- Optimality in Multivalued Optimization

In this paper we apply the directional derivative technique to characterize D-multifunction, quasi D-multifunction and use them to obtain ε-optimality for set valued vector optimization problem with multivalued maps. We introduce the notions of local and partial-ε-minimum (weak) point and study ε-optimality, ε-Lagrangian multiplier theorem and ε-duality results.


Introduction
The theory of efficiency plays an important role in various knowledge fields.It is proposed as a new frontier in mathematical physics and engineering in context of priorities concerning the alternative energies, the climate exchange and education.Pareto efficiency or Pareto optimality is a central theory in economics with broad applications in game theory, social sciences, management sciences, various industries etc.In set valued vector optimization problems, it is important to know when the set of efficient points is nonempty to establish its main properties (existence, connectedness and compactness) and to extend the concepts to set valued vector optimization in infinite dimensional ordered vector spaces.The notion of proper efficiency was first introduced by Kuhn and Tucker [1] in their well known paper on nonlinear programming and many other notions have been proposed since then.Some of the well known notions are Geoffrion proper efficiency [2], Borwein proper efficiency [3], Benson proper efficiency [4] and super efficiency [5].Chinaie and Zafarani [6] introduced the concepts of feeble multifunction minimum (weak) point, multifunction minimum (weak) point and obtained optimality conditions for set valued vector optimization problem having multivalued objective and constraints.
While it is theoretically possible to identify the complete set of solutions, finding an exact description of this set often turns out to be practically impossible or computationally too expensive.In practical situations we often stop the calculations at values that are sufficiently close to the optimal solutions, that is, we use algorithms that find approximate of the Pareto optimal set.Stability aspect in set valued vector optimization deals with the study of behaviour of the solution set under perturbations of the data.One of the approaches in this regard is the convergence of sequence of ε-solutions to a solution of the original problem.These facts justify the need of study of approximate efficiency which is equivalent to ε-optimality for set valued vector optimization problems.Some of the researchers who contributed in this area are Hamel [7], Rong and Wu [8].
Chinaie and Zafarani [9] introduced the concepts of ε-feeble multifunction minimum (weak) point and obtained optimality conditions for set valued vector optimization problem having multivalued objective and constraints.In this paper, we have given the notions of (local) partial-ε-minimum point and (local) partial-ε-weak minimum point, for set valued vector optimization problem and used them to study ε-optimality, ε-Lagrangian multiplier theorem and ε-duality results.
This paper is organized as follows: In Section 2 we have given the preliminaries and results related to quasi D-multifunction.In Section 3 we apply the directional derivative technique used by Yang [10] to characterize ε-optimality conditions for set valued vector optimization problem in terms of ε-feeble multifunction minimum point given by Chinaie and Zafarani [9].In Section 4, we introduce (local) partial-ε-minimum point and (local) partial-ε-weak minimum point, and show that it is different from ε-feeble mutifunction minimum point.Also, we prove that every local partial-ε-minimum (weak) point is a partial-ε-minimum (weak) point if the objective function of set valued vector optimization problem is strict quasi D-multifunction and constraint function is quasi D-multifunction and show that this result is not true in the case of local ε-feeble multifunction minimum point.In Section 5, we obtain ε-Lagrangian multiplier theorem in terms of partial-ε-weak minimum point and in Section 6, we establish ε-weak duality and ε-strong duality for dual problem of set valued vector optimization problem.

Preliminaries and Definitions
Let X be locally convex topological vector space, Y, Z be real locally convex Hausdorff topological vector spaces; let be pointed closed convex cones with The set of strictly positive functions in is denoted by , that is Through out this paper, we denote and , be a multifunction defined on a non empty subset U of X with values in Y, which is partially ordered by cone D. Now, for a multifunction F U Y  , denote by domF and imF the domain and the image of F, respectively.In other words is called the graph of F. Definition 2.1: [10,11] Let U be convex subset of X.Let F U Y  be a multifunction: 1) F is said to be a D-multifunction on U if, for all we have: 3) F is said to be a strictly quasi D-multifunction on U iff, for all Yang [10]  , : : there exists 0, , 0 be the cone of feasible directions.Then the limit set of F at x in the direction x in all directions The union of all limit sets of F at We need the following assumption: Assumption 2.1: be quasi D-multifunction on U and assumption 2.1 hold then, for any Then, by assumption 2.1 there is That is,

ε-Optimality in Terms of Directional Derivatives
In this section, we obtain ε-optimality conditions for set valued vector optimization problem in terms of directional derivatives given by Yang [10] for ε-feeble multifunction minimum point given by Chinaie and Zafarani [9].We consider the following set valued vector optimization problem: , are multifunctions with nonempty values.The set of feasible solutions of (VP) is denoted by V, that is Chinaie and Zafarani [9] gave the following definitions.

 
N x being neighbourhood of x , then we have local ε-f.m. m. p. and local ε-f.m. w. m. p. of problem (VP).
We now give the necessary optimal conditions for local ε-feeble weak minimum point [9] of (VP).
, for all . Now, we give the sufficient conditions for ε-feeble multifunction minimum point of problem (VP).
Theorem 3.2: Proof: Let x 0 be not ε-feeble multifunction minimum point of problem (VP), then there exists which implies that, there exists Since F is D-multifunction therefore, Thus, there exists which is contradiction to given condition (3.3).

Partial-ε-Minimum (Weak) Point
In this section we introduce the notion of partial-ε-minimum point, and partial-ε-weak minimum point.
Definition 4.1: 2) x is called a partial-ε-weak minimum point (p.-εw.m. p.) of problem (VP), if there exists, 2) The set of x V  which satisfies (4.1) or (4.2) is denoted by satisfies (4.1) then it is called partial-ε-minimizer of (VP) and if satisfies (4.2) then it is called partial-ε-weak minimizer of (VP).Now we show that partial-ε-minimum point is different from ε-feeble multifunction minimum point.
The following example illustrates that and be defined by Then, The following example illustrates that , and be defined by The following lemma can be proved as in [9].Proof: Let x be local partial-ε-minimum point of problem (VP), then there exists a neighbourhood Let if possible, x be not partial-ε-minimum point of problem (VP).Then, there exist such that, and consequently there exists


On the other hand for 0, t Since G is quasi D-multifunction, therefore, feasible set is convex and we have which contradicts (4.3).
The following example illustrates that above result is not true for ε-feeble multifunction minimum point of (VP).
, and defined by Here G is quasi D-multifunction and F is strict quasi D-multifunction.
Then, Let Thus, x is not ε-feeble multifunction minimum point of problem (VP).

ε-Lagrangian Multiplier Theorem
In this section, let L(Z, Y) be the set of continuous linear operators from Z to Y, and let Denote by (F, G) the multivalued map from X to Y  Z defined by , respectively.Lemma 5.1: [14].Let F X Y  be D-multifunction on X.Then, one and only one of the following statements is true: 1) there exists x X  such that x is partial-ε-weak minimum point of following problem: Proof: Since x is partial-ε-weak minimum point of problem (VP), therefore there exists,  , is D-multifunction on V, therefore by Lemma 5.1, there exists Hence,   (5.5) This gives that   0 p s  (5.6).Therefore we get that, which contradicts (5.2).Hence x is partial-ε-weak minimum point of problem (VP) T .

ε-Duality
Let us define a multivalued mapping Consider the following maximum problem: (VD) subject to We now establish the following ε-duality results.

4 . 1 :
show that every local partial-ε-minimum (weak) point is a partial-(weak) point if F is srictly quasi D-multifunction and G quasi D-multifunction and prove local ε-feeble multifunction minimum point is not ε-feeble multifunction minimum point of problem (VP) in above conditions.Theorem Let F be strictly quasi-D-multifunction and G be a quasi D-multifunction.Then, any local partial-ε-minimum point of problem (VP) is a partial-ε-minimum point of problem (VP).
x is partial-ε-weak minimum point of (VD) corresponding to T.It follows that, Since x is a feasible point of problem (VP) T , there exist    .
partial-ε-weak maximizer of problem (VD).Proof: Suppose x 0 is partial-ε-weak minimum point of problem (VP) and such that x 0 is partial-ε-weak minimum point of problem 0 is feasible point of (VD) and By ε-weak duality, we obtain partial-ε-weak maximizer of problem (VD).