Mixed E-duality for E-differentiable Vector Optimization Problems Under (Generalized) V-E-invexity

In this paper, a class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. The so-called vector mixed E-dual problem is defined for the considered E-differentiable vector optimization problem with both inequality and equality constraints. Then, several mixed E-duality theorems are established under (generalized) V-E-invexity hypotheses.

generalization of the notion of E-differentiable E-convexity and the notion of differentiable invexity introduced by Hanson [12]. Namely, Abdulaleem defined the concept of E-differentiable E-invexity in the case of (not necessarily) differentiable vector optimization problems with E-differentiable functions. Recently, Abdulaleem [5] introduced a new concept of generalized convexity as a generalization of the E-differentiable E-invexity notion and the concept of V-invexity given by Jeyakumar and Mond [16]. Namely, Abdulaleem defined the concept of E-differentiable V-E-invexity in the case of (not necessarily) differentiable vector optimization problems with E-differentiable functions and used this concept to prove sufficient optimality conditions for a new class of nonconvex E-differentiable vector optimization problems.
In this paper, a class of E-differentiable V-E-invex vector optimization problems with both inequality and equality constraints is considered. A mixed E-dual problem is defined for the considered E-differentiable V-E-invex vector optimization problem with both inequality and equality constraints. Then, various mixed E-duality theorems are established between the considered E-differentiable multicriteria optimization problem and its vector mixed E-dual problem under appropriate (generalized) V-E-invexity hypotheses.

Definitions and Preliminaries
Let R n be the n-dimensional Euclidean space and R n + be its nonnegative orthant. The following convention for equalities and inequalities will be used in the paper. For any vectors x = x 1 , x 2 , ..., x n T and y = y 1 , y 2 , ..., y n T in R n , x > y means that the vector x is componentwise greater than the vector y. Similarly, the same convention has been used for x = y , x ≧ y and x ≥ y. First, we recall for a common reader the definition of E-differentiable function introduced by Megahed et al. [18].
Definition 1 [18] Let E ∶ R n → R n and f ∶ R n → R be a (not necessarily) differentiable function at a given point u ∈ R n . It is said that f is an E-differentiable function at u if and only if f •E is a differentiable function at u (in the usual sense), that is, Now, we give the definition of a V-E-invex function introduced by Abdulaleem [5].
Definition 2 [5] Let E ∶ R n → R n and f ∶ R n → R k be an E-differentiable function on R n . It is said that f is a V-E-invex function (a strictly V-E-invex function) with respect to at u ∈ R n on R n if, there exist functions ∶ R n × R n → R n and i ∶ R n × R n → R + ⧵ {0}, i = 1, 2, ..., k, such that, for all x ∈ R n (E(x) ≠ E(u)) , the inequalities 1 3 hold. If inequalities (2) are fulfilled for any u ∈ R n (E(x) ≠ E(u)) , then f is a V-E-invex (strictly V-E-invex) function at u on R n . Each function f i , i = 1, ..., k, satisfying (2), is said to be i -E-invex (strictly i -E-invex) with respect to at u on R n .

Remark 1
Note that the Definition 2 generalizes and extends several generalized convexity notions, previously introduced in the literature. Indeed, there are the following special cases: (a) In the case when i (x, u) = 1 , i = 1, ..., k, then the definition of a V-E-invex function reduces to the definition of an E-invex function introduced by Abdulaleem

then the definition of a
V-E-invex function reduces to the definition of a V-invex function introduced by Jeyakumar and Mond [16]. Definition 3 Let E ∶ R n → R n and f ∶ R n → R k be an E-differentiable function on R n . It is said that f is a V-E-pseudo-invex function with respect to at u ∈ R n on R n if, there exist functions ∶ R n × R n → R n and i ∶ R n × R n → R + ⧵ {0}, i = 1, 2, ..., k, such that, for all x ∈ R n , the relations hold. If (3) are fulfilled for any u ∈ R n , then f is V-E-pseudo-invex with respect to on R n . Each function f i , i = 1, ..., k, satisfying (3) is said to be i -E-pseudo-invex with respect to at u on R n . (2) Definition 4 Let E ∶ R n → R n and f ∶ R n → R k be an E-differentiable function on R n . It is said that f is a V-E-quasi-invex function with respect to at u ∈ R n on R n if there exist functions ∶ R n × R n → R n and i ∶ R n × R n → R + ⧵ {0}, i = 1, 2, ..., k, such that, for all x ∈ R n , the relations hold. If (4) are fulfilled for any u ∈ R n , then f is V-E-quasi-invex on R n . Each function f i , i = 1, ..., k, satisfying (4) is said to be i -E-quasi-invex with respect to at u on R n .
In this paper, we consider the following (not necessarily differentiable) multiobjective programming problem (MOP) with both inequality and equality constraints defined as follows: are real-valued functions defined on R n . We shall write g ∶= g 1 , ..., g m ∶ R n → R m and h ∶= h 1 , ..., h q ∶ R n → R q for convenience. Let be the set of all feasible solutions of (MOP). Further, we denote by J(x) the set of inequality constraint indices that are active at a feasible solution x, that is, Let E ∶ R n → R n be a given one-to-one and onto operator. Throughout the paper, we shall assume that the functions constituting the considered problem (MOP) are E-differentiable at any feasible solution. [4] be satisfied at x . Then there exist Lagrange multipliers ∈ R p , ∈ R m , ∈ R q such that

Mixed E-Duality
In this section, a vector mixed E-dual problem is defined for the considered E-differentiable problem (MOP) with inequality and equality constraints. Let the index set J be partitioned into two disjoint subset J 1 and J 2 such that J = J 1 ∪ J 2 and the index set T be partitioned into two disjoint subset T 1 and T 2 such that T = T 1 ∪ T 2 . Let J 1 be an index set such that J 1 = J�J 2 and J 2 = J�J 1 , let | denote the cardinality of the index sets J 1 and J 2 , respectively. Further, let T 1 be an index set such that T 1 = T�T 2 and | denote the cardinality of the index sets T 1 and T 2 , respectively. Let us denote the set Further, let E ∶ R n → R n be a given one-to-one and onto operator. Further, let us define the following set Now, we define the following vector mixed E-dual problem (VMD E ) for the considered E-differentiable problem (MOP): where all functions are defined in the similar way as for the considered vector optimization problem (MOP). Further, let Γ E denote the set of all feasible solutions of (VMD E ), that is, subject to ∇f (E(y)) + ∇g(E(y)) + ∇h(E(y)) = 0, Further, Y E = {y ∈ R n ∶ (y, , , ) ∈ Γ E } . We call (VMD E ) the vector mixed E-dual problem for the E-differentiable multiobjective optimization problem (MOP).
Note that if set J 1 = ∅ and T 1 = ∅ in (VMD E ), then we get a vector Mond-Weir E-dual problem for (MOP) [7] and, moreover, if we set J 2 = ∅ and T 2 = ∅ in (VMD E ), then we obtain a vector Wolfe E-dual problem for (MOP) (see, for example, [1,6]). Now, we shall prove several mixed duality results between E-vector optimization problems (VP E ) and (VMD E ) under (generalized) V-E-invexity assumptions. Then, we use these duality results in proving several mixed E-duality results between vector optimization problems (MOP) and (VMD E ).
Theorem 2 (Mixed weak duality between (VP E ) and (VMD E )). Let x and (y, , , ) be any feasible solutions of the problems (VP E ) and (VMD E ), respectively. Further, assume that at least one of the following hypotheses is fulfilled: Proof Let x and (y, , , ) be any feasible solutions of the problems (VP E ) and (VMD E ), respectively. The proof of this theorem under hypothesis (a). By means of contradiction, suppose that ∇f (E(y)) + ∇g(E(y)) + ∇h(E(y)) = 0, ∑

Thus,
Multiplying each inequality (10) by i and then adding both sides of the resulting inequalities, we get Since ∑ p i=1 i = 1 , the following inequality holds. By x ∈ Ω E and (y, , , ) ∈ Γ E , we have Combining (11)−(14), we get
The proof of this theorem under hypothesis (b). We proceed by contradiction. Suppose, contrary to the result, that (10) holds. Since the function

Theorem 3 (Mixed weak E-duality between (MOP) and (VMD E )). Let E(x) and
If some stronger V-E-invexity hypotheses are imposed on the functions constituting the considered E-differentiable problem, then the following result is true.
Theorem 4 (Mixed weak duality between (VP E ) and (VMD E )). Let x and (y, , , ) be any feasible solutions of the problems (VP E ) and (VMD E ), respectively. Further, assume that at least one of the following hypotheses is fulfilled: (38) Proof Since x ∈ Ω E is a (weak) Pareto solution of the problem (VP E ) and the E-Guignard constraint qualification [4] is satisfied at x , by Theorem 1, there exist ∈ R p , ≠ 0 , ∈ R m , ≧ 0 , ∈ R q , ≧ 0 such that x, , , is a feasible solution of the problem (VMD E ). This means that the objective functions of (VP E ) and (VMD E ) are equal. If we assume that weak duality (Theorem 2) holds between (VP E ) and (VMD E ), x, , , is a (weak) maximum point for (VMD E ) in the sense of mixed. Moreover, we have, by Lemma 1, that x ∈ Ω E . Since x ∈ Ω E is a weak Pareto solution of the problem (VP E ), by Lemma 2, it follows that E x is a weak E-Pareto solution in the problem (MOP). Then, by the strong duality between (VP E ) and (VMD E ), we conclude that also the mixed strong E-duality holds between the problems (MOP) and (VMD E ). This means that if E x ∈ Ω is a weak E-Pareto solution of the problem (MOP), there exist ∈ R p , ≠ 0 , ∈ R m , ≧ 0 , ∈ R q , ≧ 0 such that x, , , is a weakly efficient solution of a maximum type in the mixed dual problem (VMD E ).
Theorem 7 (Mixed converse duality between (VP E ) and (VMD E )). Let x, , , be a (weakly) efficient solution of a maximum type in mixed E-dual problem (VMD E ) such that x ∈ Ω E . Moreover, that the objective functions f i , i ∈ I , are i -E-invex with respect to at x on Ω E ∪ Y E , the constraint functions g j , j ∈ J , are j -E-invex with respect to at x on Ω E ∪ Y E , the functions h t , t ∈ T + E x and the functions −h t , t ∈ T − E x , are t -E-invex with respect to at x on Ω E ∪ Y E . Then x is a (weak) Pareto solution of the problem (VP E ).
32 Page 12 of 18 Theorem 9 (Mixed restricted converse duality between (VP E ) and (VMD E )). Let x and y, , , be feasible solutions for the problems (VP E ) and (VMD E ), respectively, such that Proof The proof of this theorem follows directly from Lemma 2 and Theorem 9.

Conclusion
This paper analyzes mixed E-duality results for E-differentiable V-E-invex multiobjective programming problems with both inequality and equality constraints. The so-called vector mixed E-dual problem has been formulated for such nonconvex (not necessarily)differentiable multiobjective programming problems. Then, various mixed E-duality theorems between the considered E-differentiable vector optimization problem and its mixed dual problem have been proved under (generalized) V-E-invexity hypotheses. The results established in this paper for E-differentiable vector optimization problems extend and generalize similar duality results in the sense of mixed established under other concepts of E-differentiable (generalized) convexity and also duality results in the sense of Mond-Weir and in the sense of Wolfe established for such multicriteria optimization problems. However, some interesting topics for further research remain. It would be of interest to investigate whether it is possible to prove similar results for other classes of E-differentiable vector optimization problems. We shall investigate these questions in subsequent papers.