.The partially order of Y is defined by a closed convex cone C with Y  int C   .On the other hand, we assume that Y is a complete vector lattice, i.e., sup   The closure and interior of a set A are denoted by and .Let and C be the dual cone and strictly dual cone of convex cone , defined by A nonempty convex subset of the convex cone is called a base of , if and In this paper, it is always assumed that B is a base of [12]) Let K be a nonempty subset of and let be a base of .0 is said to be a Henig proper efficient point of Now, let us recall the concepts of upper and lower Dini set-valued directional derivatives given by Yang [1].
be a vector valued function and , x d  X .The limiting set of at f x in the direction is defined as follows For our approach in this note, the following assump- ; ; In addition, it has been pointed out in Ref. [1] that if Assumption 2.1 holds, then and The set is called a generalized invex set with respect to

S X 
 and  if for any , x y S  and any Suppose that is a generalized invex set with respect to It is clear that the concepts of generalized invex sets and generalized preinvex functions are the generalizations of the invex sets and preinvex functions which introduced by Weir [3,4].In addition, the function In fact, We need the next assumptions.Assumption 2.2.(See [1]) Let be defined as in (2.1).The domination property is said to hold for if The following important property of Generalized cone-preinvex functions will be used in the sequel. Proposition This means that Thus, Then, it yields from Remark 2.2 that Furthermore, if follows from Assumption 2.2 and positive homogeneous property of Dini set-valued directional derivatives that Thus, we get

Optimality Criteria
In this section, we apply the Dini set-valued directional derivatives defined in the last section to characterize optimality conditions for a vector optimization problem involving the generalized preinvex functions.We begin by presenting the following vector optimization problem , where is a nonempty open subset of S X and : f X Y  .

Definition3.1. a) The point is said to be a Henig proper efficient solution of  
VOP with respect to if there exists such that The pointis said to be a strong efficient solution of  is a Henig proper efficient of   VOP with respect to , then there exists such that In particular, 2) Assume that is a generalized invex with respect to S  and  and f is generalized   P VO Proof: 1) If (3.1) does not hold, by the Definition 2.2, then there exists x S  and small enough ˆ0 which contradicts to the assumption that x S  VOP is a Henig proper efficient solution of problem .On the other hand, It is obviously from (2.2) that the inequality (3.2) holds.


2) Suppose that there exists such that By Proposition 2.1, we get That is, x is a strong efficient solution of problem .


and  if for any , x y S and any [0,1]


and  .If (3.1) holds, then x is a Henig proper efficient solution of .