Second-order lower radial tangent derivatives and applications to set-valued optimization

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.


Introduction
In recent years, first-order optimality conditions in the set-valued optimization have attracted a great deal of attention, and various derivative-like notions have been utilized to express these optimality conditions. For example, Gong et al. [] introduced the concept of radial tangent cone and presented several kinds of necessary and sufficient conditions for some proper efficiencies, Kasimbeyli [] gave necessary and sufficient optimality conditions based on the concept of the radial epiderivatives. At the same time, second-order optimality conditions and higher-order optimality conditions for vector optimization problems have been extensively studied in the literature (see [-]). Jahn et al. [] proposed second-order epiderivatives for set-valued maps, and by using these concepts they gave second-order necessary optimality conditions and a sufficient optimality condition in set optimization. Khan and Isac [] proposed the concept of a second-order composed contingent derivative for set-valued maps, using which they established second-order optimality conditions in set-valued optimization. With a second-order composed contingent derivative, Zhu et al. [] established second-order Karush-Kuhn-Tucker necessary and sufficient optimality conditions for a set-valued optimization problem. However, in [, , -, ], in the expressions of first-order and higher-order optimality conditions, the tan-gent derivatives of the objective function and the constraint function are not separated, and thus the properties of the derivatives of the objective function are not easily obtained from those of the constraint function.
On the other hand, some efficient points exhibit certain abnormal properties. To eliminate such anomalous efficient points, various concepts of proper efficiency have been introduced [-]. Henig [] introduced the concept of Henig efficiency, which is very important for the study of set-valued optimization [, , , ].
In this paper, we introduce a new class of lower radial tangent cones and two new kinds of second-order tangent sets, using which we introduce four new kinds of second-order tangent derivatives. We discuss the properties of these second-order tangent derivatives, using which we establish second-order necessary optimality conditions for a point pair to be a Henig efficient element of a set-valued optimization problem.

Basic concepts
Throughout the paper, let X, Y , and Z be three real normed linear spaces,  X ,  Y , and  Z denote the original points of X, Y , and Z, respectively. Let M be a nonempty subset of Y . As usual, we denote the interior, closure, and cone hull of M by int M, cl M, and cone M, respectively. The cone hull of M is defined by Let C and D be closed convex pointed cones in Y and Z, respectively. A nonempty convex subset B ⊂ C is called a base of C if  / ∈ cl B and C = cone B.
Denote the closed unit ball of Y by U. Suppose that C has a base B. Let δ : for all  < ε < δ. It is clear that δ >  and C ε (B) is a pointed convex cone for all  < ε < δ (see []).
Let F : X →  Y be a set-valued map. The domain, graph, and epigraph of F are defined respectively by

Definition . (See []) Let
A be a nonempty subset of X, and letx ∈ cl A. The radial tangent cone of A atx, denoted by R(A,x), is given by Remark . Equation (.) is equivalent to R(A,x) = {u ∈ X : ∃λ n >  and u n → u such thatx + λ n u n ∈ A, ∀n ∈ N}, where N denotes the set of positive integers.

Definition . (See []) Let
A be a nonempty subset of X, and letx ∈ cl A. The contingent cone of A atx, denoted by T(A,x), is given by T(A,x) := u ∈ X : ∃t n →  + and u n → u such thatx + t n u n ∈ A, ∀n ∈ N . (.) T(A,x) := u ∈ X : ∃λ n → +∞ and x n ∈ A such that x n →x and λ n (x n -x) → u .

Definition . (See []) Let
A be a nonempty subset of X, and letx ∈ cl A. The secondorder contingent set of A atx in the direction w, denoted by T  (A,x, w), is given by In the following, we introduce a new class of lower radial tangent cones and two new kinds of second-order tangent sets.
Definition . Let Q be a nonempty subset of X × Y , and let (x,ŷ) ∈ cl Q. The lower radial tangent cone of Q at (x,ŷ) is defined by Definition . Let Q be a nonempty subset of X × Y , and let (x,ŷ) ∈ cl Q. The secondorder lower radial tangent set of Q at (x,ŷ) in the direction (û,v), denoted by R  l (Q, (x,ŷ), (û,v)), is given by Definition . Let A be a nonempty subset of X, and letx ∈ cl A. The second-order radial tangent set of A atx in the direction w, denoted by R  (A,x, w), is given by ). However, none of the inverse inclusions is necessarily true, as is shown in the following example.

The second-order lower radial tangent derivative
In this section, by virtue of the radial tangent cone, the second-order radial tangent set, the lower radial tangent cone, and the second-order lower radial tangent set, we introduce the concepts of the second-order radial composed tangent derivative, the second-order radial tangent derivative, the second-order lower radial composed tangent derivative, and the second-order lower radial tangent derivative for a set-valued map. Furthermore, we discuss some important properties of the second-order lower radial composed tangent derivative and the second-order lower radial tangent derivative.
If R(R(epi F, (x,ŷ)), (û,v)) = ∅, then F is said to be second-order radial composed derivable at (x,ŷ) in the direction (û,v) or that the second-order radial composed tangent derivative of F at (x,ŷ) in the direction (û,v) exists.
If R l (R l (epi F, (x,ŷ)), (û,v)) = ∅, then F is said to be second-order lower radial composed derivable at (x,ŷ) in the direction (û,v) or that the second-order lower radial composed tangent derivative of F at (x,ŷ) in the direction (û,v) exists.
then F is called second-order lower radial derivable at (x,ŷ) in the direction (û,v) or that the second-order lower radial tangent derivative of F at (x,ŷ) in the direction (û,v) exists.

Proposition . Suppose that E ⊂ X and the second-order lower radial composed tangent
Thus, ,x),û) it follows that there exist sequences t n >  and u n → u such that Therefore, there exist sequences t k n >  and u k n →û + t n u n such that For such t n and u n , it follows from (.) that there exists a sequence v n → v such that (û + t n u n ,v + t n v n ) ∈ R l epi F, (x,ŷ) .
Then, for the same t k n and u k n , there exists a sequence v k n →v + t n v n such that x + t k n u k n ,ŷ + t k n v k n ∈ epi F, and, consequently, and, consequently, Since v k n →v + t n v n as k → ∞, we obtain and, consequently, v n ∈ cone clcone F(E) + C -ŷ -v .
Taking n → ∞, we get v ∈ clcone clcone F(E) + C -ŷ -v . So, Proposition . Suppose that E ⊂ X and the second-order lower radial tangent derivative Thus, From u ∈ R  (E,x,û) it follows that there exist sequences t n >  and u n → u such that For such t n and u n , it follows from (.) that there exists a sequence v n → v such that is necessarily true, as is shown in the following example.  Then, the inclusion of Proposition . is true. However, Then, the inclusions of Propositions . and . are true. However,

Second-order necessary optimality conditions
Consider the following optimization problem with set-valued maps: The feasible set of (VP) is denoted byÊ, that is, , and U is the closed unit ball of Y .

Definition . (See [])
The interior tangent cone IT(S,ȳ) of S atȳ is the set of all y ∈ Y such that for any t n →  + and y n → y, we haveȳ + t n y n ∈ S.
Since -int D is a cone, we obtain Sinceŵ ∈ -D and -D is a convex cone, it follows that Since (u n , w n ) ∈ R(epi G, (x,ẑ)), there exist sequences t k n >  and (x k n , z k n ) ∈ epi G such that It follows from (.) that there exists K  (n) ∈ N such that t k n z k n -ẑ ∈ -int D, ∀n > N  , ∀k > K  (n).
Since -int D is a cone, we obtain z k n -ẑ ∈ -int D, ∀n > N  , ∀k > K  (n).
Sinceẑ ∈ -D and -D is a convex cone, it follows that Since (x k n , z k n ) ∈ epi G, we obtain z k n ∈ G(x k n ) + D. Hence, there existsz k n ∈ G(x k n ) such that z k n ∈z k n + D. Consequently, Thus, G(x k n ) ∩ (-D) = ∅, that is, x k n ∈Ê. It follows from (.) that t k n (x k n -x) → u n as k → ∞, and hence, u n ∈ R(Ê,x). It follows from (.) that t n (u n -û) →x, and hence, x ∈ R(R(Ê,x),û). By Proposition ., sinceȳ ∈ R l F(x,ŷ,û,v)(x), we conclude that Since cone(ε  U + B) is a pointed cone, it follows that and thus, Consequently, In the similar way, we conclude that Since C ⊂ cone(ε  U + B) and cone(ε  U + B) is a point cone, we obtain This is a contradiction to (.). The proof is completed.
Define the set-valued maps F : X →  Y and G : X →  Z by Then, the inclusions of Theorem . and Corollaries . and . are true.
Since C ⊂ cone(εU + B) and cone(εU + B) is a pointed cone, we obtain This is a contradiction to the assumption that (x,ŷ) is a Henig minimizer of (VP).
Proof The proof follows immediately from Theorem . and Remark .(ii). and R  l F(x,ŷ,û,v) R  (E,x,û) ⊂ clcone cone F(E) + C -ŷ -v , which are demonstrated in Propositions . and .. Just applying these properties, we established second-order necessary optimality conditions for a point pair to be a Henig efficient element of a set-valued optimization problem where the second-order tangent derivatives of the objective function and constraint function are separated.