# Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

Authors: Qilin Wang; Guolin Yu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.


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## Body

## 1. Introduction

In the last several decades, several notions of derivatives (epiderivatives) for set-valued maps have been proposed and used for the formulation of optimality conditions in set-valued optimization problems. By virtue of contingent derivative (see [1]), Corley [2] investigated first-order Fritz John type necessary and sufficient optimality conditions for set-valued optimization problems. Jahn and Rauh [3] proposed the contingent epiderivative of a set-valued map and then obtained an unified necessary and sufficient optimality condition by employing the epiderivative. The essential differences between the definitions of the contingent derivative and the contingent epiderivative are that the graph is replaced by the epigraph and the derivative is single-valued. Chen and Jahn [4] introduced a notion of a generalized contingent epiderivative of a set-valued map and then established an unified necessary and sufficient conditions for a set-valued optimization problem. Lalitha and Arora [5] introduced a notion of a weak Clarke epiderivative and used it to establish optimality criteria for a constrained set-valued optimization problem. As to other concepts of derivatives (epiderivatives) of set-valued maps and their applications, one can refer to [6–15]. Recently, second-order derivatives have also been proposed, for example, see [16, 17].Until now, there are only a few papers to deal with higher-order optimality conditions and duality of set-valued optimization problems by virtue of the higher-order derivatives or epiderivatives introduced by the higher-order tangent sets. Since higher-order tangent sets introduced in [1], in general, are not cones and convex sets, there are some difficulties in studying higher-order optimality conditions for general set-valued optimization problems. Li et al. [18] studied some properties of higher-order tangent sets and higher-order derivatives introduced in [1], and then obtained higher-order necessary and sufficient optimality conditions for set-valued optimization problems under cone-concavity assumptions. By using these higher-order derivatives, they [19] also discussed higher-order Mond-Weir duality for constrained set-valued optimization problems based on weak efficiency. Li and Chen [20] proposed higher-order generalized contingent (adjacent) epiderivatives of set-valued maps and, then obtained higher-order Fritz John type necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem. Wang and Li [21] introduced generalized higher-order contingent (adjacent) epiderivatives of set-valued maps, and then investigated both necessary and sufficient conditions for Henig efficient solutions to set-valued optimization problems by employing the generalized higher-order contingent (adjacent) epiderivatives. Chen et al. [22] introduced higher-order weak contingent epiderivative and higher-order weak adjacent epiderivative for set-valued maps, and then investigated higher-order Mond-Weir type dual, higher-order Wolfe type dual, and higher-order optimality conditions to a constrained set-valued optimization problem by employing the higher-order weak adjacent (contingent) epiderivatives and Henig efficiency.Motivated by the work reported in [5, 18–22], we first introduce the notions of higher-order weakly generalized contingent epiderivative, higher-order weakly generalized adjacent epiderivative for set-valued maps and generalized cone-convex set-valued maps. Second, we discuss some properties used in this paper and the existence of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative. Finally, based on higher-order weakly generalized contingent (adjacent) epiderivatives and Henig efficiency, we discuss higher-order optimality conditions to a constrained set-valued optimization problem.The rest of the paper is organized as follows. In Section2, we collect some of the concepts and some of their properties required for the paper. In Section 3, we define higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative of set-valued maps and study existence and some properties of them. In Section 4, we establish higher-order necessary and sufficient optimality conditions to a constrained set-valued optimization problem.


## 2. Preliminaries and Notations

Throughout this paper, letX, Y, and Z be three real normed spaces, where the spaces Y and Z are partially ordered by nontrivial pointed closed convex cones C⊂Y and D⊂Z with intC≠∅ and intD≠∅, respectively. We assume that 0X,0Y,0Z denote the origins of X,Y,Z, respectively, Y* denotes the topological dual space of Y, and C* denotes the dual cone of C, defined by C*={φ∈Y*∣φ(y)≥0,∀y∈C}. Let M be a nonempty set in Y. The cone hull of M is defined by cone(M)={ty∣t≥0,y∈M}. Let E be a nonempty subset of X, and F:E→2Y and G:E→2Z be two given nonempty set-valued maps. The effective domain, the graph, and the epigraph of F are defined, respectively, by dom(F)={x∈E∣F(x)≠∅},gph(F)={(x,y)∈X×Y∣x∈E,y∈F(x)} and epi(F)={(x,y)∈X×Y∣x∈E,y∈F(x)+C}. The profile map F+:E→2Y is defined by F+(x)=F(x)+C, for every x∈dom(F). Let y0∈Y, F(E)=⋃x∈EF(x) and (F-y0)(x) = F(x)-{y0} = {y-y0∣y∈F(x)}.A nonempty convex subsetB of the convex cone C is called a base of C, if(2.1)C=cone⁡(B),0Y∉cl⁡(B).
Suppose that C has a base B. Denote(2.2)CΔ(B)={f∈C*:inf⁡{f(b):b∈B}>0},Cϵ(B)=cone⁡(B+ϵU)∀0<ϵ<δ,
where δ=inf{∥b∥:b∈B} and U is the closed unit ball of Y. It follows from [23] that, for δ>0, cl(intCϵ(B)) is a closed convex pointed cone and C∖{0Y}⊂int(Cϵ(B)) for all 0<ϵ<δ.Definition 2.1.
LetF:E→2Y be a set-valued map, x0∈E,y0∈F(x0).(i)
F is said to be C-convex on a convex set E, if epiF is a convex set.(ii)
F is said to be generalized C-convex at (x0,y0) on E, if cone(epiF-{(x0,y0)}) is convex.Obviously, if F is C-convex on convex set E, then F is a generalized C-convex at (x0,y0) on E. But the converse does not hold. For Example, let E=[-1,1], F(x)={y∈Ry≥x2/3}, for all x∈E,(x0,y0)=(0,0)∈gph(F). F is generalized R+-convex at (x0,y0) on E, but F is not R+-convex.Definition 2.2.
An elementy∈M is said to be a minimal point (resp., weakly minimal point) of M if M∩({y}-C)={y}  (resp., M∩({y}-intC)=∅. The set of all minimal points (resp., weakly minimal point) of M is denoted by MincM  (resp., WMincM).
Suppose thatm is a positive integer, X is a normed space supplied with a distance d and K is a subset of X. We denote by d(x,K)=infy∈Kd(x,y) the distance from x to K, where we set d(x,∅)=+∞.Definition 2.3 (see [1]).
Letx belong to a subset Kof a normed space X and u1,…,um-1 be elements of X. One says the following(i)
The subset(2.3)TK(m)(x,u1,…,um-1)=lim⁡inf⁡h→0+⁡K-x-hu1-⋯-hm-1um-1hm={y∈X∣lim⁡h→0+d(y,K-x-hu1-⋯-hm-1um-1hm)=0}
is the mth-order contingent set of K at (x,u1,…,um-1).(ii)
The subset(2.4)TK(m)(x,u1,…,um-1)=lim⁡inf⁡h→0+K-x-hu1-⋯-hm-1um-1hm={y∈X∣lim⁡h→0+d(y,K-x-hu1-⋯-hm-1um-1hm)=0}
is the mth-order adjacent set of K at (x,u1,⋯,um-1).
From Proposition 3.2 in [18], we have the following result.Proposition 2.4.
IfK is convex, x∈K, and ui∈X,i=1,…,m-1, then TK♭(m)(x,u1,…,um-1) is convex.


## 3. Higher-Order Weakly Generalized Epiderivatives

Definition 3.1 (see [21]).
Letx belong to a subset K of X and u1,…,um-1 be elements of X.(i)
The subset(3.1)G-TK(m)(x,u1,…,um-1)=limsup⁡h→0+cone⁡(K-x)-hu1-⋯-hm-1um-1hm={y∈X∣lim⁡inf⁡h→0+d(y,cone⁡(K-x)-hu1-⋯-hm-1um-1hm)=0}
is said to be the mth-order generalized contingent set of K at (x,u1,…,um-1).(ii)
The subset(3.2)G-TK♭(m)(x,u1,…,um-1)=lim⁡inf⁡h→0+cone⁡(K-x)-hu1-⋯-hm-1um-1hm={y∈X∣lim⁡h→0+d(y,cone⁡(K-x)-hu1-⋯-hm-1um-1hm)=0}
is said to be the mth-order generalized adjacent set of K at (x,u1,…,um-1).Remark 3.2.
(i)
The following inclusion holds:(3.3)G-TK♭(m)(x,u1,…,um-1)⊆G-TK(m)(x,u1,…,um-1).(ii)
BothG-TK(m)(x,u1,…,um-1) and G-TK♭(m)(x,u1,…,um-1) are closed.(iii)
Ifcone(K-{x}) is convex, then G-TK♭(m)(x,u1,…,um-1) is convex.(iv)
Ifcone(K-{x}) is convex, and u1,…,um-1∈K, then(3.4)G-TK(m)(x,u1-x,…,um-1-x)=G-TK♭(m)(x,u1-x,…,um-1-x).Definition 3.3.
Let(x0,y0)∈graph(F), (ui,vi)∈X×Y,i=1,2,…,m-1.(i)
Themth-order weakly generalized contingent epiderivative Dw-g(m)F(x0,y0,u1,v1,…,um-1,vm-1) of F at (x0,y0) with respect to (in short, w.r.t.) vectors (u1,v1),…,(um-1,vm-1) is the set-valued map from X to Y defined by(3.5)Dw-g(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x)=WMin⁡c⁡{y∈Y:(x,y)∈G-Tepi⁡(F)(m)(x0,y0,u1,v1,…,um-1,vm-1)}.(ii)
Themth-order weakly generalized adjacent epiderivative Dw-g♭(m)F(x0,y0,u1,v1,…,um-1,vm-1) of F at (x0,y0) w.r.t. vectors (u1,v1),…,(um-1,vm-1) is the set-valued map from X to Y defined by(3.6)Dw-g♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x)=WMin⁡c{y∈Y:(x,y)∈G-Tepi⁡(F)♭(m)(x0,y0,u1,v1,…,um-1,vm-1)}.
To compare our derivatives with well-known derivatives, we recall some notions.Definition 3.4 (see [22]).
Let(x0,y0)∈graph(F),  (ui,vi)∈X×Y,i=1,2,…,m-1.(i)
Themth-order weakly contingent epiderivative Dw(m)F(x0,y0,u1,v1,⋯,um-1,vm-1) of F at (x0,y0) w.r.t. vectors (u1,v1),…,(um-1,vm-1) is the set-valued map from X to Y defined by(3.7)Dw(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x)=WMin⁡c{y∈Y:(x,y)∈Tepi⁡(F)(m)(x0,y0,u1,v1,…,um-1,vm-1)}.(ii)
Themth-order weakly adjacent epiderivative Dw♭(m)F(x0,y0,u1,v1,…,um-1,vm-1) of F at (x0,y0) w.r.t. vectors (u1,v1),…,(um-1,vm-1) is the set-valued map from X to Y defined by(3.8)Dw♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x)=WMin⁡c{y∈Y:(x,y)∈Tepi⁡(F)♭(m)(x0,y0,u1,v1,⋯,um-1,vm-1)}.Definition 3.5 (see [20]).
Let(x0,y0)∈graph(F), (ui,vi)∈X×Y,i=1,2,…,m-1.(i)
Themth-order generalized contingent epiderivative Dg(m)F(x0,y0,u1,v1,…,um-1,vm-1) of F at (x0,y0)∈gph(F) w.r.t. vectors (u1,v1),…,(um-1,vm-1) is the set-valued map from X to Y defined by(3.9)Dg(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x)=Min⁡c{y∈Y:(x,y)∈Tepi⁡(F)(m)(x0,y0,u1,v1,…,um-1,vm-1)},x∈dom⁡[D(m)F+(x0,y0,u1,v1,…,um-1,vm-1)].(ii)
Themth-order generalized adjacent epiderivative Dg♭(m)F(x0,y0,u1,v1,…,um-1,vm-1) of F at (x0,y0)∈gph(F) w.r.t. vectors (u1,v1),…,(um-1,vm-1) is the set-valued map from X to Y defined by(3.10)Dg♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x)=Min⁡c{y∈Y:(x,y)∈Tepi⁡(F)♭(m)(x0,y0,u1,v1,…,um-1,vm-1)},x∈dom⁡[D♭(m)F+(x0,y0,u1,v1,…,um-1,vm-1)].
We now discuss the properties of themth-order weakly generalized contingent epiderivative and adjacent epiderivative, for which we recall the following definitions.Definition 3.6 (See [5, 24]).
(i)
The coneC is called Daniell, if any decreasing sequence in Y having a lower bound converges to its infimum.(ii)
A subsetM of Y is said to be minorized, if there exists a y∈Y such that M⊆{y}+C.(iii)
The weak domination property is said to hold for a subsetH of Y if H⊂WMincH+intC∪{0Y}.
Using properties of higher-order tangent sets [1], we have the following result.Proposition 3.7.
Let(x0,y0)∈gph(F), ui∈X,v1∈Y. If the sets {y∈Y∣(x-x0,y)∈G-Tepi(F)(m)(x0,y0,u1,v1,…,um-1,vm-1)} and {y∈Y∣(x-x0,y)∈G-Tepi(F)♭(m)(x0,y0,u1,v1,…,um-1,vm-1)} fulfill the weak domination property for all x∈E, then for any x∈E,(i)
Dg(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)⊆Dw-g(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)+C,(ii)
Dw(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)⊆Dw-g(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)+C,(iii)
Dg♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)⊆Dw-g♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)+C,(iv)
Dw♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)⊆Dw-g♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)+C.Remark 3.8.
The reverse inclusions in Proposition3.7 may not hold. The following examples explain the case, where we only take m=1,2.Example 3.9.
LetX=R,Y=R2,E=R+,C=R+2, F(x)={(y1,y2)∈R2:y1≥x2/3,y2≥0}, for all x∈E, (x0,y0)=(0,(0,0))∈gph(F). Then,
(3.11)Tepi⁡(F)(x0,y0)=Tepi⁡(F)♭(x0,y0)={(x,y)∣x=0,y1∈R+,y2∈R+},G-Tepi⁡(F)(x0,y0)=G-Tepi⁡(F)♭(x0,y0)=R×(R+×R+).
Hence, for any x∈R+∖{0},
(3.12)DgF(x0,y0)(x-x0)=Dg♭F(x0,y0)(x-x0)=∅,DwF(x0,y0)(x-x0)=Dw♭F(x0,y0)(x-x0)=∅,Dw-gF(x0,y0)(x-x0)=Dw-g♭F(x0,y0)(x-x0)={(y1,y2)∈R2∣y1=0,y2∈R+}∪{(y1,y2)∈R2∣y1∈R+,y2=0}.Example 3.10.
Suppose thatX=R,Y=R2,E=X,C=R+2. Let F:E→2R2 be a set-valued map with F(x) = {(y1,y2)∈R2∣y1≥x4,y2≥x2},(x0,y0) = (0,(0,0))∈gph(F) and (u,v) = (1,(0,0)). Then Tepi(F)(2)(x0,y0,u,v) = Tepi(F)♭(2)(x0,y0,u,v) = R×(R+×[1,+∞)), G-Tepi(F)(2)(x0,y0,u,v) = G-Tepi(F)♭(2)(x0,y0,u,v) = R×(R+×R+). Therefore, for any x∈E,
(3.13)Dg(2)F(x0,y0,u,v)(x-x0)=Dg♭(2)F(x0,y0,u,v)(x-x0)={(0,1)},Dw(2)F(x0,y0,u,v)(x-x0)=Dw♭(2)F(x0,y0,u,v)(x-x0)={(0,y2)∣y2≥1}∪{(y1,1)∣y1≥0},Dw-g(2)F(x0,y0,u,v)(x-x0)=Dw-g♭(2)F(x0,y0,u,v)(x-x0)={(y1,0)∣y1≥0}∪{(0,y2)∣y2≥0}.Example 3.11.
LetX=R,Y=R2,E=X,C=R+2, F(x)={(y1,y2)∈R2:y1≥x4/3,y2∈R}, for all x∈E, (x0,y0)=(0,(0,0))∈gph(F), and (u,v)=(1,(0,0)). Then,
(3.14)Tepi⁡(F)(2)(x0,y0,u,v)=Tepi⁡(F)♭(2)(x0,y0,u,v)=∅,G-Tepi⁡(F)(2)(x0,y0,u,v)=G-Tepi⁡(F)♭(2)(x0,y0,u,v)=R×(R+×R).
Hence, for any x∈E,
(3.15)Dg(2)F(x0,y0,u,v)(x-x0)=Dg♭(2)F(x0,y0,u,v)(x-x0)=∅,Dw(2)F(x0,y0,u,v)(x-x0)=Dw♭(2)F(x0,y0,u,v)(x-x0)=∅,Dw-g(2)F(x0,y0,u,v)(x-x0)=Dw-g♭(2)F(x0,y0,u,v)(x-x0)={(y1,y2)∈R2∣y1=0,y2∈R}.
We now discuss the existence of themth-order weakly generalized contingent epiderivative and adjacent epiderivative.Theorem 3.12.
LetC be a closed convex-pointed cone and let C be Daniell.(i)
If the setP(x):={y∈Y:(x,y)∈G-Tepi(F)(m)(x0,y0,u1,v1,…,um-1,vm-1)} is minorized for every x∈dom(P), then Dw-g(m)(x0,y0,u1,v1,…,um-1,vm-1)(x) exists for all x∈dom(P).(ii)
If the setQ(x):={y∈Y:(x,y)∈G-Tepi(F)♭(m)(x0,y0,u1,v1,…,um-1,vm-1)} is minorized for every x∈dom(Q), then Dw-g♭(m)(x0,y0,u1,v1,⋯,um-1,vm-1)(x) exists for all x∈dom(Q).Proof.
From Remark3.2 (ii), we know that mth-order generalized contingent set and mth-order generalized adjacent set are closed. Then we can prove them as the proof of Theorem  3.1 in [22].Now we discuss some crucial propositions of themth-order weakly generalized contingent epiderivative and adjacent epiderivative.Proposition 3.13.
Letx0∈E, y0∈F(x0), (ui,vi)∈{0X}×C. If the set P(x-x0):={y∈Y∣(x-x0,y)∈G-Tepi(F)(m)(x0,y0,u1,v1,…,um-1,vm-1)} fulfills the weak domination property for all x∈E, then for all x∈E,
(3.16)F(x)-{y0}⊂Dw-g(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)+C.Proof.
It follows from Proposition  3.9 in [21] and the weak domination property of P(x-x0) that the result holds.From the proof process of Proposition3.13, we have the following result.Corollary 3.14.
Letx0∈E, y0∈F(x0), (ui,vi)∈{0X}×C. If the set P(x-x0):={y∈Y∣(x-x0,y)∈G-Tepi(F)♭(m)(x0,y0,u1,v1,…,um-1,vm-1)} fulfills the weak domination property for all x∈E, then for all x∈E,
(3.17)F(x)-{y0}⊂Dw-g♭(m)F(x0,y0,u1,v1,…,um-1,vm-1)(x-x0)+C.Remark 3.15.
Since the cone-convexity and cone-concavity assumptions are omitted, Proposition3.13 improves [18, Theorem 4.1], [20, Proposition 3.1] and [22, Proposition 3.1].Proposition 3.16.
LetE be a nonempty subset of X, x0∈E, y0∈F(x0), and let ui∈E,vi∈F(ui)+C,i=1,2,…,m-1. If F is generalized C-convex at (x0,y0) on E, and the set Q(x-x0):={y∈Y∣(x-x0,y)∈G-Tepi(F)(m)(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)} fulfills the weak domination property for all x∈E, then for any x∈E,
(3.18)F(x)-{y0}⊂Dw-g(m)F(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)(x-x0)+C.Proof.
Take anyx∈E, y∈F(x) and a sequence {hn} with hn→0+. Since F is generalized C-convex at (x0,y0) on E, cone(epi(F)-{(x0,y0)}) is convex, and then
(3.19)hn(u1-x0,v1-y0)+⋯+hnm-1(um-1-x0,vm-1-y0)∈cone⁡(epi⁡F-{(x0,y0)}).
It follows from hn>0 and cone(epiF-{(x0,y0)}) is a convex cone that
(3.20)(xn,yn):=hn(u1-x0,v1-y0)+⋯+hnm-1(um-1-x0,vm-1-y0)+hnm(x-x0,y-y0)∈cone⁡(epi⁡F-{(x0,y0)}).
We obtain that
(3.21)(x-x0,y-y0)=(xn,yn)-hn(u1-x0,v1-y0)-⋯-hnm-1(um-1-x0,vm-1-y0)hnm,
which implies that
(3.22)(x-x0,y-y0)∈G-Tepi⁡(F)(m)(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0),
that is, y-y0∈P(x-x0). By the definition of mth-order weakly generalized contingent epiderivative and the weak domination property, we have
(3.23)P(x-x0)⊂Dw-g(m)(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)(x-x0)+C.
Thus,
(3.24)F(x)-{y0}⊂Dw-g(m)F(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)(x-x0)+C,
and the proof is complete.Remark 3.17.
Since the cone-convexity assumptions are replaced by generalized cone-con- vexity assumptions, Proposition3.16 improves [18, Theorem 4.1], [20, Proposition 3.1] and [22, Proposition 3.1]. The following example explains the case, where we only take m=2.Example 3.18.
LetX=R,Y=R2,E=[-1,1],C=R+2, F(x)={(y1,y2)∈R:y1≥x2/3,y2≥x2/3}, for all x∈E and let (x0,y0)=(0,(0,0))∈graph(F).
Naturally,F is generalized C-convex at (x0,y0) on E, and F is not C-convex on E. Let u=1,v=(1,1)∈F(1)+C. Then(3.25)Tepi⁡(F)(2)(x0,y0,u-x0,v-y0)=∅,G-Tepi⁡(F)(2)(x0,y0,u-x0,v-y0)={(x,y)∈R×R2∣y1≥x,y2≥x}.
Hence, the conditions of Proposition 3.16 are satisfied. For any x∈X,
(3.26)Dg(2)F(x0,y0,u-x0,v-y0)(x-x0)=Dw(2)F(x0,y0,u-x0,v-y0)(x-x0)=∅,Dw-g(2)F(x0,y0,u-x0,v-y0)(x-x0)={(y1,y2)∈R2∣y1=x,y2≥x}∪{(y1,y2)∈R2∣y1≥x,y2=x}.
Thus, for any x∈E,
(3.27)F(x)-{y0}⊂Dw-g(m)F(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)(x-x0)+C.
Since F is not C-convex and C-concave on E and Tepi(F)(2)(x0,y0,u-x0,v-y0)=∅, the assumptions of [18, Theorem 4.1], [20, Proposition 3.1] and [22, Proposition 3.1] are not satisfied. Therefore [18, Theorem 4.1], [20, Proposition 3.1] and [22, Proposition 3.1] are unusable here.Corollary 3.19.
LetE be a nonempty convex subset of X, x,x0∈E, y0∈F(x0). Let ui∈E,vi∈F(ui)+C,i=1,2,…,m-1. If F is generalized C-convex at (x0,y0) on E, and the set {y∈Y∣(x-x0,y)∈G-Tepi(F)♭(m)(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)} fulfills the weak domination property for all x∈E, then
(3.28)F(x)-{y0}⊂Dw-g♭(m)F(x0,y0,u1-x0,v1-y0,…,um-1-x0,vm-1-y0)(x-x0)+C.


## 4. Higher-Order Optimality Conditions

In this section, we discuss the higher-order optimality Conditions of Henig efficient solutions for constrained set-valued optimization problems. The notation(F,G)(x) is used to denote F(x)×G(x). Firstly, we recall the definition of interior tangent cone of a set and state a result regarding it from [16].The interior tangent cone ofK at x0 is defined as
(4.1)ITK(x0)={u∈X∣∃λ>0,∀t∈(0,λ),∀u′∈BX(u,λ),x0+tu′∈K},
where BX(u,λ) stands for the closed ball centered at u∈X and of radius λ.Lemma 4.1 (see [13]).
IfK⊂X is convex, x0∈K, and intK≠∅, then
(4.2)ITint⁡K(x0)=intcone(K-{x0}).
Consider the following set-valued optimization problem:
(4.3)(SP){minF(x),s.t.G(x)∩(-D)≠∅,x∈E,
let K:={x∈E∣G(x)∩(-D)≠∅} and F(K):=⋃x∈Kf(X). Let x0∈K,y0∈F(x0), (x0,y0) is said to be a Henig efficient solution of problem (SP), if for some ϵ∈(0,δ),
(4.4)(F(K)-{y0})∩(-int⁡(Cϵ(B)))=∅.Lemma 4.2 (see [13]).
Letx0∈K,y0∈F(x0). If there exists ϕ∈CΔ(B) such that
(4.5)ϕ(y)≥ϕ(y0),∀y∈F(K),
then (x0,y0) is a Henig efficient solution of problem (SP).Theorem 4.3.
Suppose thatC has a base B. Let (ui,vi,wi)∈X×(-C)×(-D),i=1,2,…,m-1, y0∈F(x0), z0∈G(x0)∩(-D), δ=inf{∥b∥:b∈B}. If (x0,y0) is a Henig efficient solution of (SP), then there exists ϵ∈(0,δ) such that
(4.6)[Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(x)+{(0Y,z0)}Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(x)]∩-int⁡(Cϵ(B)×D)=∅.
for all x∈X.Proof.
Since(x0,y0) is a Henig efficient solution of (SP), there exists an ϵ∈(0,δ) such that (F(K)-{y0})⋂-int(Cϵ(B))=∅. Then
(4.7)cone⁡(F(K)+C-{y0})∩-int⁡(Cϵ(B))=∅.
IfDw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(x)=∅ for some x∈X, then the result (4.6) holds trivially. So we next prove that for any x∈Ω: = dom[Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)], the result (4.6) holds.
Assume that (4.6) does not hold. Then there exist x¯∈Ω and (y¯,z¯)∈X×Y such that(4.8)(y¯,z¯)∈Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(x¯),(4.9)(y¯,z¯)+(0Y,z0)∈-(int⁡(Cϵ(B))×int⁡D).
It follows from (4.8) and the definition of mth-order weakly generalized contingent epiderivative that there exist sequences {hn} with hn→0+ and {(xn,yn,zn)}⊆cone(epi(F,G)-{(x0,y0,z0)}) such that
(4.10)(xn,yn,zn)-hn(u1,v1,w1+z0)-⋯-hnm-1(um-1,vm-1,wm-1+z0)hnm⟶(x¯,y¯,z¯).
From (4.9) and (4.10), there exists a sufficiently large N1 such that
(4.11)yn-hnv1-⋯-hnm-1vm-1∈-int⁡(Cϵ(B)),forn>N1,(4.12)z¯n:=zn-hn(w1+z0)-⋯-hnm-1(wm-1+z0)hnm=hn+⋯+hnm-1hnm(zn-hnw1-⋯-hnm-1wm-1hn+⋯+hnm-1-z0)⟶z¯∈-(int⁡D+z0)⊂-intcone(D+z0).
Since v1,…,vm-1∈-C, hn>0 and C is a convex cone,
(4.13)hnv1+⋯+hnm-1vm-1∈-C.
By (4.11) and (4.13), we get
(4.14)yn∈-int⁡(Cϵ(B)),forn>N1.
According to (4.12) and Lemma 4.1, we obtain -z¯∈ITintD(-z0). Then, it follows from the definitions of ITintD(-z0) that ∃λ>0,forallt∈(0,λ),forallu′∈BZ(-z¯,λ),-z0+tu′∈intD. Since hn→0+ and (4.12), there exists a sufficiently large N2 such that
(4.15)hnmhn+⋯+hnm-1∈(0,λ),forn>N2,-z0+hnmhn+⋯+hnm-1(-z¯n)∈int⁡D,forn>N2,
that is,
(4.16)zn-hnw1-⋯-hnm-1wm-1hn+⋯+hnm-1∈-int⁡D,forn>N2.
It follows from hn>0,w1,…,wm-1∈-D and D is a convex cone that
(4.17)zn∈-int⁡D,forn>N2.
Since zn∈cone(G(xn)+D-{z0}), there exist λn≥0,z̃n∈G(xn)anddn∈D such that zn=λn(z̃n+dn-z0). It follows from (4.17) that z̃n∈G(xn)∩(-D), for n>N2, and then
(4.18)xn∈K,forn>N2.
Then it follows from (4.14) that
(4.19)yn∈cone⁡(F(K)+C-{y0})∩-int⁡(Cϵ(B)),forn>max{N1,N2},
which contradicts (4.7). Thus (4.6) holds and the proof is complete.FromD+intD⊆intD,C+intCϵ(B)⊆intCϵ(B) and Theorem 4.3, we have the following corollary.Corollary 4.4.
Suppose thatC has a base B. Let (ui,vi,wi)∈X×(-C)×(-D),i=1,2,…,m-1, y0∈F(x0), z0∈G(x0)∩(-D), δ=inf{∥b∥:b∈B}. If (x0,y0) is a Henig efficient solution of (SP), then there exists ϵ∈(0,δ) such that
(4.20)[Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(X)+C×D+{(0Y,z0)}Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)]∩(-int⁡(Cϵ(B)×D))=∅.Theorem 4.5.
Suppose thatC has a base B, δ=inf{∥b∥:b∈B}. Let y0∈F(x0), (ui,vi-y0,wi)∈X×(-C)×(-D),i=1,2,…,m-1. If (x0,y0) is a Henig efficient solution of (SP), then for some ϵ∈(0,δ) and for any z0∈G(x0)∩(-D),
(4.21)[Dw-g(m)(F,G)(x0,y0,z0,u1-x0,v1-y0,w1-z0,…,um-1-x0,vm-1-y0,wm-1-z0)(X)+C×D+{(0Y,z0)}Dw-g(m)(F,G)(x0,y0,z0,u1-x0,v1-y0,w1-z0,…,um-1-x0,vm-1-y0,wm-1-z0)]∩-int⁡(Cϵ(B)×D)=∅.Proof.
IfDw-g(m)(F,G)(x0,y0,z0,u1-x0,v1-y0,w1-z0,…,um-1-x0,vm-1-y0,wm-1-z0)(x)=∅ for some x∈X, then the result holds trivially. So we suppose that for any x∈Ω: = dom[Dw-g(m)(F,G)(x0,y0,z0,u1-x0,v1-y0,w1-z0,…,um-1-x0,vm-1-y0,wm-1-z0)]. Then the proof of the fact follows on the lines of [20, Theorem 4.1] by using mth-order weakly generalized contingent epiderivative instead of mth-order generalized adjacent epiderivative.Theorem 4.6.
Let(ui,vi,wi)∈X×(-C)×(-D),i=1,2,…,m-1, y0∈F(x0), z0∈G(x0)∩(-D), let B be base of C and δ=inf{∥b∥:b∈B}. Suppose that the following conditions are satisfied:(i)
(F,G) is generalized (C,D)-convex at (x0,y0,z0) on nonempty set E,(ii)
the pair(x0,y0) is a Henig efficient solution of (SP),(iii)
P(x):={(y,z)∈(Y,Z)∣(x-x0,y,z)∈G-Tepi(F,G)(m)(x0,y0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)} fulfills the weak domination property for all x∈E, (0Y,0Z)∈[Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)](0X).
Then there existϕ∈CΔ(B) and ψ∈D* such that(4.22)∅(y)+ψ(z)≥0,(z0)=0,
for all (y,z)∈Δ(x):=Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(x) and x∈Ω:=domΔ.Proof.
Letz0∈G(x0)∩(-D). Define
(4.23)M=⋃x∈ΩΔ(x)+C×D+(0Y,z0).
By the similar line of proof for convexity ofM in Theorem  5.1 in [20], we obtain that M is a convex set. It follows from Corollary 4.4 that(4.24)M∩(-(int⁡(cone⁡(ϵU+B))×int⁡D))=∅.
By the separation theorem of convex sets, there exist ϕ∈Y* and ψ∈Z*, not both zero functionals such that for all (y¯,z¯)∈M,(y,z)∈-(int(cone(ϵU+B))×intD), we have
(4.25)ϕ(y¯)+ψ(z¯)≥ϕ(y)+ψ(z).
It follows from (4.25) that
(4.26)ϕ(y)≤ψ(z),∀(y,z)∈-int⁡(cone⁡(ϵU+B))×int⁡D,(4.27)ϕ(y¯)+ψ(z¯)≥0,∀(y¯,z¯)∈M.
Whence
(4.28)ϕ(y)+ψ(z)≥0,
for all (y,z)∈Δ(x) and x∈Ω.
From (4.26), we obtain that ψ is bounded below on the intD. Then ψ(z)≥0, for all ∈intD. Naturally ψ∈D* and(4.29)ϕ(b)≥ϕ(u),∀b∈B,u∈ϵU.
Since U is symmetry, there exists a u0∈ϵU such that ϕ(u0)=t>0. Then ϕ∈CΔ(B).
From (4.27) and condition (iii), we have ψ(z0)≥0. Since z0∈-D and ψ∈D*, ψ(z0)≤0. So(4.30)ψ(z0)=0,
and the proof is complete.Remark 4.7.
We notice that a Kuhn-Tucker type necessary optimality condition in Theorem4.6 is obtained under weaker assumptions than those assumed in [20, Theorem 5.1] and [22, Theorem 6.1]. The following example explains the case, where we only take m=2.Example 4.8.
Suppose thatX=Y=Z=R,E=X, C=D=R+, B={2}. Let F:E→2Y be a set-valued map with
(4.31)F(x)={y∈R:y≥x2/3},x∈E,
and G:E→Z be a set-valued map with
(4.32)G(x)={z∈R:z≥x4/5},x∈E.
Let x0=0∈E,y0=0∈F(x0),z0=0∈G(x0). Naturally, (F,G) is generalized C×D-convex at (x0,y0,z0) on E. Consider the following constrained set-valued optimization problem (4.6):
(4.33)min⁡,F(x),s.t.x∈E,G(x)∩(-D)≠∅.
By definition, (x0,y0) is a Henig efficient solution of (4.6). Take (u1,v1,w1)=(-1,0,0)∈gph(Dw-g(F,G)(x0,y0,z0)). By directly calculating, we have
(4.34)G-Tepi⁡(F,G)(2)(x0,y0,z0,u1,v1,w1+z0)={(x,y,z)∈E×R2:y≥0,z≥0},(0Y,0Z)∈Dw-g(2)(F,G)(x0,y0,z0,u1,v1,w1+z0)(0X).
Then, the conditions of Theorem 4.6 are satisfied, and
(4.35)Dw-g(2)(F,G)(x0,y0,z0,u1,v1,w1+z0)(x)={(y,z)∈R2:y=0,z≥0}∪{(y,z)∈R2:y≥0,z=0}.
Simultaneously, take ϕ=1∈CΔ(B) and φ=1∈D*. Obviously, the 2nd-order necessary optimality condition of Theorem 4.6 holds.
Since neitherF nor G is R+-convex on nonempty convex set E and Tepi(F,G)(2)(x0,y0,z0,u1-x0,v1-y0,w1-z0)=∅, the assumptions of [20, Theorem 5.1] and [22, Theorem 6.1] are not satisfied. Therefore [20, Theorem 5.1] and [22, Theorem 6.1] are unusable here.
As a direct consequence of Theorem4.6, we get the following corollary.Corollary 4.9 (See [21]).
Let(ui,vi,wi)∈X×(-C)×(-D),i=1,2,…,m-1, y0∈F(x0), let B be base of C and δ=inf{∥b∥:b∈B}. Suppose that the following conditions are satisfied:(i)
F and G are C-convex and D-convex on nonempty convex set E,respectively,(ii)
the pair(x0,y0) is a Henig efficient solution of (SP),(iii)
(0Y,0Z)∈[G-D(m)(F,G)(x0,y0,z0, u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)](0X), for any z0∈G(x0)⋂(-D).
Then for anyz0∈G(x0)⋂(-D), there exist ϕ∈CΔ(B) and ψ∈D* such that(4.36)ϕ(y)+ψ(z)≥0,ψ(z0)=0,
for all (y,z)∈G-D(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0)(x) and x∈Ω: = domG-D(m)(F,G)(x0,y0,z0,u1,v1,w1+z0,…,um-1,vm-1,wm-1+z0).Theorem 4.10.
Suppose thatC has a base B. Let x,x0∈E, y0∈F(x0),z0∈G(x0)⋂(-D). Suppose that the following conditions are satisfied:(i)
ui∈E,vi∈F(ui)+C,wi∈G(ui)+D,i=1,2,…,m-1,(ii)
(F,G) is generalized C×D-convex at (x0,y0,z0) on E,(iii)
if the set{(y,z)∈(Y,Z) ∣ (x-x0,y,z)∈G-Tepi(F,G)(m)(x0, y0, z0, u1-x0, v1-y0, w1-z0, …, um-1-x0, vm-1-y0, wm-1-z0)} fulfills the weak domination property for all x∈E,(iv)
there existϕ∈CΔ(B) and ψ∈D* such that(4.37)ϕ(y)+ψ(z)≥0,ψ(z0)=0,
for all (y,z)∈Dw-g(m)(F,G)(x0,y0,z0,u1-x0,v1-y0,w1-z0,…,um-1-x0,vm-1-y0,wm-1-z0)(x-x0), for all x∈K.
Then the pair(x0,y0) is a Henig efficient solution of (SP).Proof.
It follows from Proposition3.16 that
(4.38)(y-y0,z-z0)∈Dw-g(m)(F,G)(x0,y0,z0,u1-x0,v1-y0,w1-z0,…,um-1-x0,vm-1-y0,wm-1-z0)(x-x0)+C×D,
for all (y,z)∈(F,G)(x),x∈K. Then,
(4.39)ϕ(y-y0)+ψ(z-z0)≥0,∀(y,z)∈(F,G)(x).
Thus, for any x∈K, there exists a z¯∈G(x) with z¯∈-D such that ψ(z¯)≤0. It follows from ψ(z0)=0 and (4.39) that
(4.40)ϕ(y)≥ϕ(y0),∀y∈F(K).
Whence it follows from Lemma 4.2 that (x0,y0) is a Henig efficient solution of (SP).Similarly as in the proof of Theorem4.10, it follows from Proposition 3.13 that we have the following result.Theorem 4.11.
Suppose thatC has a base B. Let x,x0∈E, y0∈F(x0),z0∈G(x0)⋂(-D). Suppose that the following conditions are satisfied:(i)
(ui,vi,wi)∈{0X}×C×D,i=1,2,…,m-1,(ii)
If the set{(y,z)∈(Y,Z)∣(x-x0,y,z)∈G-Tepi(F,G)(m)(x0,y0,u1,v1,…,um-1,vm-1)} fulfills the weak domination property for all x∈E,(iii)
there existϕ∈CΔ(B) and ψ∈D* such that(4.41)ϕ(y)+ψ(z)≥0,ψ(z0)=0,
for all (y,z)∈Dw-g(m)(F,G)(x0,y0,z0,u1,v1,w1,…,um-1,vm-1,wm-1)(x-x0), for all x∈K.
Then the pair(x0,y0) is a Henig efficient solution of (CP).Remark 4.12.
(i)
Since Theorem4.11 does not involve the convexity, it improves [20, Theorem 5.4] and [22, Theorem 6.2].(ii)
If we usemth-order weakly generalized adjacent epiderivative instead of the mth-order weak generalized contingent epiderivative in Theorems 4.3, 4.5, 4.6, 4.10 and 4.11, then the corresponding results for mth-order weakly generalized adjacent epiderivative still hold.



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