E-Univex Sets, E-Univex Functions and E-Differentiable E-Univex Programming

In this paper, we introduce a new concept of sets and a new class of functions called E-univex sets and E-univex functions, respectively. For an E-differentiable function, the concept of E-univexity is introduced by generalizing several concepts of generalized convexity earlier defined into optimization theory. In addition, some properties of E-differentiable E-univex functions are investigated. Further, also concepts of E-differentiable generalized E-univexity are introduced. Then, the sufficiency of the so-called E-Karush–Kuhn–Tucker necessary optimality conditions are proved for an E-differentiable nonlinear optimization problem in which the involved functions are E-univex and/or generalized E-univex.

One of the generalizations of convexity is the concept of univexity introduced by Bector et al. [11] which unifies various concepts of generalized convexity established in literature. In [33], Youness introduced the concept of Now, we present an example of such an E-pre-univex function which is not pre-univex.
. Then, by Definition 5, f is an E-pre-univex function on R, while f is not pre-univex, because for x = π 3 , u = π 6 , and λ = 1 2 , we have Hence, f is not pre-univex with respect to η, b, and Φ defined above. In other words, the class of E-pre-univex functions is larger than the class of pre-univex functions.
We now give the definition of an E-differentiable function introduced by Megahed et al. [23]. Definition 9. [23] 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 Edifferentiable function at u if and only if f • E is a differentiable function at u (in the usual sense), that is, (6) where θ (u, x − u) → 0 as x → u. Now, we introduce new concepts of generalized convexity for E-differentiable functions. Let M ⊆ R n be a nonempty E-invex set with respect to η.
holds. If inequality (7) holds for any u ∈ M , then f is E-univex with respect to η, b and Φ on M .
Remark 1. Note that the Definition 10 generalizes and extends several generalized convexity notions, previously introduced in the literature. Indeed, there are the following special cases: a) If b(x, u) = 1 and Φ(x) = x, then the definition of an E-univex function reduces to the definition of an E-invex function introduced by Abdulaleem [5].
b) If f is differentiable and E(x) ≡ x (E is an identity map), then the definition of an E-univex function reduces to the definition of an univex function introduced by Bector et al. [11]. c) If f is differentiable, E(x) ≡ x (E is an identity map), b(x, u) = 1 and Φ(x) = x, then the definition of an E-univex function reduces to the definition of an invex function introduced by Hanson [15].
then the definition of an E-univex function reduces to the definition of a B-invex function introduced by Bector et al. [9] and Suneja et al. [32]. e) If Φ(x) = x, then the definition of an E-univex function reduces to the definition of an E-B-invex function introduced by Abdulaleem [8].
then we obtain the definition of an E-differentiable E-convex vector-valued function introduced by Megahed et al. [23].
holds. If inequality (8) holds for any u ∈ M (E(x) = E(u)), then f is strictly E-univex with respect to η, b and Φ on M . Now, we present an example of such an E-univex function (with respect to η, b and Φ) which is not univex (with respect to the same given η, b and Φ), B-invex (with respect to the same given η and b), invex (with respect to the same given η) or convex.
Then f is an E-univex function on R with respect to η, b and Φ defined above, but it is not univex with respect to η, b and Φ defined above as can be seen by taking x = 2, u = 4, since the inequality

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holds. Moreover, by the definition of an univex function [11], it follows that f is not univex with respect to η, b and Φ given above. Also, f is not B-invex with respect to η, b defined above as can be seen by taking x = 1, u = 2, since the inequality holds. Note that, by the definition of a B-invex function (see, [9,32]), it follows that f is not B-invex with respect to η, b given above. Further, f is not invex with respect to η defined above as can be seen by taking x = 1, u = 0, since the inequality holds. Further, by the definition of an invex function [15], it follows that f is not invex on R with respect to η given above. It is not difficult to see that f is not a convex function. Now, we present an example of such an E-differentiable E-univex function (with respect to η, b and Φ) which is not E-invex (with respect to the same given η), E-B-invex (with respect to the same given η and b), or E-convex.
Note that f is nondifferentiable at x = 0, but f • E is a differentiable function at x = 0. Then, by Definition 10, f is an E-univex function on R with respect to η, b and Φ given above. Moreover, as it follows from the definition of an E-B-invex function [8], f is not an E-B-invex function on R with respect to η and b given above as can be seen by taking x = 2, u = 1, since the inequality holds. Also, f is not E-invex on R with respect to η defined above as can be seen by taking x = 0, u = −2, since the inequality holds. Note that, by the definition of an E-invex function [5], it follows that f is not E-invex on R with respect to η given above. Further, it is not difficult to see that, by the definition of an E-convex function [33], f is not E-convex at x = 4, u = 1, since the inequality holds.
holds for all x ∈ M . If relation (9) holds for any u ∈ M , then f is pseudo E-univex with respect to η, b and Φ on M .
holds for all x ∈ M . If relation (10) holds for any u ∈ M (E(x) = E(u)), then f is strictly pseudo-E-univex with respect to η, b and Φ on M .
Example 4. The function defined in Example 1 is E-univex on R but not strictly pseudo-E-univex, as can be seen by taking Hence, by the definition of a strictly pseudo-E-univex function, it follows that f is not strictly pseudo-E-univex on R.
holds for all x ∈ M . If relation (11) holds for any u ∈ M , then f is quasi E-univex with respect to η, b and Φ on M .
Remark 2. Every E-univex function is quasi-E-univex. However, the converse is not true. Now, we present an example of such a quasi-E-univex function which is not E-univex.
. Then, f is a quasi-E-univex function on R Now, we present an example of such an E-univex set with respect to η, b, and Φ which is not an univex set with respect to η, b, and Φ.
Then, by Definition 15, S is an E-univex set with respect to η, b, and Φ given above. However, as it follows from the definition of an univex set [11], S is not an univex set with respect to η, b, and Φ given above as can be seen by taking x = −1, u = 1, γ = 1, β = 1 and λ = 1 2 , we obtain (u + λη (x, u) , γ + λb(x, u, λ)Φ(β − γ)) = (0, 1) ∈ S. Thus, which completes the proof of this theorem. Now, we give the definition of E-epigraph and we discuss a characterization of an E-univex function in terms of its E-epigraph.
Now, we give an example that sets of the E-epigraph of f and the epigraph of f are not equal.
Then, by Definition 10, f is an E-univex function on R with respect to η, b and Φ given above. Further, the E-epigraph of f is E-univex on [0, 4]×R. This means that, for any (E(x), β) ∈ S, (E(u), γ) ∈ S the following inequality holds for any λ ∈ [0, 1].
Note that function f in Example 7 show that sets epi E (f ) and epi (f ) given in [11] are not equal. Thus, hence, epi E (f ) = epi (f ). Now, we give a sufficient condition for the function f to be an E-univex function.
holds for all x ∈ M .
Proof. We assume that x ∈ M is a global E-minimizer of f and let d = −∇f (E(x)). Now, we prove that there does not exist d = −∇f (E(x)) ∈ R n , d = 0, satisfying the inequality By assumption, f is E-differentiable at x. Thus, by Definition 9, we have (16), we get that the following inequality holds, which is a contradiction to the assumption that x ∈ M is a global Eminimizer of f. This means that (16) is not satisfied. Thus, ∇f (E(x)) d = 0, for all d ∈ R n . This implies that ∇f (E(x)) = 0.

Proposition 2.
Let E : R n → R n be an operator, Φ : R → R be strictly increasing with Φ(0) = 0, b(E(x), E(x)) > 0 and f : R n → R be an E-differentiable E-univex function on M with respect to η, Φ and b. If ∇f (E(x)) = 0, then x is an E-minimizer of f .
Proof. Let E : R n → R n be an operator. Further, assume that f : M → R is an E-differentiable E-univex function on M with respect to η, Φ and b. Hence, by Definition 10, the inequality (18), therefore, we have that the relation implies that the inequality

Optimality Conditions for E-Differentiable Optimization Problem
In the paper, we consider the following constrained optimization problem: where f : R n → R and g i : R n → R, i ∈ I, are E-differentiable functions on R n . We will write g := (g 1 , ..., g k ) : R n → R k for convenience. Let be the set of all feasible solutions of (P). Further, I (x) is the set of all active inequality constraints at point x ∈ D, that is,

Definition 18.
A point x is said to be an optimal solution of (P) if and only if there exists no other feasible point x such that Now, for the considered E-differentiable optimization problem (P), we define its associated differentiable E-optimization problem (P E ) as follows: We call the problem (P E ) an E-optimization problem. Let be the set of all feasible solutions of (P E ).

Definition 19. A point E(x) is said to be an E-optimal solution of (P) if and only if there exists no other feasible point E(x) such that
f (E(x)) < f (E(x)).
Lemma 1 [3]. Let E : R n → R n be a one-to-one and onto operator. Then E (D E ) = D.

Lemma 2.
Let y ∈ D be an E-optimal solution of (P). Then, there exists x ∈ D E such that y = E(x) and x is an optimal solution of (P E ).

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Proof. Let y ∈ D be an E-optimal solution of (P). By Lemma 1, it follows that there exists x ∈ D E such that y = E (x). Now, we prove that x is an optimal solution of (P E ). By means of contradiction, suppose that x is not an optimal solution of (P E ). Then, by Definition 19, there exists x ∈ D E such that f (E( x)) < f(E(x)). By Lemma 1, we have that there exists y ∈ D such that y = E ( x). Hence, the inequality above implies that f ( y) < f(y), which is a contradiction to the optimal solution of y for (P). The proof in the case when y ∈ D is an E-optimal solution of (P) is similar.
Let the objective function f, the constraint functions g i , i ∈ I, be E -differentiable at x ∈ D E . Further, let x be an optimal solution of (P E ) (and, thus, E (x) be an E-optimal solution of (P)) and the E-Guignard constraint qualification [5] be satisfied at x. Then, there exist Lagrange multiplier λ ∈ R k such that Then x is an optimal solution of the problem (P E ) and, thus, E (x) is an E-optimal solution of the problem (P).
Proof. By assumption, x, λ ∈ D E × R k is a KKT-point of the problem (P E ). Then, by Definition 20, the E-KKT necessary optimality conditions (20)- (22) are satisfied at x with Lagrange multiplier λ ∈ R k . We proceed by contradiction. Suppose, contrary to the result, that x is not an optimal solution of (P E ). Hence, by Definition 19, there exists another x ∈ D E such that From hypothesis a), by Definition 10, the following inequality holds. Combining (25) and (26), we get From hypothesis b), by Definition 10, the following inequalities hold. Multiplying inequalities (28) by the corresponding Lagrange multipliers, we obtain and, moreover, by g(E (x)) = 0, Since Φ gi , i ∈ I, is increasing and satisfies Φ gi (0) = 0, then, using (30) together with x ∈ D E and x ∈ D E , we obtain Thus, Combining (27) and (32), we obtain that the following inequality

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holds, which is a contradiction to the E-KKT necessary optimality condition (20). Since x is an optimal solution of the problem (P E ), by Lemma 2, E (x) is an E-optimal solution of the problem (P). Thus, the proof of this theorem is completed.
. Further, assume the following hypotheses are fulfilled: Then x is an optimal solution of the problem (P E ) and, thus, E (x) is an E-optimal solution of the problem (P).
In order to illustrate the sufficient optimality conditions established in Theorem 4, we now present an example of an E-differentiable optimization problem in which the involved functions are E-univex.
(P1 E ) Note that the set of all feasible solutions of the problem (P1 E ) is D E = {(x 1 , x 2 ) ∈ R 2 : (sin x 1 − 1) 2 + sin x 2 − 1 ≤ 0, 2x 1 + 3x 2 − 9 2 ≤ 0, x 2 1 + x 2 2 − 3 ≤ 0, − sin x 1 ≤ 0, − sin x 2 ≤ 0}. Note that all functions constituting the problem (P1 E ) are differentiable at (0, 0). Then, it can also be shown that the E-Karush-Kuhn-Tucker necessary optimality conditions (20)- (22) are fulfilled at (0, 0) with Lagrange multipliers λ = ( 1 4 , 0, 0, 1 2 , 5 4 ). Further, all hypotheses of Theorem 4 are fulfilled, it can be proved that f, is E-univex at x on D E with respect to η, b f and Φ f given above, each function g i , i = 1, 2, ..., 5, is E-univex at x on D E with respect to η, b gi and Φ gi given above. Hence, x = (0, 0) is an optimal solution of the problem (P1 E ) and, thus, E (x) = (0, 0) is an E-optimal solution of the problem (P). Further, that the sufficient optimality conditions under univexity are not applicable since not all functions constituting problem (P1) are differentiable univex with respect to η, b given above (see, Bector et al. [11]). Moreover, that the sufficient optimality conditions under E-invexity are not applicable since not all functions constituting problem (P1) are E-invex with respect to η given above. Now, under generalized E-univexity notions, we prove the sufficient optimality conditions for the problem (P E ). Theorem 6. Let x, λ ∈ D E × R k be a KKT-point of the E-optimization problem (P E ). Further, assume the following hypotheses are fulfilled: a) the function f , is pseudo-E-univex at x on D E with respect to η, Φ f and b f , b) each constraint function g i , i ∈ I E (x), is quasi-E-univex at x on D E with respect to η, Φ gi and b gi , c) Φ f is strictly increasing and Φ gi , i ∈ I is increasing with Φ f (0) = 0 and Φ gi (0) = 0, Then x is an optimal solution of the problem (P E ) and, thus, E (x) is an E-optimal solution of the problem (P).