Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order ( ) , B V σ φ − − type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.


Introduction
The field of multiobjective programming, also called vector programming, has grown remarkably in different directions in the settings of optimality conditions and duality theory since the 1980s.It has been enriched by the applications of various types of generalizations of convexity theory, with and without differentiability assumptions.The Clarke subdifferential [1] (also called the Clarke generalized gradient) is an important tool to derive optimality conditions for nonsmooth optimization problems.Together with the Clarke's subdifferential, many generalized convexity or invexity functions were generalized to locally Lipschitz functions.Based upon the generalized functions, several sufficient optimality conditions and duality results were established for the optimization problems.We can see for examples [2]- [8].In [9] Upadhyay introduced some G.An new generalizations of the concept of ( ) , φ ρ -invexity and established the neces- sary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems.In [10] the new concepts of ( ) , V φ ρ − −type I were introduced.Sufficient optimality conditions and Mond-Weir duality results were obtained for nonsmooth multiobjective programming problems.Recently, many researchers have been interested in other types of solution concepts.One of them is higher order strict minimizer.In [11] and [12] some sufficient conditions and duality results were obtained for the new concept of strict minimizer of higher order for a multiobjective optimization problem.
In this paper, we consider the nonsmooth multiobjective programming including the locally Lipschitz functions.The new concepts of invex of order ( )

Preliminaries and Definitions
Let n R be the n-dimensional Euclidean space and let X be a nonempty open where ( ) where , ⋅ ⋅ is the inner product in n R .Consider the following nonsmooth multiobjective programming problem: .
be the set of feasible solutions of (MP), and with respect to a nonlinear function : Throughout the paper, we suppose that x X ∈ , if there exist ( ) such that for all x X ∈ the following inequalities hold: Definition 2.6.( ) , , , , , , , ,

Optimality Condition
In this section, we establish sufficient optimality conditions for a strict minimizer of (MP).
Theorem 3.1.Let 0 x X ∈ .Suppose that 1) There exist 0, Then x is a strict minimizer of order σ for (MP).
Proof: Since ( ) , there exists ( ), whence Suppose that x is not a strict minimizer of order σ for (MP).Then there Since ( ) ( ) , and hypothesis 3), we get , 0 In view of the hypothesis 1), one finds from ( 12) and (13) that .

Wolfe Type Duality
In this section, we consider the Wolfe type dual for the primal problem (MP) and establish various duality theorems.Let e be the vector of k R whose components are all ones.
Then the following can hold: Proof: Suppose contrary to the result that , .
By hypothesis 2), we have with hypothesis 1) and 2), the above inequality yields ( On the other hand, by using the constraint conditions of (MD), there exist ( ), which contradicts (26).Then the result is true.
Then the following can hold: Proof: Suppose contrary to the result that It follows from hypothesis 2) that In the view of hypothesis 1), one finds from (33) that For ( ) Since ( ) λ µ ∈ is a feasible solution for (MD), there exist ( ), ( ), It follows from ( From hypothesis 1), it follows that ( ) ( ) which contradicts (33).Then the result is true.
The following definition is needed in the proof of the strong duality theorem.
Definition 4.1.A point u X ∈ is called a strict maximizer of order σ for (MD) with respect to a nonlinear function : x λ µ is a strict maximizer of order σ for (MD) with respect to ψ .
Proof: The hypothesis implies that ( ) , , x λ µ is a feasible solution of (MD).
By Theorem 4.1, for any feasible ( ) , , y λ µ of (MD), we have  x λ µ is a strict maximizer of order σ for (MD) with respect to ψ .

Conclusion
In this paper, we have defined a class of new generalized functions.By using the new functions, we have presented a sufficient optimality condition and Wolfe type duality results for a nondifferentiable multiobjective problem.The present results can be further generalized for other programming problems.