Generalized composed radial epiderivatives

In this paper, the generalized composed radial epiderivati ve for set-valued maps is introduced and some of its properti es are investigated. Existence conditions for the generalized co mposed radial epiderivative are established.


Introduction
In the last thirty years the notion of derivatives or epiderivatives has been formulated in different ways. (see [1,2,7,8,10,13,19,23,24,25,28,31]). Aubin first introduced the notion of the contingent derivative for set-valued map by using the contingent cone [1]. Corley established the sufficient and necessary optimality conditions for set-valued optimization problems by virtue of the concept of contingent and circatangent derivative [13]. The contingent derivative play an important role for set-valued optimization problems. But, necesary and sufficient optimality conditions do not coincide unified under standart assumptions. To overcome the difficulty, another of differentiability concept which is based on using epigraphs of set-valued maps was proposed Jahn and Rauh [23].
Kasımbeyli introduced in [24] the notion of the radial epiderivative of a nonconvex set-valued map. This definition of the radial epiderivative given by Kasımbeyli is different from that of Flores-Bazan [7] and is similar to the definition of the contingent epiderivative given by Jahn and Rauh [23]. He derived the formulation of optimality conditions in the single valued and set-valued optimization without convexity assumption and investigated relationships between this kind of epiderivative and weak subdifferentials and directional derivatives for real-valued nonconvex functions.
Kasımbeyli and Inceoglu introduced in [25] the notion of generalized radial epiderivative for set-valued maps and investigated existence conditions for generalized radial epiderivative. They established the relationship between the radial epiderivative and the generalized radial epiderivative. By using the generalized radial epiderivative, Kasımbeyli and Inceoglu presented the necessary and sufficient optimality conditions for set-valued optimization.
Recently, there has been an increasing interest in second-order and higher-order optimality research for set-valued map [3,4,5,6,9,11,12,14,15,16,21,22,27,20,26,29,30,32]. Jahn et al. proposed the second-order epiderivatives in terms of the second-order contingent set [21], introduced by Aubin and Frankowska [2]. They obtained the second-order optimality conditions by using these dervatives in set-valued optimization.It can be seen that a second-order contingent set, introduced by Aubin and Frankowska [2], and a second-order asymptotic contingent cone, introduced by Penot [30], play a important role in establishing second-order optimality conditions. Li et al. proposed a generalized second-order composed contingent epiderivative for a set-valued map and investigated some of its properties. By virtue of the generalized second-order composed contingent epiderivative, they also establised a unified second-order sufficient and c 2018 BISKA Bilisim Technology necessary optimality conditions for set-valued optimization [29]. Isac and Khan employed the second-orer optimality conditions in set-valued optimization problems, using which they introduced new kind of second-order tangent epiderivative [16].
Anh and Khanh introduced the higher-order radial sets and corresponding derivatives. They proposed their properties and basic calculus rules. They established both necessary and sufficient higher-order conditions for weak efficiency in set-valued vector optimization problem [4]. Anh and Khanh gived both necessary and sufficient higher-order conditions for various kinds of proper solutions to nonsmooth vector optimization problem in terms of higher-order radial sets and radial derivatives [5]. In [25], Inceoglu introduce the concepts of second-order radial epiderivative and second-order generalized radial epiderivative for nonconvex set-valued maps. They also investigate in [25] some of their properties and give existence theorems for the second-order generalized radial epiderivatives. In [25], Inceoglu propose second-order optimality conditions by using second-order radial epiderivatives.It is wort noting that higher-order radial derivative or radial epiderivative by a higher-order radial set, in general, is not a cone and a convex set. Therefore, there are some difficulties in studying higher-order optimality conditions for set-valued optimization problems.
Motivated by this problem, we intend to give a new generalized second-order composed radial epiderivatives for set-valued maps and investigate some of its properties. By using this concept, we give a unified second-order sufficient and necessary optimality conditions for set-valued optimization problems, which is a generalization of the corresponding in [25]. This paper is divided into two sections. In Section 2 ,we give the second-order radial epiderivatives and prove the existence conditions of one of them.

Preliminaries
Throughout this paper, Let (X, . X ) and (Y, . Y ) real normed spaces and S be a nonempty subset of X. Let C ⊂ Y a pointed, closed and convex cone with apex at the origin and a nonempty interior intC, and let Y be partially ordered by C. Let F : S ⇒ Y be a set-valued map. Let a pair (x,ȳ) ∈ graph (F) be given. In this section, we recall the concept of the radial epiderivative and the generalized radial epiderivative introduced by Kasımbeyli [24], and Kasımbeyli and Inceoglu [25], respectively, together with some standart notions. Definition 1. Let U be a nonempty subset of a real normed space (Z, . Z ) , and letz ∈ cl (U) (closure of U) be a given element. The closed radial cone R (U,z) of U atz ∈ cl (U) is the set of all z ∈ Z such that there are λ n > 0 and a sequence (z n ) n∈N ⊂ Z with lim n→∞ z n = z so thatz + λ n z n ∈ U, for all n ∈ N.
Note that the closed radial cone can equivalently be also defined as the following definition.

Definition 2. Let U be a nonempty subset of a real normed space
It follows from these definitions that where cone denotes the conic hull of a set, which is the smallest cone containing U −z.
is called the domain of F. F is said to be proper if dom (F) = / 0; (iii) Let Y be partially ordered by a proper, convex, and pointed cone C ⊂ Y. The set Now we give the definition of the radial epiderivative given by Kasımbeyli without convexity and boundedness [24].
is called radial epiderivative of F at (x,ȳ) .

Definition 5.
Let F : X ⇒ Y be a set valued map and let (x,ȳ) ∈ graph (F) . A set valued map D R F (x,ȳ) : X ⇒ Y whose graph coincides with the contingent cone to graph of F at (x,ȳ) , that is is called radial derivative of F at (x,ȳ) [7,31].
To give the definition of the generalized radial epiderivative, we recall the minimality concept.

Definition 7. A set valued map D gr F
where MinD is the set of all minimal elements of D [25,Definition 8].

Generalized composed radial epiderivatives
In this section we introduce a generalized composed radial epiderivative for set-valued maps, and give some of properties of its and existence theorems. The following concept extends a characterization of Definition 7 given by Kasımbeyli and Inceoglu [25]. , (x,ȳ)) , (ū,v))} .

Definition 8. Let F : S ⇒ Y be a set-valued map and (x,ȳ) ∈ graphF, and (ū,v) ∈ X × Y. The generalized second-order composed radial epiderivative of F at (x,ȳ) in the direction (ū,v) is the set-valued map D
(2)

Lemma 1.
Let F : S ⇒ Y be a set-valued map and (x,ȳ) ∈ graphF, and (ū,v) ∈ X × Y . Then we have the following statement: Proof. We need to prove that Since (x,ŷ) ∈ R (R(epiF, (x,ȳ)), (ū,v)), by the definiton of radial cone, by the definition of radial cone, there are sequence (x n , y n ) → (x,ŷ) and t n > 0 such that (ū,v) + t n (x n , y n ) ∈ R (epiF, (x,ȳ)), ∀n ∈ N. Moreover, ∀n ∈ N, there exist sequences x k n , y k n → (ū,v) + t n (x n , y n ) and t k n > o such that (x,ȳ) + t k n x k n , y k n ∈ epiF, ∀k ∈ N. Then we havē y + t k n y k n ∈ F x + t k n x k n + C, ∀n, k ∈ N.
Since C is a cone and c ∈ C, together with (3), we havē y + t k n y k n + t n c =ȳ + t k n y k n + t k n t n c ∈ F x + t k n x k n + C, ∀n, k ∈ N, that is, (x,ȳ) + t k n x k n , y k n + t n c ∈ epiF ∀n, k ∈ N. Since x k n , y k n → (ū,v) + t n (x n , y n ) as k → ∞. Thus (ū,v) + t n (x n , y n + c) ∈ R (epiF, (x,ȳ)), ∀n ∈ N. Simultaneously, (x n , y n + c) → (x,ŷ + c) since (x n , y n ) → (x,ŷ) as n → ∞, that is, (x, y) = (x,ŷ + c) ∈ R (R(epiF, (x,ȳ)), (ū,v)). This completes the proof. Now, we define a strictly positive homogeneous and subadditive map and then show under appropriate assumption that the generalized second-order composed radial epiderivative is strictly positive homogeneous and subadditive [10,25].
If the properties a) holds with α ≥ 0 and b) holds, F is called sublinear [10].
Proof. Firstly, we prove the strictly positive homogenity. For every λ > 0 ve x ∈ X, we have For every x 1 , x 2 ∈ X and y 1 ∈ D 2 , (x,ȳ)) , (ū,v)) and (x 2 , y 2 ) ∈ R (R (epiF, (x,ȳ)) , (ū,v)) . Since R (R (epiF, (x,ȳ)) , (ū,v)) is a closed and convex cone, we have It follows from the domination property and definition of the generalized second-order composed radial epiderivative that Therefore, we have Definition 10. Let Y be a partially ordered by a pointed, closed, and convex cone C ⊂ Y wiht apex at origin and a nonempty interior int (C) .
(a) The cone C is Daniell, if any decreasing sequence in Y having a lower bound, converges to its infimum.
The domination property is said to be hold for a sunset D of Y if D ⊂ min (D,C) + C [28]. Now we need to establish an existence theorem for the generalized second-order radial epiderivative.
Proof. Since the radial cone is always closed in a normed space, then for every x ∈ S the setĜ (x) is and closed and minorized. From the existence theorems of minimal elements (see proposition 3.16. page in Luc) minĜ (x) is nonempty, i.e. D 2 grc F (x,ȳ,ū,v) is well defined.  , (x,ȳ)) , (ū,v)) .

Conclusion
In this article, we introduce a new second-order radial epiderivative by taking radial epiderivative of a radial epiderivative. We obtain the some properties of these second-order radial epideivative.