Nondifferentiable Multiobjective Programming Problem under Strongly K - G f -Pseudoinvexity Assumptions

: In this paper we consider the introduction of the concept of (strongly) K - G f -pseudoinvex functions which enable to study a pair of nondifferentiable K - G - Mond-Weir type symmetric multiobjective programming model under such assumptions.


Introduction
Duality mathematical programming is used in Economics, Control Theory, Business and other diverse fields. In mathematical programming, a pair of primal and dual problems are said to be symmetric when the dual of the dual is the primal problem, i.e., when the dual problem is expressed in the form of the primal problem, then it does happen that its dual is the primal problem. This type of dual problem was introduced by Dorn [1], later on Mond and Weir [2] studying them under weaker convexity assumptions.
Antczak [3] introduced the notion of G-invex function obtaining some optimality conditions which he himself [4] comprehends to be a G f -invex function, deriving optimality conditions for a multiobjective nonlinear programming problem. Ferrara and Stefaneseu [5] also discussed the conditions of optimality and duality for multiobjective programming problem, and Chen [6] considered multiobjective fractional problems and its duality theorems under higher-order (F, α, ρ, d)convexity.
In recent years, several definitions such as nonsmooth univex, nonsmooth quasiunivex, and nonsmooth pseudoinvex functions have been introduced by Xianjun [7]. By introducing these new concepts, sufficient optimality conditions for a nonsmooth multiobjective problem were obtained and, a fortiori, weak and strong duality results were established for a Mond-Weir type multiobjective dual program.
Jiao [8] introduced new concepts of nonsmooth K − α − d I -invex and generalized type I univex functions over cones by using Clarke's generalized directional derivative and d I -invexity for a Remark 1. If G f (t) = t and G g j (t j ) = t j , j = 1, 2, 3, ..., m, then the vector minimization problem (KGMP) reduces to vector minimization problem (KMP).

Definition 3.
A pointx ∈ X 0 is said to be a weak minimum of (KGMP) if there exists no other Lemma 2. Ifȳ ∈ X 0 is a weak minimum of (KGMP), then there exist α ∈ K * , β ∈ Q * which are not simultaneously zero such that Let C 1 ⊆ R n and C 2 ⊆ R m be two closed convex cones with non-empty interiors, and let S 1 and S 2 be two non-empty open sets in R n and R m , respectively, so that C 1 × C 2 ⊆ S 1 × S 2 . Given a vector valued differentiable function f = ( f 1 , f 2 , ..., f k ) : S 1 × S 2 → R k we consider the following definitions.

Definition 4. The function f is said to be
Definition 6. The function f is said to be K-G f -pseudoinvex at u ∈ S 1 (with respect to η) if ∀ x ∈ S 1 and for fixed v ∈ S 2 , we have Remark 2. If G f i (t) = t, i = 1, 2, 3, ..., k, then Definition 2.6 becomes K − η-pseudoinvex (Definition 4).
Finally we recall that [12] given a compact convex set C in R n , the support function of C is defined by The subdifferential of s(x|C) is given by For any convex set S ⊂ R n , the normal cone to S at a point x ∈ S is defined by It is readily verified that for a compact convex set S, y ∈ N S (x) if and only if s(y|S) = x T y.

K-G-Mond-Weir Type Primal Dual Model
In this section, we consider a multiobjective K-G-Mond-Weir type primal-dual model over arbitrary cones: where, for i = 1, 2, 3, ..., k, it holds that: (I) K * , C * 1 and C * 2 are the positive polar cones of K, C 1 and C 2 , respectively. (II) Given f i : → R as a strictly increasing function on its domain, G f is a differentiable function.
Next we prove weak, strong and converse duality theorems for (KGMPP) and (KGMDP), respectively. Let Z 0 and W 0 be the set of feasible solutions of (KGMPP) and (KGMDP), respectively.
.., f k (., v)} be strongly K-G f -pseudoinvex at u with respect to η 1 , Then, Proof. The proof is given by contradiction. Suppose that (7) does not hold. Then, For the dual constraint (4) and assumption (iii), we get Using the dual constraint (5) in the above inequality, we deduce that Taking into account that λ ∈ intK * , By hypothesis (i), it holds that Having in mind (8), we obtain On the similar lines to the above proof, we have By using now generalized convexity assumptions, it follows that , ..., −G f 1 ( f 1 (x, y)) a contradiction with (9). Hence, the conclusion follows.
Thanks to the fact that under symmetric duality, the converse duality theorem proof works in the same as for the strong duality theorem, Theorem 1 infers the following result. Theorem 3 (Converse duality theorem). Let (ū,v,λ) be a weak efficient solution of (KGMDP); fix λ =λ in (KGMPP) and suppose that Then (ū,v,λ) ∈ Z 0 and the objective values of (KGMPP) and (KGMDP) coincide. Moreover, if the assumptions of Theorem 1 are satisfied for every feasible solution of (KGMPP) and (KGMDP), then (ū,v,λ) is a weak efficient solution of (KGMPP).

K-N-G-Mond-Weir Type Nondifferentiable Dual Model
Herein, we consider a nondifferentiable multiobjective K-N-G-Mond-Weir primal-dual model over arbitrary cones.
where for i = 1, 2, 3, ..., k, it holds that: (I) K * , C * 1 and C * 2 are the positive polar cones of K, C 1 and C 2 , respectively. (II) Given f i : G f 2 , ..., G f k ) : R → R k has any of its components G f i :

Remark 4.
In the primal-dual model (K-N-G-Mond-Weir type nondifferentiable dual model), we used support function for a nondifferentiable term. Now we are ready to provide three duality theorems for (KGNMPP) and (KGNMDP). Their proofs are easily obtained by mimicking the ones of the three theorems obtained in the previous section.
Let X 0 and Y 0 be the set of feasible solutions of (KGNMPP) and (KGNMDP), respectively.
Finally, the following result becomes the sibling result of the last one obtained in the previous section.

Conclusions
By using the notion of K-G f -pseudo-invex/ strongly K − G f -pseudo-invex functions we have established duality results for (KGMPP) /(KGNMPP)-Mond-Weir dual models applied in multiobjective nondifferentiable symmetric programming problems with objective cone and cone constraints, too. This work may be inspirational for extension to nondifferentiable higher-order symmetric fractional programming.