E-invexity and generalized E-invexity in E-differentiable multiobjective programming

In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems. For an E-differentiable function, the concept of E-invexity is introduced as a generalization of the E-differentiable E-convexity notion. In addition, some properties of E-differentiable E-invex functions are investigated. Furthermore, the so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered E-differentiable vector optimization problems with both inequality and equality constraints. Also, the sufficiency of the E-Karush-Kuhn-Tucker necessary optimality conditions are proved for such E-differentiable vector optimization problems in which the involved functions are E-invex and/or generalized E-invex.


Introduction
Multiobjective programming, has grown remarkably in studying the development of optimality conditions. This is a consequence of the fact that many real world problems can be modeled as optimization problems with several objectives conflicting with one another, that is, by vector optimization problems. Although the concept of convexity plays a vital real in proving the fundamental results in optimization theory, however, not all real life problems can be analyzed as convex multiobjective programming problems. Therefore, various classes of nonconvex vector optimization problems have been defined in optimization literature. One of such important generalizations of the convexity notion is the concept of invexity introduced by Hanson [12]. In the case of differentiable scalar optimization problems. Namely, Hanson showed that, instead of the usual convexity assumption, if all functions are assumed to be invex (with respect to the same function η), then the sufficient optimality conditions and weak duality can be proved. Over the years, many generalizations of this concept have been introduced in the literature (see, for example, [2], [3], [4], [5], [6], [7], [8], [10], [11], [14], [15], [16], [20], [21], [22], [23], and others).
Craven and Glover [9] characterized the cone-invexity property, for differentiable functions, in terms of Lagrange multipliers. They also established Kuhn-Tucker type optimality conditions and duality theorems for cone invex programs. Jeyakumar and Mond [13] generalized Hanson's definition to the vectorial case. They defined V-invexity of * e-mail: nabbas985@gmail.com differentiable vector-valued functions which preserve the sufficient optimality conditions and duality results as in the scalar case and avoid the major difficulty of verifying that the inequality holds for the same function η for invex functions in multiobjective programming problems.
Another generalization of convexity was derived by Youness [24]. Namely, he introduced the definition of an E-convex set and the definition of an E-convex function and analyzed some properties of these nonconvex sets and functions. Moreover, the results established by Youness [24] were improved by Yang [25]. Further, Megahed et al. [19] presented the concept of an E-differentiable convex function which transforms a (not necessarily) differentiable convex function to a differentiable function based on the effect of an operator E : R n → R n .
In this paper, a new class of nonconvex Edifferentiable vector optimization problems with both inequality and equality constraints is considered in which the involved functions are E-invex. Therefore, the concept of a so-called E-differentiable E-invex function for Edifferentiable vector optimization problems is introduced. Further, under the introduced E-Guignard constraint qualification, the so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered E-differentiable vector optimization problems with both inequality and equality constraints. It is also given an example of such a vector optimization problems with Edifferentiable E-invex functions for which the E-Guignard constraint qualification is satisfied but the E-Abadie constraint qualification is not satisfied. It turns out that the E-Karush-Kuhn-Tucker necessary optimality conditions es-tablished for such a nonsmooth vector optimization problem are not satisfied in such a case. Moreover, the sufficient optimality conditions are derived for the considered E-differentiable vector optimization problem under E-invexity and/or generalized E-invexity. This result is illustrated by the example of nonconvex E-differentiable vector optimization problem in which the involved functions are E-invex (with respect to the same function η). Thus, in the present paper, tools of differentiable optimization problems are used in proving optimality conditions for (weakly) efficiency of nonsmooth multiobjective programming problems.

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 , we define: (i) x = y if and only if x i = y i for all i = 1, 2, ..., n; (ii) x > y if and only if x i > y i for all i = 1, 2, ..., n; (iii) x y if and only if x i y i for all i = 1, 2, ..., n; (iv) x ≥ y if and only if x y and x y. Now, we introduce the definition of an E-invex set as a generalization of an E-convex set given by Youness [24] and the definition of an invex set (with respect to η) given by Mohan and Neogy [21].
Definition 1 Let E : R n → R n . A set M ⊆ R n is said to be an E-invex set (with respect to η : M×M → R n ) if and only if there exists a vector-valued function η : M × M → R n such that the relation holds for all x, u ∈ M and any λ ∈ [0, 1].

Remark 2
If η is a vector-valued function defined by η(z, y) = z − y, then the definition of an E-invex set reduces to the definition of an E-convex set (see Youness [24]).

Remark 3
If E(a) = a, then the definition of an E-invex set with respect to the function η reduces to the definition of an invex set with respect to η (see Mohan and Neogy [21]). Now, we present an example of such an E-invex set which is not E-convex.
and E : R → R be an operator defined by and η : M × M → R be defined by Then, by Definition 1, M is an E-invex set with respect to the function η given above. However, it is not E-convex as can be seen by taking x = 1, u = 4, and λ = 1 2 , we have Hence, by the definition of an E-convex set (see Remark 2), it follows that M is not E-convex. Now, we present of an example of such an E-invex set with respect to η which is not invex with respect to η.
and η : M × M → R be defined by Then, by Definition 1, M is an E-invex set with respect to the function η given above. However, it is not invex with respect to η as can be seen by taking x = −1, u = 1, and λ = 1 2 , we have Hence, by the definition of an invex set (see Remark 3), it follows that M is not an invex set with respect to η.
Definition 6 [19] Let E : R n → R n and f : M → R be a (not necessarily) differentiable function at a given point u ∈ M. 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) and, moreover, where θ (u, x − u) → 0 as x → u.
Now, we introduce a new concept of generalized convexity for E-differentiable vector-valued functions.
(2) If inequalities (2) hold for any u ∈ M, then f is E-invex with respect to η on M. b) If η : M × M → R n is defined by η(x, u) = x − u, then we obtain the definition of an E-differentiable Econvex vector-valued function introduced by Megahed et al. [19].
c) If f is differentiable, E(x) = x and η(x, u) = x − u, then the definition of an E-invex function reduces to the definition of a differentiable convex vectorvalued function.
d) If f is differentiable and η(x, u) = x − u, then we obtain the definition of a differentiable E-convex function introduced by Youness [24].
hold. If inequalities (3) are fulfilled for any u ∈ M (E(x) E(u)), then f is strictly E-invex with respect to η on M. Now, we present an example of such an E-invex function which is not E-convex.
Example 10 Let f : [0, π 2 ) → R be defined by f (x) = cos x and E : [0, π 2 ) → [0, π 2 ) be an operator defined by E(x) = π 2 − x and η be defined by Then f is E-invex on [0, π 2 ), but it is not E-convex as can be seen by taking x = 0, u = π 4 , and λ = 1 2 , since the inequality holds. Hence, by the definition of an E-convex function [24], it follows that f is not E-convex on [0, π 2 ). Now, we give the necessary condition for Edifferentiable E-invexity.

Proposition 11
Let E : R n → R n and f : M → R k be an E-invex (strictly E-invex) function with respect to η on M and u ∈ M. Further, assume that f is E-differentiable at u. Then, the following inequality holds for all x, u ∈ M (E(x) E(u)). Now, we prove a sufficient condition for an Edifferentiable E-invex function with respect to η.
Theorem 12 Let E : R n → R n and f : M → R be an E-differentiable function at u ∈ M on M. Further, assume that there exists η : M × M → R n such that holds for any λ ∈ [0, 1]. Then, f is an E-invex function with respect to η.
Proof. By (5), we have that the inequality holds for any λ ∈ [0, 1]. Thus, the above inequality yields for any λ ∈ [0, 1], By assumption, f is E-differentiable at u. Hence, by Definition 6, it follows that f • E is differentiable at u. Therefore, letting λ → 0, we obtain the inequality (2). Now, we introduce various classes of generalized Edifferentiable E-invex functions as a generalization of appropriate generalized E-convex functions and, thus, pseudo-convex and quasi-convex functions. It is said that f is strictly pseudo E-invex with respect to η at u ∈ M on M if, there exists η : M × M → R n such that, for all x ∈ M, x u, and i = 1, ..., k, Note that every strictly pseudo E-invex function is pseudo E-invex and every E-differentiable pseudo Econvex function is pseudo E-invex. Also, every pseudo E-convex function is E-invex and every E-invex function is pseudo E-invex function for the same function η, but the converse is not true. Now, we present an example of such an E- Hence, by Definition 2, it follows that f is not E-invex.
Definition 16 Let E : R n → R n and f : M → R k be an E-differentiable function on a nonempty open set M ⊂ R n . It is said that f is quasi-E-invex with respect to η at u ∈ M on M if, there exists η : M × M → R n such that, for all x ∈ M and i = 1, ..., k, (10) If (10) holds for any u ∈ M, then f is quasi-E-invex with respect to η on M.
Note that E-differentiable quasi E-convex is trivially quasi E-invex and every pseudo E-invex function is quasi E-invex. Now, we present an example of such a quasi E-invex function but not a quasi E-convex function.

E-optimality conditions for E-differentiable multiobjective programming
In some cases, the multiobjective programming problem can be represented as the following unconstrained vector optimization problem: where M is a nonempty set of R n , and f denotes a vectorvalued E-differentiable function on M. Now, we give the definitions of a weak Pareto solution and a Pareto solution of the considered vector optimization problem (VP).
Let E : R n → R n be an one-to-one and onto operator. For the considered multiobjective programming problem (VP), we define the vector optimization problem (VP E ) as follows where E(M) is a nonempty set of R n , and f • E denotes a vector-valued differentiable function on E(M). Now, we give the definitions of a weak Pareto (weakly efficient) solution and a Pareto (efficient) solution of the vector optimization problem (VP E ), which are at the same time a weak E-Pareto solution (weakly E-efficient solution) and an E-Pareto solution (E-efficient solution) of the considered multiobjective programming problem (VP).
As it is known [6], a characteristic property of a scalar invex function with respect to η is the fact that each its stationary point is also its global minimum. It turns out that this property can be generalized to the class of vector E-invex functions with respect to η. For this purpose, we have to define adequately an E-critical point concept for vector-valued functions.

Definition 22
Let E : R n → R n . A point u ∈ M is said to be a vector E-critical point of an E-differentiable vectorvalued function f : M → R k , (or, in other words, for the problem (VP)) if there exists a vector λ ∈ R k with λ ≥ 0 such that λ T ∇( f • E)(u) = 0. Now, we prove that every weakly efficient point is also an E-vector critical point.
By assumption, the objective function f , is Edifferentiable at x. Thus, by Definition 6, we get Using θ (x, x − x) → 0 and x−x x−x → d as x → x together with (11), we get that the following inequality holds, which is a contradiction to the assumption that E(x) is a weakly E-efficient solution of the vector optimization problem (VP). Hence, there exists a vector λ ∈ R k with The proof of this theorem is completed. Now, we prove the converse of the above theorem using the concept of vectorial E-invexity introduced in the paper.
Theorem 24 Let E : R n → R n , x be a vector E-critical point of (VP), and let f • E be a vector E-invex function at x with respect to η. Then x is a weak E-Pareto solution of (VP).
Proof. Let x be a vector E-critical point. Then, there exists a vector λ ∈ R k with λ ≥ 0 such that λ We proceed by contradiction. Suppose that x is not a weak E-Pareto solution of (VP). Then, there exists another point z ∈ M such that Thus, by E-invexity of f , we get that Combining (13) and (14), we get that the inequality holds, which is a contradiction to the assumption that x is a weak Pareto solution for (VP). The proof of this theorem is completed. In general, a vector optimization problem is considered with the set of inequality and equality constraints as follows where X is a nonempty convex open subset of R n , f i : X → R, i ∈ I = {1, ..., p} , g j : X → R, j ∈ J, h t : X → R, t ∈ T , are real-valued E-differentiable functions defined on X.
For the purpose of simplifying our presentation, we will next introduce some notations which will be used frequently throughout this paper. We will write g := (g 1 , ..., g m ) : X → R m and h := (h 1 , ..., h s ) : X → R s for convenience. Let Ω := x ∈ R n : g j (x) 0, j ∈ J, h t (x) = 0, t ∈ T be the set of all feasible solutions of (CVP). Further, let us denote by J (x), the set of inequality constraint indices that are active at a feasible solution x, that is, J (x) = j ∈ J : g j (x) = 0 Let E : R n → R n be an one-to-one and onto operator. For the considered constrained multiobjective programming problem (CVP), we define its associated constrained vector optimization problem (CVP E ) with both inequality and equality constraints as follows where the set X and the functions f i , i ∈ I, g j , j ∈ J, h t , t ∈ T , are defined in the similar way as for (CVP). We call (CVP E ) the E-vector optimization problem (associated to the multiobjective programming problem (CVP)). Let be the set of all feasible solutions of (CVP E ).
In this section, we derive both necessary and sufficient optimality conditions for a new class of nonconvex multicriteria optimization problems. Namely, we consider a class of E-differentiable multiobjective programming problems. Throughout this section, E : R n → R n is assumed to be an one-to-one and onto operator. By Definition 6, the functions constituting the E-vector optimization problem (CVP E ) are differentiable at any its feasible solution (in the usual sense). Further, we denote by J E (x), the set of inequality constraint indices that are active at a feasible solution E(x), that is, J E (x) = j ∈ J : g j • E (x) = 0 . Moreover, it can be proved [1] (see Lemma 25 below) that if x is a (weak) Pareto solution of the E-vector optimization problem (CVP E ), then E (x) is a (weak) Pareto solution of the original multiobjective programming problem (CVP). We call E (x) a (weak) E-Pareto solution of the problem (CVP).
Lemma 25 [1] Let E : R n → R n be a one-to-one and onto and z ∈ Ω E be a weak Pareto (Pareto) solution of the constrained E-vector optimization problem (CVP E ). Then E (z) is a weak E-Pareto solution (E-Pareto solution) of the considered constrained multiobjective programming problem (CVP).
Before we establish the Karush-Kuhn-Tucker necessary optimality conditions for problem (CVP), we re-call the Motzkin's theorem of the alternative.
Theorem 26 [18] (Motzkin's theorem of the alternative). Let A, C, D be given matrices, with A being nonvacuous. Then either the system of inequalities Ax < 0, Cx 0, Dx = 0 has a solution x, or the system A T y 1 + C T y 2 + D T y 3 = 0, y 1 ≥ 0, y 2 0 has solution y 1 , y 2 and y 3 , but never both. Now, we give the definition of the minimal element of a given set (with respect to an order relation).
Definition 27 [17] Let Y be a given set in R k ordered by or by <. Specifically, we call the minimal element of Y defined by ≤ a minimal vector, and that defined by < a weak minimal vector. Formally speaking, a vector y ∈ Y is called a minimal vector in Y if there exists no vector y in Y such that y ≤ y; it is called a weak minimal vector if there exists no vector y in Y such that y < y.
Definition 28 Let Y ⊆ R k . The Bouligand contingent cone of Y at y ∈ Y is the set T Y (y) of all vectors q ∈ R k such that there exist a sequence {y n } ∈ Y and a sequence β n of strictly positive real number such that lim n→∞ y n = y, lim n→∞ β n = 0, lim n→∞ y n − y β n = q.
In other words, the Bouligand contingent cone of Y at y is defined by Remark 29 Note that Lin [17] named any Bouligand contingent vector, that is, any vector q ∈ T Y (y), a convergence vector for the set Y at y. Now, we extend the result established by Lin [17] for the E-vector optimization problem (CVP E ).
Theorem 30 If x ∈ Ω E is locally (weak) minimal for f • E on Ω E then no Bouligand contingent vector for f (E(Ω E )) at y = f (E(x)) is strictly negative.

Definition 31
The tangent cone (also called contingent cone or Bouligand cone) of Ω E at x ∈ cl Ω E is defined by T Before we prove the Karush-Kuhn-Tucker necessary optimality conditions for the differentiable constrained Evector optimization problem with inequality and equality constraint (CVP E ), we introduce the so-called E-Guignard constraint qualification. In order to do this, for the constrained E-vector optimization problem (CVP E ), we introduce the E-linearized cone L E (x).
Definition 32 For the constrained E-vector optimization problem (CVP E ), the E-linearized cone at x ∈ Ω E , denoted by L E (x), is defined by It is easy to see that L E (x) is a nonempty closed convex cone.
The following lemma shows the relationship between the Bouligand contingent cone T Ω E (x) and the Elinearizing L E (x) cone.
Lemma 33 If x ∈ Ω E is a Pareto solution of the constrained E-vector optimization problem (CVP E ), then Proof. Let x ∈ Ω E be given and d ∈ T Ω E (x). Then, by Definition 31, there exists a sequence {x n } ∈ Ω E such that lim n→∞ x n = x, lim n→∞ β n = 0, lim n→∞ x n −x β n = d. By assumption, all constraint functions g j , j ∈ J and h t , t ∈ T , are E-differentiable at x. Hence, by Definition 6, it follows that where Assume that d n := x n −x β n . Then x n = x + β n d n and we obtain that (19) From (18), (19), and Definition 31, it follows that By (20), (21) and Definition 32, we have Since L E (x) is closed and convex, we get Now, we give the so-called E-Abadie constraint qualification for the E-differentiable vector optimization problem (CVP) with both inequality and equality constraints which was introduced in [1].
Definition 34 It is said that the so-called E-Abadie constraint qualification (ACQ E ) holds at x ∈ Ω E for the differentiable E-vector optimization problem (CVP E ) with both inequality and equality constraints if Now, we introduce the so-called E-Guignard constraint qualification for the E-differentiable constrained vector optimization problem (CVP) with both inequality and equality constraints.
Definition 35 It is said that the so-called E-Guignard constraint qualification (GCQ E ) holds at x ∈ Ω E for the differentiable constrained E-vector optimization problem (CVP E ) with both inequality and equality constraints if Now, we present an example of such a nondifferentiable vector optimization problem for which the E-Guignard constraint qualification is satisfied but E-Abadie constraint qualification introduced in [1] does not hold.
Example 36 Consider the following nonconvex nondifferentiable vector optimization problem R 2 → R 2 be an one-to-one and onto mapping defined as follows E (x 1 , x 2 ) = x 3 1 , x 2 . For the considered vector optimization problem (CVP1), we define its associated Evector optimization problem (CVP E 1) as follows Then, by the definition of the E-linearized cone, we have that L E (x) = (d 1 , d 2 ) ∈ R 2 : d 2 d 1 . Further, by the definition of the Bouligand contingent cone, we have that T Ω E (x) . Hence, the E-Guignard constraint qualification is satisfied at x but E-Abadie constraint qualification is not satisfied. Now, we prove the Karush-Kuhn-Tucker necessary optimality conditions for the differentiable constrained Evector optimization problem (CVP E ) and, thus, the socalled E-Karush-Kuhn-Tucker necessary optimality conditions for not necessarily differentiable constrained multiobjective programming problem (CVP) in which the involved functions are E-differentiable and for which the E-Guignard constraint qualification holds.
Theorem 37 (E-Karush-Kuhn-Tucker necessary optimality conditions). Let x ∈ Ω E be a weak Pareto solution of the constrained E-vector optimization problem (CVP E ) (and, thus, E (x) be a weak E-Pareto solution of the considered constrained multiobjective programming problem (CVP)). Further, let f , g, h be E-differentiable at x and the E-Guignard constraint qualification be satisfied at x. Then there exist Lagrange multipliers λ ∈ R p , µ ∈ R m , ξ ∈ R s such that Proof. By assumption, x ∈ Ω E is a weak Pareto solution in the E-vector optimization problem (VP E ) (and, thus, E (x) is a weak E-Pareto solution of the considered multiobjective programming problem (VP)). Let d ∈ T Ω E (x) and x n be the corresponding sequence of feasible solutions in E-vector optimization problem (VP E ) converging to x and {β n } be the corresponding sequence of scalars such that β n > 0 for each integer n converging to 0 (see Definition 31). We denote by f (E(Ω E )) ⊂ R k and y = f (E(x)). Since x ∈ Ω E is a weak Pareto point in the E-vector optimization problems (VP E ), y = f (E(x)) is a weak minimal vector in f (E(Ω E )) (see Theorem 30). Further, we consider the sequence of vectors {y n } ∈ f (E(Ω E )), where y n = f (E(x n )). By assumption, the objective functions f i , i ∈ I, are E-differentiable at x, Thus, by Definition 6, we have where θ i (x, x n − x) → 0, i ∈ I, as x n → x. By the above equality, we obtain, for any i ∈ I, By assumption, {x n } is a sequence of feasible solutions in the constrained E-vector optimization problems (CVP E ) converging to x. In view of E-differentiability of the functions f i , i ∈ I, at x, it follows that ( f • E) is differentiable at x and, hence, it is also a continuous function at x. Therefore, the sequence {y n } converges to y = f (E(x)). Hence, by (28), it follows that Then, by Definition 31, q is a Bouligand contingent vector for f (E(Ω E )) at y. From the E-Guignard constraint qualification, it follows that d is a Bouligand contingent vector (convergence vector) for Ω E at x if and only if d is a solution to the system Since y is a weak Pareto of f (E(Ω E )), there is no Bouligand contingent (convergence vector) for f (E(Ω E )) at y strictly negative (see Theorem 30). Therefore, the system is inconsistent. From Motzkin's theorem of the alternative (see Theorem 26), it follows that the system is consistent. Let (λ, µ, ξ) be a solution to the above system. Then, we define ζ ∈ R q + as follows Thus, we conclude that (λ, µ, ξ) satisfies the E-Karush-Kuhn-Tucker necessary optimality conditions (24)-(26). Hence, the proof of this theorem is completed.
In order to show that the E-Karush-Kuhn-Tucker necessary optimality conditions cannot be fulfilled without the E-Guignard constraint qualification, we present the example of such an E-differentiable vector optimization problem.
Example 38 Consider the following nondifferentiable vector optimization problem Note that the set of all feasible solutions of the considered constrained vector optimization problem (CVP2) is Further, note that the functions constituting problem (CVP2) are nondifferentiable at (0, 0). It can be shown by Definition 19 that the feasible solution x = (0, 0) is an E-Pareto solution of the considered nondifferentiable constrained multiobjective programming problem (CVP2). Let E : R 2 → R 2 be defined as follows: For the considered constrained vector optimization problem (CVP2), we define its associated constrained E-vector optimization problem (CVP E 2) as follows Note that the set of all feasible solutions of the constructed constrained E-vector optimization problem (CVP E 2) is Ω E = {(x 1 , x 2 ) ∈ R 2 : sin x 1 − sin x 2 0 ∧ x 1 x 2 = 0} and η(E(x), E(u)) = (2 sin x 1 , 2 sin x 2 ). Then, by the definition of the E-linearized cone, we have that L E (x) = (d 1 , d 2 ) ∈ R 2 : d 2 d 1 . Further, by the definition of the Bouligand contingent cone, we have that cl conv T Ω E (x) . Hence, the E-Guignard constraint qualification is not fulfilled at x. Now, we show that E-Karush-Kuhn-Tucker necessary optimality conditions are not satisfied at x. Indeed, we have ∇ ( However, note that the E-Karush-Kuhn-Tucker necessary optimality conditions are not satisfied at x = (0, 0). Namely, by (24), it follows that λ 1 + λ 2 = 0, are fulfilled only in the case when λ 1 = 0 and λ 2 = 0, what is impossible.
Definition 40 x, λ, µ, ξ ∈ Ω E ×R p ×R m ×R s is said to be a Karush-Kuhn-Tucker point for the considered constrained E-vector optimization problem (CVP E ) if the Karush-Kuhn-Tucker necessary optimality conditions (24)-(26) are satisfied at x with Lagrange multiplier λ, µ, ξ. Now, we prove the sufficiency of the E-Karush-Kuhn-Tucker necessary optimality conditions for constrained vector optimization problem (CVP) under E-invexity hypotheses.
Theorem 41 Let x, λ, µ, ξ ∈ Ω E × R p × R m × R s be a Karush-Kuhn-Tucker point of the constrained Evector optimization problem (CVP E ). Let T + E (E (x)) = t ∈ T : ξ t > 0 and T − E (E (x)) = t ∈ T : ξ t < 0 . Furthermore, assume the following hypotheses are fulfilled: a) each objective function f is an E-invex with respect to η at x on Ω E , b) each inequality constraint g j , j ∈ J (E (x)), is an E-invex function with respect to η at x on Ω E , c) each equality constraint h t , t ∈ T + (E (x)), is an Einvex function with respect to η at x on Ω E , , is an E-invex function with respect to η at x on Ω E . Then x is a weak Pareto solution of the problem (CVP E ) and, thus, E(x) is a weak E-Pareto solution of the problem (CVP).
Proof. By assumption, x, λ, µ, ξ ∈ Ω E ×R p ×R m ×R s is a Karush-Kuhn-Tucker point of the constrained E-vector optimization problem (CVP E ). Then, by Definition 40, the Karush-Kuhn-Tucker necessary optimality conditions (24)-(26) are satisfied at x with Lagrange multipliers λ ∈ R p , µ ∈ R m and ξ ∈ R s . We proceed by contradiction. Suppose, contrary to the result, that x is not a weak Pareto solution of the problem (CVP E ). Hence, by Definition 18, there exists another x ∈ Ω E such that Using hypotheses a)-d), by Definition 7 and Theorem 12, the following inequalities hold, respectively. Combining (35)-(36) and then multiplying the resulting inequalities by the corresponding Lagrange multipliers and adding both their sides, we get Multiplying inequalities (37)-(39) by the corresponding Lagrange multipliers, respectively, we obtain Using the E-Karush-Kuhn-Tucker necessary optimality condition (25) together with x ∈ Ω E and x ∈ Ω E , we get, respectively, Adding both sides of the above inequalities, by (40), we obtain that the following inequality holds, which is a contradiction to the the E-Karush-Kuhn-Tucker necessary optimality condition (24). By assumption, E : R n → R n is an one-to-one and onto operator. Since x is a weak Pareto solution of the problem (CVP E ), by Lemma 25, E (x) is a weak E-Pareto solution of the problem (CVP). Thus, the proof of this theorem is completed.
Remark 42 As it follows from the proof of Theorem 41, the sufficient conditions are also satisfied if all or some of the functions g j , j ∈ J (E (x)), h t , t ∈ T + (E (x)), −h t , t ∈ T − (E (x)) are E-differentiable quasi E-invex function at x on Ω with respect to η.
Theorem 43 Let x, λ, µ, ξ ∈ Ω E × R p × R m × R s be a Karush-Kuhn-Tucker point of the constrained E-vector optimization problem (CVP E ). Furthermore, assume that the following hypotheses are fulfilled: a) each objective function f is strictly E-invex with respect to η at x on Ω E , b) each inequality constraint g j , j ∈ J (x), is an Einvex function with respect to η at x on Ω E , c) each equality constraint h t , t ∈ T + (E (x)), is an Einvex function with respect to η at x on Ω E , , is an E-invex function with respect to η at x on Ω E . Then x is a Pareto solution of the problem (CVP E ) and, thus, E (x) is an E-Pareto solution of the problem (CVP).
Now, under the concepts of generalized E-invexity, we prove the sufficient optimality conditions for a feasible solution to be a weak E-Pareto solution of problem (CVP).
Theorem 44 Let x, λ, µ, ξ ∈ Ω E × R p × R m × R s be a Karush-Kuhn-Tucker point of the constrained E-vector optimization problem (CVP E ). Furthermore, assume that the following hypotheses are fulfilled: a) each objective function f is an pseudo E-invex function with respect to η at x on Ω E , b) each inequality constraint g j , j ∈ J (E (x)), is an quasi E-invex function with respect to η at x on Ω E , c) each equality constraint h t , t ∈ T + (E (x)), is an quasi E-invex function with respect to η at x on Ω E , d) each function −h t , t ∈ T − (E (x)), is an quasi Einvex function with respect to η at x on Ω E . Then x is a weak Pareto solution of the problem (CVP E ) and, thus, E (x) is a weak E-Pareto solution of the problem (CVP).
Proof. By assumption, x, λ, µ, ξ ∈ Ω E ×R p ×R m ×R s is a Karush-Kuhn-Tucker point in the considered constrained E-vector optimization problem (CVP E ). Then, by Definition 40, the Karush-Kuhn-Tucker necessary optimality conditions (24)-(26) are satisfied at x with Lagrange multipliers λ ∈ R p , µ ∈ R m and ξ ∈ R s . We proceed by contradiction. Suppose, contrary to the result, that x is not a weak Pareto solution in problem (CVP E ). Hence, by Definition 19, there exists another x ∈ Ω E such that f i (E( x)) ≤ f i (E (x)) , i ∈ I.

Concluding remarks
In this paper, a new class of nonconvex nondifferentiable vector optimization problems has been defined. Namely, an E-differentiable multiobjective programming problem with both inequality and equality constraint has been considered. Further, the so-called E-Karush-Kuhn-Tucker necessary optimality conditions with both inequality and equality constraints under the introduced E-Guignard constraint qualification have been established for the considered E-differentiable vector optimization problem. Moreover, the sufficient optimality conditions have been derived for such nonconvex nonsmooth vector optimization problems under the introduced concepts of E-invexity and generalized E-invexity. In order to illustrate the optimality results established in the paper, the examples of Edifferentiable multiobjective programming problems have been given.
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 Edifferentiable vector optimization problems. We shall investigate these questions in subsequent papers.