OPTIMALITY CONDITIONS OF SECOND-ORDER RADIAL EPIDERIVATIVES

In this paper, we introduce the concepts of second-order radial epiderivative and second-order generalized radial epiderivative for nonconvex set-valued maps. We give existence theorems for the second-order generalized radial epiderivatives. We also establish the second-order optimality conditions by using second-order radial epiderivatives.


Introduction
In the last years, the second-order optimality conditions have a great deal of attention in scalar and vector-optimization problems and have been widely investigated [2,3,4,5,8,9,10,11,12,13,14,15,16,17,19,22,24,26].It can be seen that a second-order contingent set, introduced by Aubin and Frankowska [1], and a second-order asymptotic contingent cone, introduced by Penot [24], play a important role in establishing second-order optimality conditions.Jahn et al. proposed the second-order contingent derivative and the second-order contingent epiderivative in terms of the second-order contingent set [15], introduced by Aubin and Frankowska [1].They obtained the second-order optimality conditions by using these derivatives in set-valued optimization.In [22], Khan and Tammer gave new second-order optimality conditions in set-valued optimization.They presented an extension of the well-known Dubovitski-Milutin approach to set-valued optimization.In [3], Anh and Khanh introduced the higher-order radial sets and corresponding derivatives.They established both necessary and sufficient higher-order conditions for weak efficiency in set-valued vector optimization problem .In [4], Anh and Khanh gave 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.In [18], İnceoglu introduce the concepts of second-order radial epiderivative and second-order generalized radial epiderivative for nonconvex set-valued maps.They also investigate in [18] some of their properties and give existence theorems for the second-order generalized radial epiderivatives.
Motivated by the work above, we study the second-order radial epiderivatives and the secondorder generalized radial epiderivative.We also propose second-order optimality conditions by using second-order radial epiderivatives.This paper is divided into four sections.In Section 2, we recall some basic concepts.In Section 3, we introduce the second-order radial epiderivative and the second-order generalized radial epiderivative and give the existence theorems and some of their basic properties.In Section 4, we establish the second-optimality conditions for weak minimizers.

Preliminaries
Throughout this paper, let (X, .X ) and (Y, .Y ) be real normed spaces and let Y be partially ordered by a closed convex pointed cone C ⊂ Y .Let F : X → 2 Y be a set-valued map, let We recall the concept of the radial epiderivative and the generalized radial epiderivative introduced by Kasımbeyli [20], and Kasımbeyli and İnceoglu [21], respectively, together with some standard notions.Definition 2.1.Let U be a nonempty subset of a real normed space (Z, .Z ) , and let z ∈ cl (U) (closure of U) be a given element.The closed radial cone R (U, z) of U at z ∈ 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 that z + λ n z n ∈ U, for all n ∈ N [6], [20,21], [25].
(i) The set is called the domain of F; (iii) Let Y be partially ordered by a proper, convex, and pointed cone C ⊂ Y.The set is called the epigraph of F, (iv) Let C ⊂ Y a proper, convex and pointed cone.The profile map P F : X → 2 Y is defined by with the contingent cone to graph of F at ( x, ȳ) , that is is called radial derivative of F at ( x, ȳ) , [6], [25].
Now, we give the definition of the radial epiderivative given by Kasımbeyli without convexity and boundedness [20].
Definition 2.3.Let Y be partially ordered by a convex cone C ⊂ Y , let S be a nonempty subset of X and let F : S → 2 Y be a set-valued map.Let a pair ( x, ȳ) ∈ graph (F) be given.A singlevalued map D r F ( x, ȳ) : X → Y whose epigraph equals the radial cone to the epigraph of F at To give the definition of the generalized radial epiderivative, we recall the minimality concept Now, we recall the generalized radial epiderivative for set-valued maps given by Kasımbeyli and İnceoglu in [21].
where G : X → 2 Y is the set-valued map given by 3.Second-Order Radial Set and Second-Order Radial Epiderivatives In this section, we propose the definitions of the second-order radial epiderivatives.By using these definitions, we prove existence theorem and give some of their properties and optimality conditions.
Anh and Khanh defined m-th-order radial set and m-th-order radial derivative [4].Based on this, we give the following definitions of second-order radial set and second-order radial derivative.
Definition 3.1.Let (X, .X ) be a real normed space, let S be a nonempty subset of X, let x ∈ cl (S) and let w ∈ X The second-order radial set of S at x with respect to w is It is also clear that R 2 (S, x, 0 X ) = R (S, x), 0 X the zero element of X.
The following definition was presented by Ha in [13].
The second-order radial derivative of F at ( x, ȳ) with respect to ( ū, v) is the set-valued map The relation (1) can be expressed equivalently by The following definition is a generalization given by Kasımbeyli and Kasımbeyli and İnceoglu, respectively [20], [21].
(i) A single-valued map D 2 r F ( x, ȳ, ū, v) : X → Y whose epigraph equals the second-order radial set to the epigraph of F at ( x, ȳ) with respect to ( ū, v), i.e., is called the second-order radial epiderivative.
(ii) A set-valued map D 2 gr F ( x, ȳ, ū, v) : X → 2 Y is called the second-order generalized radial epiderivative of F at ( x, ȳ) with respect to ( ū, v) if where G 2 : X → 2 Y is a set-valued map defined by Example 3.1.Let F : R → 2 R be a set-valued map given by The condition is equivalent to hence, Consequently, we have for every x ∈ R. On the other hand, for every x ∈ R.
Theorem3.1.Let the convex cone C ⊂ Y be regular.For every x ∈ dom G 2 , let the set D 2 gr F ( x, ȳ, ū, v) (x) have a C-lower bound.Then for every x ∈ dom G 2 , D 2 gr F ( x, ȳ, ū, v) (x) exists.Moreover, the following equality holds: Propositon3.2.Let the convex cone C ⊂ Y be regular.Let F : X → 2 Y be a set-valued map, let ( x, ȳ) ∈ graph (F), let ( ū, v) ∈ X ×Y.For every x ∈ dom G 2 (x) , let the set G 2 (x) have a C− lower bound.The following assertion is satisfied: )) It follows from the definition of the second-order generalized radial epiderivative that there exist sequences Therefore we have x + t n ū + t 2 n x n ∈ dom (F).This implies that (x, y) ∈ R 2 (dom (F) , x, ū) ×Y.

4.Optimality Conditions
Now, we obtain the optimality conditions for set-valued maps in terms of second-order radial epiderivativatives.Let F : S → 2 Y be a set-valued map.
Consider the following set-valued optimization problem: where Here we present a second-order optimality condition by using the second-order radial derivative.
Theorem 4.1.Let ( x, ȳ) ∈ graph (F) be a weak minimizer of the problem (P) and let ū ∈ dom (DP F ( x, ȳ)) be arbitrary.Then, for every v ∈ D R P F ( x, ȳ) ( ū) ∩ (−∂C), Proof.Let ( x, ȳ) ∈ graph (F) and let ȳ ∈ W − Min (F (S) ,C).Assume to the contrary that there exist an element x ∈ dom D 2 R P F ( x, ȳ, ū, v) with By the definition of the second-order radial epiderivative Then ∃t n > 0, ∃ (x n , y n ) ⊂ epi (F) , with By the definition of epi (F), we get From t n > 0, we get By using the above equality (2), we have We set Because of the equalities ( 5) and ( 2), we have Therefore, there exists some ϑ n ∈ F (α n ) +C with β n ∈ ϑ n +C.From here Because of the inclusion int (C) +C ⊂ int (C) and the equality Therefore, we have shown that which is a contradiction to the assumption that ( x, ȳ) is a weak minimizer.Now we propose some important properties of the second-order radial epiderivative.
The following sufficient optimality condition for the weak minimizer will be proved by using the Lemma 4.1.

Conclusion
In this paper two new concept of second-order epiderivative are presented.The relationship between the second-order radial epiderivative and the second-order generalized radial epiderivative are discussed.Some of their properties are investigated also.In set-valued optimization, second-order optimality conditions are obtained by using these epiderivatives.

[23]. Definition 2 . 4 .
Let (Y, .Y ) be a real normed space partially ordered by a convex cone C ⊂ Y. Let D be a subset of Y and let ȳ ∈ D.(i) The element ȳ is said to be a minimal element of D, if D ∩ ({ ȳ} −C) = { ȳ}.(ii) Let the ordering cone have a nonempty interior int (C).The element ȳ is said to be a weakly minimal element of D, if D ∩ ({ ȳ} − int (C)) = / 0 .The set of all minimal, weakly minimal elements of D with respect to the ordering cone C is denoted by MinD, W − MinD, respectively.

Definition 4 . 1 .
Let the ordering cone C have a nonempty interior int (C).A pair ( x, ȳ) ∈ graph (F) is called weak minimizer of (), if ȳ is a weakly minimal element of the set F (S)