Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function

In this paper, a nonconvex vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. The functions constituting it are not necessarily differentiable, but they are \begin{document}$ E $\end{document} -differentiable. The so-called \begin{document}$ E $\end{document} -Karush-Kuhn-Tucker necessary optimality conditions are established for the considered \begin{document}$ E $\end{document} -differentiable vector optimization problem with the multiple interval-valued objective function. Also the sufficient optimality conditions are derived for such interval-valued vector optimization problems under appropriate (generalized) \begin{document}$ E $\end{document} -convexity hypotheses.


1.
Introduction. In mathematical programming, we usually deal with real numbers which are assumed to be fixed. However, in many real-life situations, the coefficients of decision support models are not exactly known and, therefore, data suffer from inexactness. In other words, we often encounter cases where the information items can't be determined with certainty. The interval-valued optimization problems are closely related to optimization problems with inexact data. According to the decision maker's point of view under changeable conditions, we may replace the real numbers by the interval numbers to formulate optimization problems more appropriately. Therefore, the interval-valued optimization problems have been of much interest in recent past and thus explored the extent of optimality conditions and duality applicability in different areas (see, for example, [1], [2], [4], [5], [7], [13], [14], [15], [16], [17], [18], [19], [25], [26], [29], [30], [35], [36], and others).
Several generalizations of the definition of a convex function have been introduced to optimization theory in order to weaken an assumption of convexity for establishing optimality and duality results for new classes of nonconvex optimization problems, including vector optimization ones. Youness [32] brought forward the concepts of E-convex sets, E-convex functions and E-convex mathematical programming by relaxing the definition of convex sets and convex functions, discussed some of their basic properties and established some optimality results for optimization problems under E-convexity hypotheses. This kind of generalized convexity is based on the effect of an operator E : R n → R n on the sets and the domain of the definition of functions. Unfortunately, some of them turned out to be incorrect as it was noted by Yang [31]. The initial results of Youness [32] inspired a great deal of subsequent works, which greatly expanded the role of E-convexity in optimization theory (see, for instance, [6], [8], [9], [11], [12], [22], [24], [27], [28], [33], [34], and others). Youness [33] established some properties of E-convex functions and some necessary and sufficient optimality criteria for nonlinear E-convex mathematical programming problems. Later, Syau and Lee [28] presented the concept of E-quasiconvex functions and discussed some properties of E-convex and E-quasiconvex functions. Emam and Youness [10] introduced a new class of Econvex sets and E-convex functions, which are called strongly E-convex sets and strongly E-convex functions, respectively, by taking the images of two points under an operator E : R n → R n besides the two points themselves. Youness [34] gave a characterization of efficient solutions for multiobjective programming problems involving E-convex functions. Soleimani-damaneh [27] established some properties of E-convex and generalized E-convex functions. Recently, Megahed et al. [21] introduced a combined interactive approach for solving nonlinear generalized E-convex multiobjective programming problems.
In this paper, the class of E-differentiable multiobjective programming problems with multiple interval-valued objective functions and with both inequality and equality constraints is considered. The so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for such nondifferentiable vector optimization problems under the constraint qualification introduced in the paper. The case is illustrated in which these necessary optimality conditions are not fulfilled if the introduced constraint qualification is not satisfied. Under E-convexity and/or generalized E-convexity assumptions, the sufficient optimality conditions for so-called (weak) LU -E-Pareto optimality are also established for the considered Edifferentiable interval-valued multiobjective programming problem. It is also illustrated that the optimality conditions established in the paper are also applicable for such vector optimization problems with multiple interval-valued objective functions for which the classical optimality conditions can be avoided. Namely, an example of a nondifferentiable vector optimization problem with the multiple interval-valued objective function is presented for which both the classical Karush-Kuhn-Tucker necessary optimality conditions and the sufficient optimality conditions for differentiable interval-valued multiobjective programming problems can not be applied. However, the E-Karush-Kuhn-Tucker necessary optimality conditions and the sufficient optimality conditions established in the paper may be used in such a case to analyze (weak) LU -E-Pareto optimality of feasible solutions.

2.
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.
x y if and only if x i y i for all i = 1, 2, ..., n; 4. x ≥ y if and only if x y and x = y. Let I (R) be a class of all closed and bounded intervals in R. Throughout this paper, when we say that A is a closed interval, we mean that A is also bounded in R. If A is a closed interval, we use the notation A = [a L , a U ], where a L and a U mean the lower and upper bounds of A, respectively. In other words, if A = [a L , a U ] ∈ I (R), Then, by definition, we have: For more details on the topic of interval analysis, we refer to Moore [23] and Alefeld and Herzberger [3].
In interval mathematics, an order relation is often used to rank interval numbers and it implies that an interval number is better than another but not that one is larger than another. For It means that A is inferior to B, or B is superior to A. It is easy to see that LU is a partial ordering on I (R Throughout this section, let X be a nonempty open subset of R n . A function ψ : Now, we shall consider the differentiation of an interval-valued function. Namely, we use a very straightforward concept of differentiation introduced by Wu [29]. We now give the definition of an E-differentiable function introduced by Megahed et al. [21]. Definition 2.2. Let f : R n → R be a (not necessarily) differentiable function on R n , x ∈ R n and E : R n → R n . It is said that f is E-differentiable at x if and only if f • E is a differentiable function at x (in the usual sense) and In other words, (9) is equivalent to the fact that at least one of the inequalities (7) and (8) is strict.
Let f be a real-valued differentiable function on an E-convex set. The definition of a differentiable E-convex function was introduced by Piao et al. [24]. The next result characterizes an E-differentiable E-convex interval-valued function with respect to the gradients of its lower and upper functions.
be an E-convex (strictly E-convex) function on S and u ∈ S. Further, assume that f is E-differentiable at u. Then, the inequalities hold for all x ∈ S (E(x) = E(u)).
Proof. Assume that S is an E-convex set, f : S → R is an E-convex function on S and u ∈ S. By Definition 2.6, it follows that the inequalities hold for all x, u ∈ S and any λ ∈ [0, 1]. Thus, the above inequalities yield, respectively, Letting λ → 0, we obtain the inequalities (10) and (11), respectively. Now, we define the concepts of generalized E-convexity for E-differentiable interval-valued functions.
Definition 2.8. Let E : R n → R n , S be a nonempty E-convex subset of R n and f : S → I (R) be an E-differentiable function at u ∈ S. Then f is said to be a pseudo E-convex function at u on S if the relations hold for all x ∈ S. If (12) and (13) are satisfied for each u ∈ S, then f is said to be a pseudo E-convex function on S.
Definition 2.9. Let E : R n → R n , S be a nonempty E-convex subset of R n and f : S → I (R) be an E-differentiable function at u ∈ S. Then f is said to be a strictly pseudo E-convex function at u on S if the relations hold for all x, u ∈ S, E(x) = E(u). If (14) and (15) are satisfied for each u ∈ S, E(x) = E(u), then f is said to be a strictly pseudo E-convex function on S.
Definition 2.11. Let E : R n → R n , S be a nonempty E-convex subset of R n and f : S → I (R) be an E-differentiable function at u ∈ S. Then f is said to be a quasi E-convex function at u on S if the relations hold for all x ∈ S. If (16) and (17) are satisfied for each u ∈ S, then f is said to be a quasi E-convex function on S.
3. E-differentiable interval-valued multiobjective programming. In this paper, consider the following (not necessarily differentiable) interval-valued multiobjective programming problem with both inequality and equality constraints: where each objective function f k : R n → I (R), k ∈ K = {1, ..., p} is an intervalvalued function defined on R n , each function g j : R n → R, j ∈ J and each function h t : R n → R, t ∈ T , are real-valued functions defined on R n . 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 ) : R n → R m and h := (h 1 , ..., h s ) : R n → R s for convenience. Let be the set of all feasible solutions of (IVP).
For such interval-valued multicriterion optimization problems, Wu [30] proposed the following different concepts of (weak) Pareto optimal solutions in terms of a weak LU -Pareto (weakly LU -efficient) solution and a LU -Pareto (LU -efficient) solution in the following sense: Further, 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 intervalvalued multiobjective programming problem (IVP) are E-differentiable. Now, we prove the so-called E-Karush-Kuhn-Tucker necessary optimality conditions for the problem (IVP). Therefore, for the considered E-differentiable intervalvalued multiobjective programming problem (IVP), we construct the following associated vector optimization problem (IVP E ) with the multiple interval-valued objective function: We call the problem (IVP E ) an E-vector optimization problem with the multiple interval-valued objective function or an interval-valued E-vector optimization problem. Let be the set of all feasible solutions of (IVP E ). Now, we give the definitions of a weak Pareto optimal solution and a Pareto solution in terms of a weak LU -Pareto (weakly LU -efficient) solution and a LU -Pareto (LU -efficient) solution of the differentiable (in the usual sense) interval-valued Evector optimization problem (IVP E ) which are, at the same time, (weak) E-Pareto optimal solutions in terms of a weak LU -E-Pareto (weakly LU -E-efficient) solution and a LU -E-Pareto (LU -E-efficient) solution of the considered E-differentiable interval-valued multiobjective programming problem (IVP).
Before proving the optimality conditions for (weakly) LU -efficiency of the intervalvalued E-vector optimization problem (IVP E ) defined above and, thus, for (weakly) LU -E-efficiency of the considered (not necessarily differentiable) multiobjective programming problem (IVP) with the multiple interval-valued objective function, we establish some useful results which show the equivalency between these intervalvalued vector optimization problems.
Lemma 3.5. Let E : R n → R n be a one-to-one and onto operator and Proof. We now assume that x ∈ E (Ω E ). By assumption, E is a one-to-one and onto operator. Hence, by the definition of the set Ω E , there exists z ∈ Ω E such that x = E (z). By means of contradiction, suppose that x / ∈ Ω. Then there exists at least one j ∈ J such that g j (x) > 0 or at least one t ∈ T such that h t (x) = 0. Thus, by x = E (z), we have, for at least one j ∈ J, that (g j • E) (z) > 0 or, for at least one t ∈ T , that (h t • E) (z) = 0, which contradicting z ∈ Ω E . Thus, E (Ω E ) ⊂ Ω. On the other hand, let x ∈ Ω. We proceed by contradiction. Suppose that x / ∈ E (Ω E ). By assumption, this means that E −1 (x) / ∈ Ω E . By the definition of Ω E , it follows that there exists at least one j ∈ J such that (g j • E) (E −1 (x)) > 0 or at least one t ∈ T such that (h t • E) (E −1 (x)) = 0. Therefore, there exists at least one j ∈ J such that g j (x) > 0 or at least one t ∈ T such that h t (x) = 0, contradicting x ∈ Ω. Thus, Ω ⊂ E (Ω E ). Hence, by E (Ω E ) ⊂ Ω and Ω ⊂ E (Ω E ), we conclude that E (Ω E ) = Ω. Now, we prove the relationship between (weak) LU -Pareto optimal solutions in both interval-valued vector optimization problems (IVP) and (IVP E ).
Lemma 3.6. Let E : R n → R n be a one-to-one and onto operator and x ∈ Ω be a weak LU -Pareto solution (a LU -Pareto solution) of the considered multiobjective programming problem (IVP) with the multiple interval-valued objective function. Then, there exists z ∈ Ω E such that x = E (z) and z is a weak LU -Pareto solution (a LU -Pareto solution) of the E-vector optimization problem (IVP E ) with the multiple interval-valued objective function.
Proof. Let x ∈ Ω be a weak LU -Pareto solution for (IVP). Moreover, E : R n → R n is assumed to be a one-to-one and onto operator. Hence, by Lemma 3.5, there exists z ∈ Ω E such that x = E (z). Now, we prove that z is a weak LU -Pareto solution of the interval-valued E-vector optimization problem (IVP E ). By means of contradiction, suppose that z is not a weak LU -Pareto solution of the problem (IVP E ). Then, by the definition of a weak LU -Pareto solution, there exists Hence, by the definition of the relation < LU , it follows that, for any k ∈ K, ) . By Lemma 3.5, we have that there exists x ∈ Ω such that x = E ( z). Taking also that x = E (z), the above inequalities yield, respectively, . Hence, by the definition of the relation < LU , inequalities (18) imply that the inequality f ( x) < LU f (x) is fulfilled, which is a contradiction to weakly LU -efficiency of x for the problem (IVP). The proof for the case, in which x ∈ Ω is a LU -Pareto solution of the problem (IVP), is analogous and, therefore, it is omitted. Lemma 3.7. Let E : R n → R n be a one-to-one and onto operator and let z ∈ Ω E be a weak LU -Pareto solution (a LU -Pareto solution) of the interval-valued E-vector optimization problem (IVP E ). Then E (z) is a weak LU -Pareto solution (a LU -Pareto solution) of the considered multiobjective programming problem (IVP) with the multiple interval-valued objective function.
Proof. Assume now that z ∈ Ω E is a weak LU -Pareto solution of the interval-valued E-vector optimization problem (IVP E ). Note that, by Lemma 3.5, it follows that E (z) ∈ Ω. We proceed by contradiction. Suppose, contrary to the result, that E (z) is not a weak LU -Pareto solution of the considered interval-valued vector optimization problem (IVP). Then, by Definition 3.3, there exists x ∈ Ω such that f k ( x) < LU f k (E (z)) for each k ∈ K. Hence, by Lemma 3.5, there exists z ∈ Ω E such that x = E ( z). Thus, the inequality above implies that (f k • E) ( z) < LU (f k • E) (z) for each k ∈ K, which is a contradiction to weakly LU -efficiency of z for the interval-valued E-vector optimization problem (VP E ). The proof when it is assumed that z ∈ Ω E is a LU -Pareto solution of the problem (VP E ) is similar and, therefore, it is omitted. Remark 1. As it follows from Lemma 3.7, if z ∈ Ω E is a weak LU -Pareto solution (a LU -Pareto solution) of the E-vector optimization problem (IVP E ), then E (z) is a weak LU -Pareto solution (a LU -Pareto solution) of the considered multiobjective programming problem (IVP) with the multiple interval-valued objective function. Therefore, we call E (z) a weak LU -E-Pareto solution (a LU -E-Pareto solution) of the E-differentiable interval-valued multiobjective programming problem (IVP).
As it follows from the above lemmas, there is an equivalence between the intervalvalued vector optimization problems (IVP) and (IVP E ). This means that, if we prove optimality results for the differentiable interval-valued E-vector optimization problem (IVP E ), they will be applicable also for the original (not necessarily differentiable) multiobjective programming problem (IVP) with the multiple intervalvalued objective function in which the involved functions are E-differentiable.
In order to prove the E-Karush-Kuhn-Tucker necessary optimality conditions for a weak LU -E-Pareto solution of (IVP), we introduce the so-called E-Kuhn-Tucker constraint qualification for E-differentiable optimization problems with both inequality and equality constraints.
Definition 3.8. Let E : R n → R n and x ∈ Ω E be given. Further, assume that the constraint functions g = (g 1 , ..., g m ) and h = (h 1 , ..., h s ) are E-differentiable at x. It is said that the E-Kuhn-Tucker constraint qualification (E-CQ) is satisfied at x if, for any d ∈ R n , d = 0, such that ∇ (g j • E) (x) T d 0 for all j ∈ J (E (x)) := {j ∈ J : g j (E (x)) = 0}, and ∇ (h k • E) (x) T d = 0, t ∈ T , there exist a function ϕ : [0, 1] → R n , which is continuously differentiable at 0, and some real scalar β > 0 such that Before we establish the E-Karush-Kuhn-Tucker necessary optimality conditions for the E-differentiable interval-valued multiobjective programming problem (IVP), we re-call the Motzkin's theorem of the alternative. 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 a solution y 1 , y 2 , and y 3 .
In [30], Wu proved the Karush-Kuhn-Tucker necessary optimality conditions for a differentiable scalar optimization problem with the multiple interval-valued objective function under the Kuhn-Tucker constraint qualification. Now, we generalize this result to the case of an E-differentiable vector optimization problem with multiple interval-valued function and with both inequality and equality constraints. Theorem 3.10. (E-Karush-Kuhn-Tucker necessary optimality conditions). Let E : R n → R n be a given one-to-one and onto operator and let x ∈ Ω E be a weak LU -Pareto solution of the interval-valued E-vector optimization problem (IVP E ) (and, thus, E (x) be a weak LU -E-Pareto solution of the considered multicriteria optimization problem (IVP) with the multiple interval-valued objective function). Further, assume that the E-Kuhn-Tucker constraint qualification is satisfied at x. Then there exist Lagrange multipliers λ Proof. Let x ∈ Ω E be a weak LU -Pareto solution of the E-vector optimization problem (IVP E ) with the multiple interval-valued objective function. Hence, by Lemma 3.7, E(x) is a weak LU -E-Pareto solution of the considered multicriteria optimization problem (IVP) with the multiple interval-valued objective function. Further, assume that the E-Kuhn-Tucker constraint qualification is satisfied at x. Now, we prove that there does not exist a vector d ∈ R n , d = 0, satisfying the following system of inequalities: Suppose, contrary to the result, that there exists any d ∈ R n , d = 0, satisfying (23), (24) and (25). By assumption, the E-Kuhn-Tucker constraint qualification is satisfied at x. Hence, there exist a function ϕ : [0, 1] → R n which is continuously differentiable at 0, and some real scalar β > 0 such that (18) and where θ L k (x, ϕ (α) − x) → 0 and θ U k (x, ϕ (α) − x) → 0 as ϕ (α) − ϕ (0) → 0. Thus, we re-write (26) and (27) as follows By the E-Kuhn-Tucker constraint qualification, it follows that ϕ is a differentiable function at 0. Hence, we have Also it follows from the E-Kuhn-Tucker constraint qualification, for any d ∈ R n , d = 0, there exists a real scalar β > 0 such that Combining (28)-(31) and ϕ (α) − ϕ (0) → 0 as α → 0, we get, for sufficiently small α, respectively, By assumption, (23) is satisfied for d ∈ R n , d = 0. Therefore, by α > 0 and β > 0, (32) and (33) yield, respectively, that the inequalities , k ∈ K hold for sufficiently small α. This is a contradiction to the assumption that x ∈ Ω E is a weak LU -Pareto solution of (IVP E ). This means that there does not exist any d ∈ R n satisfying the system of inequalities (23)- (25). Hence, by Motzkin's theorem of the alternative (see Theorem 3.9), there exist Lagrange multipliers λ L ∈ R p , λ U ∈ R p , µ j , j ∈ J (E (x)), and ξ ∈ R s such that If we set µ j = 0 for all j ∈ J\J (E (x)), then (34) implies (20). Further, note that also the complementary slackness condition (21) is satisfied. Indeed, if g j (E(x)) < 0, then µ j = 0 for j ∈ J\J (E (x)).

Remark 2.
It is assumed in Theorem 3.10 that x ∈ Ω E is a (weak) LU -Pareto solution of the differentiable interval-valued vector optimization problem (IVP E ). Then, the fact that E (x) is a (weak) LU -E-Pareto solution of the considered Edifferentiable multicriteria optimization problem (IVP) with the multiple intervalvalued objective function follows directly from Lemma 3.7. Hence, the Karush-Kuhn-Tucker necessary optimality conditions established in Theorem 3.10 for the problem (IVP E ) are also applicable for the problem (IVP). In such a case, we call them the E-Karush-Kuhn-Tucker necessary optimality conditions for the considered E-differentiable multiobjective programming problem (IVP) with the multiple interval-valued objective function.
Definition 3.11. x, λ, µ, ξ ∈ Ω E × R 2p × R m × R s is said to be a Karush-Kuhn-Tucker point for the E-vector optimization problem (IVP E ) with the multiple interval-valued objective if the Karush-Kuhn-Tucker necessary optimality conditions (20)- (22) are satisfied at x with Lagrange multipliers λ, µ, and ξ.
In order to illustrate the above result, we present an example of such a nondifferentiable vector optimization problem with the multiple interval-valued objective function for which the E-Kuhn-Tucker constraint qualification is not satisfied. Note that in such a case Lagrange multipliers λ L and λ U corresponding to the objective functions f L and f U can be equal to 0.
Example 3.12. Consider the following nondifferentiable vector optimization problem with the multiple interval-valued objective function: Note that the set of all feasible solutions in the considered vector optimization problem (IVP1) with the multiple interval-valued objective function is Ω 0 . Further, note that some of the functions constituting (IVP1) are nondifferentiable at feasible solutions (x 1 , 0), where x 1 0. Let E : R 2 → R 2 be defined as follows: E (x 1 , x 2 ) = 1 2 x 1 , x 3 2 . Then, it can be shown by Definition 2.3 that the objective function f is E-differentiable at x = (0, 1) and, by Definition 2.2, the constraints g 1 and g 2 are E-differentiable at x = (0, 1). Indeed, we have that the functions f L Also it can be established by Definition 3.2 that x = (0, 1) ∈ Ω is a LU -E-Pareto solution of (IVP1). However, note that the E-Karush-Kuhn-Tucker necessary optimality conditions are not satisfied at x = (0, 1).
Theorem 3.13. Let E : R n → R n be a given one-to-one and onto operator and x, λ, µ, ξ ∈ Ω E × R p × R m × R s be a Karush-Kuhn-Tucker point of the E-vector optimization problem (IVP E ) with the multiple interval-valued objective function, where λ = λ L , λ U > 0. Let T + (E (x)) = t ∈ T : ξ t > 0 and T − (E (x)) = t ∈ T : ξ t < 0 . Furthermore, assume that the following hypotheses are fulfilled: a) the objective function f is an E-differentiable E-convex interval-valued function at x on Ω E , b) each inequality constraint function Then x is a LU -Pareto solution of the problem (IVP E ) and, thus, E (x) is a LU -E-Pareto solution of the problem (IVP).
(48) Adding both sides of the above inequalities, by (42), we obtain that the inequality holds, which is a contradiction to the E-Karush-Kuhn-Tucker necessary optimality condition (20). Hence, x is a LU -Pareto solution of (IVP E ). Thus, by Lemma 3.7, it follows directly that E (x) is a LU -E-Pareto solution of the considered multicriteria optimization problem (IVP) with the multiple interval-valued objective function. Then, the proof of this theorem is completed.

Remark 3.
As it follows from the proof of Theorem 3.13, 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-convex functions at x on Ω E .
Theorem 3.14. Let E : R n → R n be a given one-to-one and onto operator and x, λ, µ, ξ ∈ Ω E × R 2p × R m × R s be a Karush-Kuhn-Tucker point of the intervalvalued E-vector optimization problem (IVP E ). Further, assume that the following hypotheses are fulfilled: a) the objective function f is an E-differentiable strictly E-convex interval-valued function at x on Ω E , b) each inequality constraint function g j , j ∈ J (E (x)), is an E-differentiable E-convex function at x on Ω E , c) each equality constraint function h t , t ∈ T + (E (x)), is an E-differentiable E-convex function at x on Ω E , d) each function −h t , t ∈ T − (E (x)), is an E-differentiable E-convex function at x on Ω E . Then x is a weak LU -Pareto solution of the problem (IVP E ) and, thus, E (x) is a weak LU -E-Pareto solution of the considered interval-valued multicriteria optimization problem (IVP).
Now, under the concepts of generalized E-convexity, we prove the sufficient optimality conditions for a feasible solution to be a weak LU -E-Pareto solution of the problem (IVP).
Theorem 3.15. Let E : R n → R n be a given one-to-one and onto operator and x, λ, µ, ξ ∈ Ω E × R 2p × R m × R s be a Karush-Kuhn-Tucker point of the E-vector optimization problem (IVP E ) with the multiple interval-valued objective function. Let T + (x) = t ∈ T : ξ t > 0 and T − (x) = t ∈ T : ξ t < 0 . Furthermore, assume the following hypotheses: a) the objective function f is an E-differentiable strictly pseudo E-convex intervalvalued function at x on Ω E , b) each inequality constraint function g j , j ∈ J (E (x)), is E-differentiable quasi E-convex at x on Ω E , c) each equality constraint function h t , t ∈ T + (E (x)), is E-differentiable quasi E-convex at x on Ω E ,