Optimality conditions and duality for E-differentiable multiobjective programming involving V-E-type I functions

In this paper, we introduce a new concept of sets and a new class of functions called α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-E-invex sets and V-E-preinvex functions. Furthermore, a new concept of generalized convexity is introduced for (not necessarily) differentiable vector optimization problems. Namely, the concept of V-E-type I functions is defined for E-differentiable vector optimization problem. A number of sufficiency results are established under various types of (generalized) V-E-type I requirements. Moreover, several E-duality theorems in the sense of Mond–Weir are proved under appropriate (generalized) V-E-type I functions.

One of the generalized convex functions is an invex function, which was introduced by Hanson [17].Hanson and Mond [15] considered a new generalization called type I function.Rueda and Hanson [16] introduced pseudo-type I and quasitype I functions.Kaul et al. [21] obtained optimality and duality results for multiobjective nonlinear programming problems involving type I and generalized type I functions.Megahed et al. [22] introduced a new definition of an E-differentiable function which transforms a (not necessarily) differentiable function to a differentiable function.Abdulaleem [6] introduced the so-called E-type I functions for not necessarily differentiable multicriteria optimization problems, namely for E-differentiable multiobjective programming problems involving E-type I functions.Recently, Abdulaleem [3] introduced a new concept of generalized convexity.Namely, he defined the concept of E-differentiable V-E-invexity in the case of (not necessarily) differentiable vector optimization problems with E-differentiable functions.
In this paper, new classes of nonconvex E-differentiable multiobjective programming problems.Namely, the concept of the so-called V-E-type I functions for E-differentiable multiobjective programming problems is introduced.Moreover, several classes of generalized V-E-type I functions are also defined.Further, the sufficient optimality conditions are derived for the considered E-differentiable multiobjective programming problem under appropriate V-E-type I and/or generalized V-E-type I hypotheses.Optimality results established in the paper are illustrated by examples of E-differentiable multiobjective optimization problems with (generalized) V-E-type I functions.Furthermore, the so-called vector Mond-Weir E-duality is defined for E-differentiable problems.Then, several Mond-Weir E-duality results are established between the considered E-differentiable multicriteria optimization problem and its Mond-Weir vector dual problem also under appropriate (generalized) V-E-type I functions

New classes of nonconvex E-differentiable functions
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: Now, we introduce the definition of a -E-invex set as a generalization of a -invex set given by Noor [25] and the definition of an E-invex set given by Abdulaleem [5].
, such that the relation holds for all x, x 0 ∈ S and any ∈ [0, 1].
Remark 1 From Definition 3, there are the following special cases: a) If E(x) = x , then the definition of a -E-invex set reduces to the definition of a -invex set (see Noor [25]).b) If x, x 0 = 1, (x, x 0 ) = x − x 0 , then the definition of a -E-invex set reduces to the definition of an E-convex set (see Youness [29] ).c) If x, x 0 = 1, then the definition of a -E-invex set reduces to the definition of an E-invex set (see Abdulaleem [5]).d) If x, x 0 = 1, E(x) = x , then the definition of a -E-invex set reduces to the definition of an invex set (see Mohan and Neogy [23]).e) If E(x) = x, x, x 0 = 1 and (x, x 0 ) = x − x 0 , then the definition of a -E-invex set reduces to the definition of a convex set.Now, we present an example of such a -E-invex set which is not -invex.
. Then, by Defini- tion 3, S is a -E-invex set with respect to given above.However, it is not -invex set as can be seen by taking x = −1 , x 0 = 1 , and = 1 2 , we have Hence, by the definition of a -invex set (see Noor [25]), it follows that S is not -invex.
and E ∶ R → R be an operator defined by where ∶ S × S → R + ⧵ {0} and ∶ S × S → R be defined by Then, by Definition 3, S is a -E-invex set with respect to the function given above.However, it is not E-convex as can be seen by taking x = 1 , x 0 = 16 , and = 1 2 , we have Hence, by the definition of an E-convex set (see Youness [29]), it follows that S is not E-convex.Also, S is not E-invex as can be seen by taking x = 10 , x 0 = 4 , and = 1 2 , we have Hence, by the definition of an E-invex set (see Abdulaleem [5]), it follows that S is not E-invex.Moreover, S is not an invex set with respect to the function given above.
Now, we introduce new concept of generalized convexity.
such that the following inequality holds for all x, x 0 ∈ S and any ∈ [0, 1].
It is clear that every convex function is V-E-preinvex, but the converse is not true.
, such that the following inequality holds for all x, x 0 ∈ S, x ≠ x 0 , and any ∈ [0, 1].
, such that the following inequality holds for all x, x 0 ∈ S and any ∈ [0, 1].
Definition 7 [22] Let E ∶ R n → R n and f ∶ S → R be a (not necessarily) differenti- able function at a given point x 0 ∈ S .It is said that f is an E-differentiable function at x 0 if and only if f •E is a differentiable function at x 0 (in the usual sense), that is, where x 0 , x − x 0 → 0 as x → x 0 .
Let E ∶ R n → R n be a given one-to-one and onto operator.Now, for the prob- lem (VP), we define its associated differentiable vector optimization problem ( VP E ) as follows: We call the problem ( VP E ) an E-vector optimization problem associated to (VP).Let OPSEARCH (2023) 60:1824-1843 be the set of all feasible solutions of ( VP E ).Further, we denote by J E (x) the set of inequality constraint indices that are active at a feasible solution x ∈ Ω E , that is, Definition 8 A feasible point E(x 0 ) is said to be a (weak E-Pareto solution) weakly E-efficient solution of (VP) if and only if there exists no feasible point E(x) such that Definition 9 A feasible point E(x 0 ) is said to be an (E-Pareto solution) E-efficient solution of (VP) if and only if there exists no feasible point E(x) such that Lemma 1 [1] Let E ∶ R n → R n be a one-to-one and onto.Then E Ω E = Ω.
Lemma 2 [1] Let x 0 ∈ Ω be a (weak Pareto solution) Pareto solution of the problem (VP).Then, there exists z ∈ Ω E such that x 0 = E z and z is a (weak Pareto) Pareto solution of the problem ( VP E ).
Lemma 3 [1] Let z ∈ Ω E be a (weak Pareto) Pareto solution of the problem ( VP E ).
Then E z is a (weak Pareto solution) Pareto solution of the problem (VP).Now, we introduce new concept of generalized convexity for E-differentiable vector-valued functions.
Remark 2 From Definition 10, there are the following special cases: a) If f and g are differentiable functions and E(x) ≡ x (E is an identity map), then the definition of V-E-type I functions reduces to the definition of V-type I functions introduced by Hanson et al. [18] functions and E(x) ≡ x (E is an identity map), then the definition of V-E-type I functions reduces to the definition of type I functions introduced by Rueda and Hanson [27]. (2) I functions reduces to the definition of E-type I functions introduced by Abdulaleem [6].
If the inequalities (4, 5) are satisfied at each x 0 ∈ R n , then (f, g) are said to be strictly Now, we present an example of such V-E-type I functions which are not E-type I functions nor type I functions with respect to the same function .
. However, (f, g) is not E-type I as can be seen by taking x = ln 2 , x 0 = ln 6 , since the inequality holds, it follows that (f, g) are not E-type I nor E-invex with respect to given above.Moreover, (f, g) is neither type I nor invex with respect to given above.Now, we introduce various classes of generalized V-E-type I functions.If (6, 7) are satisfied at each x 0 ∈ R n , then (f, g) are said to be quasi-V-E-type I on R n .
If (8,9) are satisfied at each x 0 ∈ R n , then (f, g) are said to be pseudo-V-E-type I on R n .Now, we present an example of such a pseudo-V-E-type I function which is not and g ∶ R → R be defined by Therefore, (f, g) is pseudo-V-E-type I with respect to However, (f, g) is not V-E-type I. Indeed, if we set x = 1 2 , x 0 = 0 , then we have Hence, by Definition 10, it follows that (f, g) is not V-E-type I.
If (10,11) are satisfied at each x 0 ∈ R n , then (f, g) are said to be quasi-pseudo-V-E- type I on R n .If (11) is replaced in the above definition by the inequality then we say (f, g) is strictly quasi-pseudo-V-E-type I at x 0 .
If (13,14) are satisfied at each x 0 ∈ R n , then (f, g) are said to be pseudo-quasi-V-E- type I on R n .If (13) is replaced in the above definition by the inequality then we say (f, g) is strictly pseudo-quasi-V-E-type I at x 0 .

E-optimality conditions
In this section, we prove the sufficient E-optimality conditions for (weak E-Pareto) E-Pareto optimality in the considered E-differentiable vector optimization problem (VP) under assumption that the functions constituting it belong to the classes of E-differentiable functions defined in the preceding section. ( OPSEARCH (2023) 60:1824-1843 Theorem 1 (E-Karush-Kuhn-Tucker necessary optimality conditions).Let x 0 ∈ Ω E be (a weak Pareto solution) a Pareto solution of the problem ( VP E ) (and, thus, E x 0 be (a weak E-Pareto solution) an E-Pareto solution of the problem (VP)).Further, f, g are E-differentiable at x 0 and the E-Guignard constraint qualification [5] be satis- fied at x 0 .Then there exist Lagrange multipliers 0 ∈ R k , 0 ∈ R m such that Definition 16 x 0 , 0 , 0 ∈ Ω E × R k × R m is said to be a Karush-Kuhn-Tucker point for the problem ( VP E ) if the Karush-Kuhn-Tucker necessary optimality conditions (16-18) are satisfied at x 0 with Lagrange multiplier 0 , 0 .
Theorem 2 Let x 0 , 0 , 0 ∈ Ω E × R k × R m be a Karush-Kuhn-Tucker point of the problem ( VP E ).If (f, g) is V-E-type I at x 0 , then x 0 is a weak Pareto solution of the problem ( VP E ) and, thus, E(x 0 ) is a weak E-Pareto solution of the problem (VP).
Proof Suppose that x 0 ∈ Ω E is not a weak Pareto solution of the problem ( VP E ) (and, thus, E x 0 is not a weak E-Pareto solution of the considered multiobjective programming problem (VP)).Then, there exists a feasible solution x for ( VP E ) such that By  0 i > 0 , i ∈ I , we have Since (f i , g j ), i ∈ I, j ∈ J is V-E-type I at x 0 , we have for all x ∈ Ω E Combining ( 19) and ( 21), we have Since  i (E(x), E(x 0 )) > 0, i ∈ I, the above inequality yields (16) By  0 i > 0 , i ∈ I, 0 j ≧ 0, j ∈ J E x 0 , using (24, 25), we obtain that the inequality holds, contradicting (16).By assumption, E ∶ R n → R n is a one-to-one and onto operator.Since x 0 is a weak Pareto solution of the problem ( VP E ), by Lemma 3, E x 0 is a weak E-Pareto solution of the problem (VP).Thus, the proof of this theo- rem is completed.
◻ Proof Suppose contrary to the result, that x 0 is not a Pareto solution in problem ( VP E ) (and, thus, E x 0 is not an E-Pareto solution of the considered multiobjective programming problem (VP)).Then, there exists a feasible solution x • for ( VP E ) such that Since  0 i > 0 , i ∈ I , above inequalities give By ( 0 f , 0 g) is pseudo-quasi-V-E-type I at x 0 and by inequalities ( 29) and (30) we have Adding these above inequalities, we obtain the inequality holds, contradicting (16).Since x 0 is a Pareto solution of the problem ( VP E ), by Lemma 3, E x 0 is an E-Pareto solution of the problem (VP).Thus, the proof of this theorem is completed.E-type I with respect to at x 0 , then x 0 is a weak Pareto solution of the problem ( VP E ) and, thus, E(x 0 ) is a E-Pareto solution the problem (VP).
Proof Suppose contrary to the result, that x 0 is not a weak Pareto solution in prob- lem ( VP E ) (and, thus, E x 0 is not a weak E-Pareto solution of the considered mul- tiobjective programming problem (VP)).Then, there exists a feasible solution x • for ( VP E ) such that Since  0 i > 0 , i ∈ I , above inequality give The remaining part of the proof is similar to that of Theorem 3. ◻ Proof Suppose contrary to the result, that x 0 is not a Pareto solution in problem ( VP E ) (and, thus, E x 0 is not an E-Pareto solution of the considered multiobjective programming problem (VP)).Then, there exists a feasible solution x for ( VP E ) such that (31 By ( 0 f , 0 g) is quasi-pseudo-V-E-type I with respect to at x 0 and by inequalities (38) and (39) we have Adding these above inequalities, we obtain the inequality holds, contradicting (16).By assumption, E ∶ R n → R n is a one-to-one and onto operator.Since x 0 is a Pareto solution of the problem ( VP E ), by Lemma 3, E x 0 is an E-Pareto solution of the problem (VP).Thus, the proof of this theorem is completed. ◻ Example 5 Consider the following nondifferentiable vector optimization problem x ≦ 0} is the set of all feasible solutions of the prob- lem (VP1).Further, note that the functions constituting problem (VP1) are nondifferentiable at x 0 = 0 .Let E(x) = x 3 .For the considered vector optimization problem (VP1), we define its associated constrained E-vector optimization problem ( VP E 1) as follows (37 OPSEARCH (2023) 60:1824-1843 is the set of all feasible solutions of the problem ( VP E 1) and (E(x), E(x 0 )) = x + x 0 , (E(x), E(x 0 )) = e x + x 0 and (E(x), E(x 0 )) = 2e x − x 0 .Then, by Definition 10, it can be shown that the objec- tive functions f and the constraint function g are V-E-type I at x 0 on Ω E .Thus, all hypotheses of Theorem 2 are fulfilled and, therefore, we conclude that x 0 = 0 is a weak Pareto solution of the E-vector optimization problem ( VP E 1) and, thus, E(x 0 ) is a weak E-Pareto solution of the considered multiobjective programming problem (VP1).

Mond-Weir E-duality
In this section, for the problem ( VP E ), we define its vector E-dual problem ( VD E ) in the sense of Mond-Weir [15].Further, let be the set of all feasible solutions of the problem ( VD E ).Let us denote, Theorem 6 (Weak duality between ( VP E ) and ( VD E )).Let x ∈ Ω E and z, 0 , 0 ∈ W E such that (f, g) is V-E-type I at z. Then Proof Let x ∈ Ω E and z, 0 , 0 ∈ W E .We proceed by contradiction.Suppose, con- trary to the result, that the inequality holds.By assumption, x and z, 0 , 0 are feasible solutions of ( VP E ) and ( VD E ), respectively.Since (f, g) is V-E-type I with respect to at x and by Definition 10, the following inequalities hold, respectively.Combining (48, 49), we get Since  i (E(x), E(z)) > 0, i ∈ I, the above inequalities yield Multiplying (52) by the corresponding Lagrange multipliers and then adding both sides of the obtained inequalities, we get that the following inequality Using the condition (17), together with z ∈ Ω E and (50), we get Since  j (E(x), E(z)) > 0, j ∈ J, the above inequalities yield Multiplying (55) by the corresponding Lagrange multipliers and then adding both sides of the obtained inequalities, we get that the following inequality Adding (53) and (56), we obtain that the inequality holds, contradicting (44).This means that the proof of weak duality theorem between the problems ( VP E ) and ( VD E ) is completed.
OPSEARCH (2023) 60:1824-1843 Theorem 7 (Weak E-duality between (VP) and (VD E )).Let E(x) ∈ Ω and z, 0 , 0 ∈ W E .Further, assume that all hypotheses of Theorem 6 are fulfilled.Then, weak E-duality between (VP) and ( VD E ) holds, that is, Proof Let E(x) ∈ Ω and z, 0 , 0 ∈ W E .Then, by Lemma 1. it follows that x ∈ Ω E .Since all hypotheses of Theorem 6 are fulfilled, the weak E-duality theorem between the problems (VP) and ( VD E ) follows directly from Theorem 6. ◻ Theorem 8 (Weak duality between ( VP E ) and , then the weak duality trivially holds.Now, we prove the weak duality theorem when x ≠ z .We proceed by contradiction.Suppose, contrary to the result, that the inequality holds.By the feasibility of (z, , ) ∈ W E ,  i (E(x), E(z)) > 0 and  > 0 , the above inequality yields By (z, , ) ∈ W E , we have j g j (E(z)) = 0, for all j ∈ J implies that Since ( f , g) is pseudo-quasi-V-E-type I with respect to at z and together with inequalities (61), it follows that Using (62) and (z, , ) ∈ W E we have Since ( f , g) is pseudo-quasi-V-E-type I with respect to at z and by inequalities (63), the following inequalities hold, which contradicts (60).This means that the proof of the Mond-Weir weak duality theorem between the problems ( VP E ) and ( VD E ) is completed.◻ Theorem 9 (Weak E-duality between (VP) and (VD E )).Let E(x) ∈ Ω and (z, , ) ∈ W E .Further, assume that all hypotheses of Theorem 8 are fulfilled.Then, Mond-Weir weak E-duality between (VP) and ( VD E ) holds, that is, Proof Let E(x) ∈ Ω and (z, , ) ∈ W E .Then, by Lemma 1. it follows that x ∈ Ω E .Since all hypotheses of Theorem 8 are fulfilled, the Mond-Weir weak E-duality theorem between the problems (VP) and ( VD E ) follows directly from Theorem 8. ◻ Theorem 10 (Weak duality between ( VP E ) and ( E )).Let x ∈ Ω E and (z, , ) ∈ W E such that ( f , g) is strictly pseudo-quasi-V-E-type I at z. Then Theorem 11 (Weak E-duality between (VP) and ( VD E )).Let E(x) ∈ Ω and (z, , ) ∈ W E .Further, assume that all hypotheses of Theorem 10 are fulfilled.Then, weak E-duality between (VP) and ( VD E ) holds, that is, Theorem 12 (Mond-Weir strong duality between ( VP E ) and ( VD E ) and also strong E-duality between (VP) and ( VD E )).Let x 0 ∈ Ω E be a weak Pareto solution (Pareto solution) of the problem ( VP E ) (and, thus, E(x 0 ) ∈ Ω be a weak E-Pareto solution (E-Pareto solution) of the problem (VP)).Further, assume that the Abadie constraint qualification ( ACQ E ) be satisfied at x 0 .Then there exist ∈ R p , ∈ R m , ≧ 0 such that x 0 , , ∈ W E .If all hypotheses of (Theorem 8) Theorem 10 are satisfied, then x 0 , , is a (weak) efficient solution of a maximum type in the problem ( VD E ).In other words, if E(x 0 ) ∈ Ω is a (weak) E-Pareto solution of the problem (VP), then x 0 , , is a (weak) efficient solution of a maximum type in the dual problem ( VD E ).
Proof Since x 0 ∈ Ω E is a weak Pareto solution of the problem ( VP E ) and the Abadie constraint qualification ( ACQ E ) is satisfied at x 0 , there exist ∈ R p , ∈ R m , ≧ 0 such that Thus, x 0 , , is a feasible solution for ( VD E ).If x 0 , , is not a (weak) efficient solution for ( VD E ), then there exists a feasible solution x, ̃ , ̃ of ( VD E ) such that f (E(x)) ≮ f (E(z)).
(65) f (E(x)) ≮ f (E(z)).f (E(� x)) < f (E(x 0 )), which contradicts the Theorem 8. Hence x 0 , , is a (weak) efficient solution for ( VD E ).Moreover, we have, by Lemma 1, that E x 0 ∈ Ω .Since x 0 ∈ Ω E is a weak Pareto solution of the problem ( VP E ), by Lemma 3, it follows that E x 0 is a weak E-Pareto solution in the problem (VP).Then, by the Mond-Weir strong duality between ( VP E ) and ( VD E ), we conclude that also the Mond-Weir strong E-duality holds between the problems (VP) and ( VD E ).This means that if E x 0 ∈ Ω is a weak E-Pareto solution of the problem (VP), there exist ∈ R p , ∈ R m , ≧ 0 such that x 0 , , is a weakly efficient solution of a maximum type in the Mond-Weir dual problem ( VD E ). ◻

Concluding remarks
In this paper, new classes of (not necessarily) differentiable nonconvex multiobjective programming problems have been considered.Specifically, the concept of V-E-type I and/or generalized V-E-type I have been introduced for (not necessarily) differentiable multiobjective programming problems.Further, the sufficiency of the E-Karush-Kuhn-Tucker optimality conditions have been established for these (not necessarily) differentiable vector optimization problems under (generalized) V-E-type I hypotheses.Additionally, the vector Mond-Weir E-dual problems have been formulated for such E-differentiable multiobjective programming problems.Furthermore, various E-duality theorems between the considered E-differentiable vector optimization problem and its Mond-Weir vector dual problem have been proved under (generalized) V-E-type I hypotheses.
Nevertheless, there are still some interesting topics that warrant further research.It would be worthwhile to explore whether similar results can be proven for other classes of E-differentiable vector optimization problems.We intend to investigate these questions in future papers.

1 3 OPSEARCH
(2023) 60:1824-1843 -type I with respect to at x 0 , then x 0 is a Pareto solution of the problem ( VP ) and, E(x 0 ) is an E-Pareto solution of the problem (VP).
type I with respect to , then x 0 is a Pareto solution of the problem ( VP E ) and, thus, E(x 0 ) is an E-Pareto solution of the problem (VP).