On the Existence of Super Efficient Solutions and Optimality Conditions for Set-Valued Vector Optimization Problems

In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality terms for a general type of super efficiency.


Introduction
During the past more than 20 years, extending and characterizing definitions/properties of generalized convexity from the real-valued to the multi-valued mappings had been investigated by many scholars; the readers are referred to Benoist and Popovici [1,2], Jabarootian and Zafarani [3], Oveisiha and Zafarani [4], Sach and Yen [5], Yang [6] and the references cited therein. In particular, Sach and Yen [5] provided certain necessity and sufficiency terms for a multi-valued F to be K-convex by using a contingent derivative of the epigraphical multifunction of F w.r.t. an ordering cone K. Subsequently, Yang [6] introduced Dini direction derivative for multifunctions and it was used to derive certain properties of K-convexity mappings.
Set-valued vector optimization problems received great attention from many authors, who considered and studied them via various kinds of methods and approaches; see, for example, Chen and Jahn [7], Chinaie and Zafarani [8,9], Chinchuluun and Pardalos [10], Durea and Strugariu [11], Floudas and Pardalos [12], Ha [13], Mohan and Neogy [14] and Mordukhovich [15,16]. Super efficiency was first put forth by Borwein and Zhuang [17] in linear normed space and then explored in a few papers: Bao and Mordukhovich [18], Huang [19], Rong and Wu [20], Zaffaroni [21], Zheng et al. [22]. Such a concept extracts and puries the notion of efficiency and other types of proper efficiency. So, Rong and Wu [20] gave certain characterizations of super efficiency by virtue of super duality with cone-convexlike conditions, Lagrange multipliers and scalarizing procedure. Zaffaroni [21] applied various scalarizing functions to present some characterizations of super minimizers and other different solutions of vector optimization issues. Bao and Mordukhovich [18] derived some necessity terms of super efficiency in constrained issues of multiobjective optimization by advanced techniques of variational analysis; also, see [23,24]. In 2013, Oveisiha and Zafarani [25] obtained a characterization of K-preinvexity mappings by virtue of the normal subdifferential notion and marginal functions. Meanwhile, two sufficiency conditions for which super minimal points exist were deduced using the K-preinvexity hypothesis. Certain necessity optimality conditions for a general type of super efficiency were also obtained. For the concepts of generalized invexity and invariant monotonicity w.r.t. a function, the reader is referred to: Jabarootian and Zafarani [26], Soleimani and Damaneh [27], Weir and Mond [28], Yang et al. [29,30], and the references cited therein. In addition, by virtue of the scalarizing technique, Oveisiha and Zafarani investigated Stampacchia variational-like inequalities by using a normal subdifferential for multifunctions and built their relations with set-valued vector optimization issues. Besides, they attained certain characterizations of the solution sets of pseudoinvexity extremum issues. Subsequently, motivated by Oveisiha and Zafarani [31], Ceng and Latif [32] considered Stampacchia equilibrium-like problems by virtue of a normal subdifferential for multifunctions, established their relations with set-valued vector optimization issues, and attained certain characterizations of a solution set of a set-valued generalized K-pseudoinvexity program. Very recently, Atarzadeh et al. [33] considered the nonsmooth composite minimization problem (NCMP) with inequality constraints and obtained some equivalent conditions for the Karush-Kuhn-Tucker (KKT) optimality condition of the NCMP. Atarzadeh et al. [34] studied Fritz John (FJ) and KKT multiplier rules of orders one and two for a set-valued vector optimization problem with inequality constraints and established sufficient conditions for the equivalence between the disjunction and multiplier rules in various cases.
In this paper, inspired by the above research works, we first deduce some properties for K-preinvex set-valued maps using their marginal functions, equilibrium-like function and normal subdifferential concept. Secondly, in terms of this equilibrium-like function we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality conditions for a general type of super efficiency. The structure of this paper is assigned below: Some concepts and basic tools are contained in Section 2. Certain properties of K-preinvexity mappings are obtained by virtue of a normal subdifferential and an equilibrium-like function in Section 3. Certain necessity and sufficiency optimality conditions are established for efficiency and a general type of super efficiency in Section 4.

Concepts and Basic Tools
Suppose that X * is the topological dual space of a Banach space X. We denote by the same notation · the norms in X and X * . Let the ·, · , [υ, ω] and (υ, ω) represent the duality pairing between X * and X, the line segment for υ, ω ∈ X and the interior of [υ, ω], respectively. Recall now certain notions of coderivatives and subdifferentials below.
Next, we always suppose that Ξ ⊂ X is an invex subset w.r.t. η : Ξ × Ξ → X. Motivated by Theorem 1, we define a mean-value condition for limiting subdifferential ∂φ w.r.t. ψ. Definition 1. Suppose that the space X is Asplund one, and ψ : X * × Ξ × Ξ → R. Given υ, ω ∈ Ξ and Lipschitz-continuous φ : Ξ → R on some open subset containing [υ, ω]. Then φ is referred to as satisfying mean-value condition for limiting subdifferential ∂φ w.r.t. ψ iff ∃u ∈ [υ, ω) and ∃ξ * ∈ ∂φ(u) Given a multi-valued map Γ : X → 2 Y with Y being partially ordered by a convex and closed cone K = ∅. We denote by "≤ K " the ordering relation on Y, that is, In what follows, we recall some definitions and results involving coderivatives and subdifferentials of set-valued mappings.
On basis of the coderivative of epigraphical multifunction, Bao and Mordukhovich [23] formulated the mild extensions of subdifferential concept from extended-real-valued func-tions to vector-valued and multi-valued mappings with values in partially ordered spaces, and provided certain applications to multiobjective optimization issues in [18,24].
Suppose that Γ : Recall that F is referred to as being (see [16]): The relationship between Γ and normal coderivatives of Γ Ξ is formulated in [16] (Proposition 3.12).

Proposition 1 ([16]
). Suppose that the spaces Y, X are Asplund ones, the closed set Ξ ⊂ X and the local closedness map Γ : X → 2 Y is Lipschitz-like around (ῡ,ω) ∈ grΓ. Then for each ω * ∈ Y * the inclusion holds below: Suppose that K is a pointed convex closed cone in Y enjoying K + := {y * ∈ Y * : y * (u) ≥ 0 ∀u ∈ K}. Next, we aim to marginal functions associated with Γ. Associated with Γ and ω * ∈ Y * , the marginal function and minimum set are formulated, successively, as In what follows, we give the proposition below for certain properties of φ ω * and M ω * . Because the demonstration is simple, we omit it. Proposition 2 ([25]). Assumeῡ ∈ domΓ. In case Γ(ῡ) is of weak compactness, one has M ω * (ῡ) = ∅ for any ω * ∈ Y * . Moreover, in case M ω * (υ) is nonempty aroundῡ and multi-valued Γ is u.s.c., the real-valued φ ω * is l.s.c. at this point.
Then ψ is referred to as an equilibrium function if the following holds: Inspired by Lemma 1, we present the following definition concerning the basic normal cone w.r.t. η and ψ.

Condition B.
Suppose that X is a Banach space and Ξ ⊂ X is a closed and invex set w.r.t. η s.t. η is continuous in the second variable. Letῡ ∈ Ξ and ψ : X * × Ξ × Ξ → R. Then for any υ * ∈ N(ῡ; Ξ), one has In addition, ∂ M φ is referred to as being invariant monotone on Ξ w.r.t. ψ if for any υ i ∈ Ξ and ξ i ∈ ∂φ(υ i ), (i = 1, 2), one has and Definition 2 reduce to Lemma 2.12 and Definition 2.13 in [25], respectively. Moreover, for the concepts of generalized invexity and invariant monotonicity w.r.t. η, the reader is referred to [26,27,29] and the references cited therein.
The following conditions will be used in the proof of our main results later on.
Condition A (see [3]). A mapping Γ : Ξ → 2 Y from an invex set Ξ w.r.t. η to an ordered Banach space is referred to as enjoying Condition A if It should be noted that in case K = R + and Γ = φ : Ξ → R, we attain Condition A for real-valued functions of [29]: It is clear that in Γ : Ξ → 2 Y suits to Condition A, for each ω * ∈ K + , φ ω * suits to Condition A for real-valued functions.
Motivated by the Condition C of [14], we put forth the novel one below.

Definition 3.
Suppose that ∅ = Ξ ⊂ X and Ξ is of invexity w.r.t. η. A compact-valued T : Ξ → 2 L(X,Y) is referred to as being of H-hemicontinuity iff the map t → T(υ + tη(ω, υ)) is continuous at 0 + , where L(X, Y) is the family of all linear bounded mappings of X into Y and CB(L(X, Y)) is endowed with the metric topology derived by H.
For unspecified terms, we are referred to [16].

K-Preinvexity Mappings
Using a normal subdifferential, we first put forth the notions of K-invexity w.r.t. φ, weak K-invexity w.r.t. φ and invariant K-monotonicity w.r.t. φ for set-valued maps, and then establish the relations between them and K-preinvex maps.

Lemma 3 ([25]). Let
Next, let us recall the relationship between limiting subdifferential of marginal functions for Γ and its normal coderivative.
Applying the last theorem for E Γ , we obtain the conclusion below.
Proof. Via certain mild corrections in the demonstration of [1] (Lemma 1.1 and Proposition 2.1), we can obtain the desired conclusion.
Proof. The conclusion follows directly from Definition 4.
If we replace these relations in (3), we obtain: Now, from (1) we have: In a similar way, we can derive: By adding these two inequalities, we obtain: Since it is clear that φ ω * is the real-valued which suits to Condition A, one gets for all ω i ∈ M ω * (υ i ), (i = 1, 2) and w ∈ ∂Γ(υ 2 , ω 2 )(ω * ). Consequently, This completes the proof.
Theorem 5. Suppose that Γ : X → 2 Y is a locally epi-Lipschitz set-valued map satisfying Condition A. Let E Γ be closed convex-valued for every υ, and η satisfy Condition C w.

Super Efficient Solutions
In this section, we aim to establish sufficiency terms for the existence of super minimizers to set-valued vector optimization issues. Moreover, we put forth an extension of super minimizers and obtain certain necessity optimality terms for it.
For recent research on set-valued vector optimization problems, the reader is referred to [25,31,32] and the references cited therein.
Borwein and Zhuang [17] put forward the concept of super minimal points to any set of partially ordered space. Let the Banach space Y be ordered by a convex and closed cone with the closed unit ball B Y ⊂ Y. Given (ῡ,ω) ∈ grΓ withῡ ∈ Ξ. Then (ῡ,ω) is said to be a local super minimizer to problem (9) iff, ∃ (neighborhood) U atῡ s.t.ω ∈ SE(Γ(Ξ ∩ U); K). Recall a necessity term for super minimizers that had been demonstrated in [18] (Theorem 3.8).
Proof. Via mild corrections in the demonstration of [3] (Theorem 6.1 and Remark 6.1), we can derive the desired conclusion. [20] had shown that, in case there is a bounded closed base in a pointed convex cone K, one has SE(A; K) = SE(A + K; K) for all nonempty sets A ⊂ Y. By this hypothesis and conditions of Theorem 8, we can obtain that

Remark 4. Rong and Wu
is a necessity optimality term for super minimal points. Recently, it was proven in [18] that the relationship (10) is true under the normality property of the ordering cone K.
Next, under the K-preinvexity of Γ w.r.t. η, we demonstrate that the converse of Theorem 8 in the presence of (10) is true.
Proof. Note thatῡ ∈ Ξ and there is a ω * ∈ intK + such that (15) holds. By Lemma 4 we know that φ ω * is preinvex w.r.t. η. Utilizing a similar inference to that of Theorem 9, we can derive the desired conclusion.