# ϵ-Henig Saddle Points and Duality of Set-Valued Optimization Problems in Real Linear Spaces

Authors: Zhi-Ang Zhou
Journal: The Scientific World Journal
 (2013)

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## Abstract

We studyϵ-Henig saddle points and duality of set-valued optimization problems in the setting of real linear spaces. Firstly, an equivalent characterization of ϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between the ϵ-Henig saddle point of the Lagrangian set-valued map and the ϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.


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## Body

## 1. Introduction

Saddle points and duality of set-valued optimization, which have close relationship to the generalized convexity of set-valued maps and the efficiency of solutions of set-valued optimization, are two important topics in optimization theory. Recently, many new researches involving saddle points and duality of set-valued optimization have appeared in the literature. Li [1] introduced Benson proper saddle point of the Lagrangian set-valued maps for a set-valued optimization problem in locally convex spaces and established saddle point theorems and duality theorems under the assumption of the cone subconvexlikeness of set-valued maps. Zhao and Rong [2] and Li et al. [3] investigated ϵ-strict saddle points and duality of set-valued optimization problems under the assumption of cone convexlikeness and ic-cone convexlikeness of set-valued maps, respectively. Xia and Qiu [4] obtained saddle point theorems and duality theorems of set-valued optimization problems in the sense of super efficiency under the nearly subconvexlikeness of set-valued maps.However, in the above mentioned references, saddle points and duality of set-valued optimization were studied in locally convex spaces. How to generalize saddle point theorems and duality theorems of set-valued optimization from locally convex spaces to linear spaces is interesting. Adán and Novo [5] studied saddle points and duality for convexlike vector optimization problems in real linear spaces. In the ϵ-global prober efficiency, Zhou et al. [6] introduced the concept of the ϵ-global prober saddle point and obtained the relationships between the ϵ-global proper saddle points of Lagrangian set-valued maps and theϵ-global properly efficient element of set-valued optimization problems.This paper is a continuation of the research work [7]. The aim of this paper is to investigate ϵ-Henig proper saddle points and duality of set-valued optimization problems in linear spaces. This paper is organized as follows. In Section 2, some preliminaries, including notations and lemmas, are given. In Section 3, we introduce a new notion of ϵ-Henig proper saddle point of the Lagrangian set-valued map in linear spaces and obtain several saddle point theorems. In Section 4, we present three duality theorems including ϵ-weak duality, ϵ-converse duality, and ϵ-strong duality.


## 2. Preliminaries

LetX be a real linear space, and let Y and Z be two real ordered linear spaces. 0 denotes the zero element of every space. Let K be a nonempty subset in Y. The generated cone of K is defined as cone(K):={λk∣k∈K,λ≥0}. K is called a cone if and only if λK⊆K for any λ≥0. A cone K is said to be pointed if and only if K∩(-K)={0}.K is said to be nontrivial if and only if K≠{0} and K≠Y.The algebraic dual ofY and Z is denoted by Y* and Z*, respectively. Let C and D be two nontrivial pointed convex cones in Y and Z, respectively. The algebraic dual cone C+ and strictly algebraic dual cone C+i of C are, respectively, defined as
(1)C+:={y*∈Y*∣〈y,y*〉⩾0,∀y∈C},C+i:={y*∈Y*∣〈y,y*〉>0,∀y∈C∖{0}},
where 〈y,y*〉 denotes the value of the linear functional y* at the point y. The meaning of D+ is similar to that of C+.Definition 1 (see [8]).
LetK be a nonempty subset in Y. The algebraic interior of Kis the set(2)cor(K):={k∈K∣∀h∈Y,∃λ′>0,∀λ∈[0,λ′],k+λh∈K}.Definition 2 (see [9]).
LetK be a nonempty subset in Y. K is balanced if and only if, for allx∈K, for all λ∈[-1,1],λk∈K.  K is called absorbent if and only if 0∈cor(K).Definition 3 (see [10]).
LetK be a nonempty subset in Y. The vector closure of K is the set
(3)vcl(K):={k∈Y∣∃h∈Y,∀λ′>0,∃λ∈]0,λ′],hk+λh∈Kk∈Y∣∃h∈Y,∀λ′>0,∃λ∈]0,λ′]}.Following Adán and Novo [10], K is vectorially closed (v-closed) if vcl(K)=K.Definition 4 (see [7]).
LetBbe a nonempty convex subset in Y. B is a base of C if and only if C=cone(B) and there exists a balanced, absorbent, and convex set V such that 0∉B+V in Y. Write CV(B):=cone(V+B).From now on, we suppose thatcor(C)×cor(D)≠∅ and B is a base of C. We recall a notion ofϵ-Henig properly efficient point introduced by Zhou et al. [7] in linear spaces.Definition 5 (see [7]).
LetK⊆Y and ϵ∈C. y¯∈K is called an ϵ-Henig properly minimal efficient point with respect to B (denoted by y¯∈ϵ-Hmin(K,B)) if and only if there exists a balanced, absorbent, and convex set V with 0∉B+V such that cone(K-y¯+ϵ)∩(-CV(B))={0}.y¯∈K is called an ϵ-Henig properly maximal efficient point with respect toB(denoted by y¯∈ϵ-Hmax(K,B)) if and only if there exists a balanced, absorbent, and convex set V with 0∉B+V such that cone(K-y¯-ϵ)∩CV(B)={0}.
LetA be a nonempty set in X and F:A⇉Y and G:A⇉Z be two set-valued maps on A. Write F(A):=⋃x∈AF(x),〈F(x),y*〉:={〈y,y*〉∣y∈F(x)}, and 〈F(A),y*〉:=⋃x∈A〈F(x),y*〉. The meanings of G(A),〈G(x),z*〉, and 〈G(A),z*〉 are similar to those of F(A),〈F(x),y*〉, and 〈F(A),y*〉, respectively.Definition 6 (see [11]).
A set-valued mapF:A⇉Y is called generalized C-subconvexlike on A if and only if cone(F(A))+cor(C)is a convex set in Y.Lemma 7 (see [6]).
LetZ be a linear space, and let M,N⊆Z be two nonempty sets such that M-N is a convex set in Z. Ifcor(M-N)≠∅ and 0∉vcl(M-N), then there exists z*∈Z*∖{0} such that supz2∈N〈z2,z*〉<infz1∈M〈z1,z*〉.


## 3.ϵ-Henig Saddle Points

In this section, we will establish approximate saddle point theorems of set-valued optimization problems in the sense ofϵ-Henig proper efficiency.LetF:A⇉Y and G:A⇉Z be two set-valued maps on A. We consider the following vector optimization problem with set-valued maps:
(4)(VP)minF(x)subjecttoG(x)∩(-D)≠∅.
The feasible set of (VP) is defined by S:={x∈A∣G(x)∩(-D)≠∅}.Definition 8 (see [7]).
Letϵ∈C. x¯∈S is called an ϵ-Henig properly efficient solution of (VP) if and only if there exists y¯∈F(x¯) such that y¯∈ϵ-Hmin(F(S),B). The pair (x¯,y¯) is called an ϵ-Henig properly efficient element of (VP).We denote byL(Z,Y) the set of all linear operators from Z to Y. A subset L+(Z,Y) of L(Z,Y) is defined as L+(Z,Y):={T∈L(Z,Y)∣T(D)⊆C}. The Lagrangian set-valued map of (VP) is defined by
(5)L(x,T):=F(x)+T(G(x)),∀(x,T)∈A×L+(Z,Y).Consider the following unconstrained vector optimization problem with set-valued maps:(6)(UVP)TminL(x,T)subjectto(x,T)∈A×L+(Z,Y).Lemma 9 (see [7]).
Letϵ∈C,  x¯∈S, and y¯∈F(x¯). If there exists T¯∈L+(Z,Y) such that (x¯,y¯) is an ϵ-Henig properly efficient element of (UVP)T¯, then (x¯,y¯) is an ϵ-Henig properly efficient element of (VP).Now, we will introduce a new notion calledϵ-Henig proper saddle point of the Lagrangian set-valued map L(x,T) in linear spaces.Definition 10.
(
x
¯
,
T
¯
)
∈
A
×
L
+
(
Z
,
Y
) is called anϵ-Henig proper saddle point of the Lagrangian set-valued map L(x,T) if and only if
(7)L(x¯,T¯)∩ϵ-Hmin(⋃x∈AL(x,T¯),B)∩ϵ-Hmax(⋃T∈L+(Z,Y)L(x¯,T),B)≠∅.The following proposition is an important equivalent characterization for anϵ-Henig proper saddle point of the Lagrangian set-valued map L(x,T).Proposition 11.
LetD be v-closed and ϵ∈C. Then, (x¯,T¯)∈A×L+(Z,Y) is anϵ-Henig proper saddle point of the Lagrangian set-valued map L(x,T¯) if and only if there exist y¯∈F(x¯),  z¯∈G(x¯), and a balanced, absorbent, and convex set V with 0∉B+V such that (i)
y¯+T¯(z¯)∈ϵ-Hmin(⋃x∈AL(x,T¯),B);(ii)
G(x¯)⊆-D;(iii)
-T¯(z¯)∈C∖(ϵ+C∖{0});(iv)
cone(F(x¯)-y¯-T¯(z¯)-ϵ)∩CV(B)={0}.Proof.
Necessity. Let (x¯,T¯) be an ϵ-Henig proper saddle point of L(x,T). Then, there exist y¯∈F(x¯) and z¯∈G(x¯) such that
(8)y¯+T¯(z¯)∈ϵ-Hmin(⋃x∈A‍L(x,T¯),B),(9)y¯+T¯(z¯)∈ϵ-Hmax(⋃T∈L+(Z,Y)‍L(x¯,T),B).
Equation (8) shows that (i) holds. By (9), there exists a balanced, absorbent, and convex set V with 0∉B+V such that
(10)cone(⋃T∈L+(Z,Y)‍L(x¯,T)-y¯-T¯(z¯)-ϵ)∩CV(B)={0}.
Taking T=0 in (10), we obtain
(11)cone(F(x¯)-y¯-T¯(z¯)-ϵ)∩CV(B)={0}.
Therefore, (iv) holds. Since y¯∈F(x¯) and V is absorbent in Y, it follows from (11) that
(12)cone(T¯(z¯)+ϵ)∩(-B)=∅.
Because cone(B)=C, it follows from (12) that cone(T¯(z¯)+ϵ)∩(-C∖{0})=∅. Clearly, (T¯(z¯)+ϵ)∩(-C∖{0})=∅. Therefore,
(13)-T¯(z¯)∉ϵ+C∖{0}.
We assert that -z¯∈D. Otherwise, by Lemma 7, it is easy to prove (see the proof of Proposition  4.1 in [6]) that there exists z1*∈D+∖{0} such that 〈z¯,z1*〉>0. Taking b1∈B, we define a vector-valued map T1:Z→Y as follows:
(14)T1(z)=〈z,z1*〉〈z¯,z1*〉(b1+ϵ)+T¯(z),∀z∈Z.
Clearly, T1∈L+(Z,Y) and T1(z¯)-T¯(z¯)-ϵ=b1∈CV(B). On the other hand, T1(z¯)-T¯(z¯)-ϵ∈cone(⋃T∈L+(Z,Y)L(x¯,T)-y¯-T¯(z¯)-ϵ). Since 0∉B+V,b1≠0. Therefore,
(15)cone(⋃T∈L+(Z,Y)‍L(x¯,T)-y¯-T¯(z¯)-ϵ)∩CV(B)≠{0},
which contradicts (10). Hence,-z¯∈D. Since T¯∈L+(Z,Y), we have
(16)-T¯(z¯)∈C.
It follows from (13) and (16) that (iii) holds. We assert that G(x¯)⊆-D. Otherwise, there exists z1∈G(x¯) such that z1∉-D. Similar to the above proof, there exists z2*∈D+∖{0} such that 〈z1,z2*〉>0. Taking b2∈B, we define a vector-valued map T2:Z→Y as follows:
(17)T2(z)=〈z,z2*〉〈z1,z2*〉(b2+ϵ),∀z∈Z.
Clearly,
(18)T2∈L+(Z,Y),(19)T2(z1)-T¯(z¯)-ϵ=b2-T¯(z¯)∈B+C⊆C∖{0}.
Since y¯∈F(x¯) and z1∈G(x¯), it follows from (10) and (18) that
(20)cone(T2(z1)-T¯(z¯)-ϵ)∩CV(B)={0}.
By (20), it is easy to check that T2(z1)-T¯(z¯)-ϵ∉C∖{0}, which contradicts (19). Therefore, (ii) holds.
Sufficiency. Since y¯+T¯(z¯)∈L(x¯,T¯), by condition (i), we only prove that
(21)y¯+T¯(z¯)∈ϵ-Hmax(⋃T∈L+(Z,Y)‍L(x¯,T),B).
We assert
(22)cone(⋃T∈L+(Z,Y)‍L(x¯,T)-y¯-T¯(z¯)-ϵ)∩CV(B)={0}.
Otherwise, there exists r1>0,   r2>0,  T3∈L+(Z,Y),   y′∈F(x¯),  z′∈G(x¯),  v∈V,  and  b3∈B such that r1(y′+T3(z′)-y¯-T¯(z¯)-ϵ)=r2(v+b). Clearly,
(23)r1(y′-y¯-T¯(z¯)-ϵ)=r2(v+b3)-r1T3(z′).
By condition (ii),  z′∈-D. Since T3∈L+(Z,Y),  -T3(z′)∈C. Therefore, there exist r3≥0 and b4∈B such that
(24)-T3(z′)=r3b4.
It follows from (23) and (24) that
(25)r1(y′-y¯-T¯(z¯)-ϵ)=r2(v+b3)+r1r3b4=r2v+(r2b3+r1r3b4)=(r2+r1r3)×[r2r2+r1r3v+(r2r2+r1r3b3+r1r3r2+r1r3b4)].
Since 0<r2/(r2+r1r3)≤1, it follows from the balance ofVthat
(26)r2r2+r1r3v∈V.
It follows from the convexity of B that
(27)r2r2+r1r3b3+r1r3r2+r1r3b4∈B.
Since 0∉V+B and r2+r1r3>0, it follows from (25)–(27) that
(28)0≠r1(y′-y¯-T¯(z¯)-ϵ)∈cone(F(x¯)-y¯-T¯(z¯)-ϵ)∩CV(B),
which contradicts condition (iv). Therefore, (22) holds. Thus, (21) holds.Remark 12.
According to Theorem 1 in [12], the notion of ϵ-strictly efficient point is equivalent to the notion of ϵ-Henig properly efficient point in locally convex spaces. Moreover, the generalized subconvexlikeness of the set-valued map F is equivalent to ic-cone convexlikeness of the set-valued map F introduced by Sach [13] when the topological interior int(C)≠∅. Therefore, Proposition 11 generalizes Proposition 5.1 in [3] from locally convex spaces to real linear spaces.Theorem 13.
LetD be v-closed and ϵ∈C. If (x¯,T¯)∈A×L+(Z,Y) is an ϵ-Henig proper saddle point of the Lagrangian set-valued map L(x,T), then there exist y¯∈F(x¯) and  z¯∈G(x¯) such that (x¯,y¯) is an ϵ¯-Henig properly efficient element of (VP), where ϵ¯=ϵ-T¯(z¯).Proof.
Since(x¯,T¯)∈A×L+(Z,Y) is an ϵ-Henig proper saddle point of the Lagrangian set-valued map L(x,T), it follows from Proposition 11 that there exist y¯∈F(x¯),  z¯∈G(x¯), and a balanced, absorbent, and convex set V with0∉B+V such that conditions (i)–(iv) hold. By condition (ii), x¯∈S. By condition (iii),  ϵ¯=ϵ-T¯(z¯)∈C+C∖(ϵ+C∖{0})⊆C. We assert that (x¯,y¯) is an ϵ¯-Henig properly efficient element of (VP). Otherwise, for any balanced, absorbent, and convex set U with 0∉B+U,cone(F(S)-y¯+ϵ¯)∩(-CU(B))≠{0}. Therefore, there exist r1>0,  r2>0,  x1∈S,  y1∈F(S),  b1∈B, and  u1∈U such that
(29)r1(y1-y¯+ϵ¯)=-r2(b1+u1).
It follows from (29) that
(30)r1(y1+T¯(z)+ϵ-(y¯+T¯(z¯)))=-r2(b1+u1)+r1T¯(z),∀z∈G(x¯)∩(-D).
Since T¯∈L+(Z,Y), for any z∈G(x¯)∩(-D), it follows from (30) that there exists c∈C such that
(31)r1(y1+T¯(z)+ϵ-(y¯+T¯(z¯)))=-r2(b1+u1)-r1c.
Because B is a base of C, there exist r3≥0 and b2∈B such that c=r3b2. By (31), we obtain
(32)r1(y1+T¯(z)+ϵ-(y¯+T¯(z¯)))=-r2(b1+u1)-r1r3b2=-[r2u1+(r2b1+r1r3b2)]=-(r2+r1r3)×[r2r2+r1r3u1+(r2r2+r1r3b1+r1r3r2+r1r3b2)]∈-CU(B).
Clearly, r1(y1+T¯(z)+ϵ-(y¯+T¯(z¯)))≠0forallz∈G(x¯)∩(-D). Therefore, we obtain
(33)cone(⋃x∈A‍L(x,T¯)+ϵ-(y¯+T¯(z¯)))∩(-CU(B))≠{0},
which contradicts y¯+T¯(z¯)∈ϵ-Hmin(⋃x∈AL(x,T¯),B). Hence, (x¯,y¯) is an ϵ¯-Henig properly efficient element of (VP).Remark 14.
Comparing Theorem13 with Theorem  4.1 in [6], the notion of ϵ-global proper efficiency has been replaced by the notion of ϵ-Henig proper efficiency and the condition 0∈G(x¯) has been dropped.In order to obtain sufficient conditions ofϵ-Henig proper saddle point under the assumption of the generalized cone subconvexlikeness, we need the following lemma.Lemma 15 (see [7]).
Letϵ∈C,  x¯∈S, and 0∈G(x¯). Suppose that the following conditions hold: (i)
(x¯,y¯) is an ϵ-Henig properly efficient element of (VP);(ii)
I¯(x) is generalized C×D-subconvexlike on A, where I¯(x)=(F(x)-y¯+ϵ)×G(x);(iii)
vcl(cone(G(A)+D))=Z. Then, there exists T¯∈L+(Z,Y) such that (x¯,y¯) is anϵ-Henig properly efficient element of (UVP)T¯.By Lemma15, we easily obtain the following theorem involving the generalized cone subconvexlikeness of set-valued maps.Theorem 16.
LetD be v-closed, ϵ∈C,  x¯∈S, and 0∈G(x¯). Suppose that the following conditions hold: (i)
(x¯,y¯) is anϵ-Henig properly efficient element of (VP);(ii)
I¯(x) is generalized  C×D-subconvexlike on A, where I¯(x)=(F(x)-y¯+ϵ)×G(x);(iii)
vcl(cone(G(A)+D))=Z;(iv)
y¯∈ϵ-Hmax(⋃T∈L+(Z,Y)L(x¯,T),C). Then, there exists T¯∈L+(Z,Y) such that (x¯,T¯) is an ϵ-Henig proper saddle point ofL.


## 4.ϵ-Duality

In this section, we will give several duality theorems characterized byϵ-Henig proper efficiency of set-valued optimization problems in linear spaces.Definition 17.
Letϵ∈C and let B be a base of C. The set-valued map Φ:L+(Z,Y)⇉Y, defined by Φ(T)=ϵ-Hmin(⋃x∈AL(x,T),B), is called an ϵ-Henig properly dual map of (VP).Now, we construct the following duality problem of the primal problem (VP):(34)(VD)max⋃T∈L+(Z,Y)‍Φ(T).Definition 18.
Letϵ∈C.y¯∈⋃T∈L+(Z,Y)Φ(T) is called anϵ-efficient point of (VD) if and only if
(35)(⋃T∈L+(Z,Y)‍Φ(T)-y¯-ϵ)∩(C∖{0})=∅.Theorem 19 (ϵ-weak duality).
Letϵ∈C,  x¯∈S, and y¯∈⋃T∈L+(Z,Y)Φ(T). Then, (y¯-F(x¯)-ϵ)∩(C∖{0})=∅.Proof.
Sincey¯∈⋃T∈L+(Z,Y)Φ(T), there exists T¯∈L+(Z,Y) such that y¯∈Φ(T¯). Clearly, y¯∈ϵ-Hmin(⋃x∈AL(x,T¯),B). Thus, there exists a balanced, absorbent, and convex set V with 0∉B+V such that cone(⋃x∈AL(x,T¯)-y¯+ϵ)∩(-CV(B))={0}. It is easy to check C∖{0}⊆CV(B)∖{0}. Therefore, (y¯-⋃x∈AL(x,T¯)-ϵ)∩(C∖{0})=∅. Since x¯∈S⊆A,(y¯-F(x¯)-T¯(G(x¯))-ϵ)∩(C∖{0})=∅. Because x¯∈S, there exists z¯∈G(x¯)∩(-D) such that (y¯-F(x¯)-T¯(z¯)-ϵ)∩(C∖{0})=∅. Since T¯(z¯)∈C, it is easy to check that (y¯-F(x¯)-ϵ)∩(C∖{0})=∅.Theorem 20 (ϵ-converse duality).
Letϵ∈C andx¯∈S. If y¯∈F(x¯)∩⋃T∈L+(Z,Y)Φ(T) and 0∈G(x¯), then (x¯,y¯) is an ϵ-Henig properly efficient element of (VP) and y¯ is ϵ-efficient point of (VD).Proof.
Sincey¯∈F(x¯)∩⋃T∈L+(Z,Y)Φ(T), there exists T¯∈L+(Z,Y) such that y¯∈Φ(T¯). It follows from 0∈G(x¯) and the definition ofΦthat(x¯,y¯)is anϵ-Henig properly efficient element of (UVP)T¯. According to Lemma 9,(x¯,y¯)is anϵ-Henig properly efficient element of (VP). Because x¯∈S and y¯∈⋃T∈L+(Z,Y)Φ(T), using Theorem 19, we have (⋃T∈L+(Z,Y)Φ(T)-F(x¯)-ϵ)∩(C∖{0})=∅. Clearly, (⋃T∈L+(Z,Y)Φ(T)-y¯-ϵ)∩(C∖{0})=∅. Therefore, y¯ is ϵ-efficient point of (VD).Theorem 21 (ϵ-strong duality).
Letϵ∈C,x¯∈S, and 0∈G(x¯). Suppose that the following conditions hold: (i)
(x¯,y¯) is anϵ-Henig properly efficient element of (VP);(ii)
I¯(x) is generalized C×D-subconvexlike on A, where I¯(x)=(F(x)-y¯+ϵ)×G(x);(iii)
vcl(cone(G(A)+D))=Z. Then,y¯isϵ-efficient point of (VD).Proof.
According to Lemma15, there exists T¯∈L+(Z,Y) such that (x¯,y¯) is anϵ-Henig properly efficient element of (UVP)T¯. Since 0∈G(x¯), we have
(36)y¯∈ϵ-Hmin(⋃x∈A‍L(x,T¯),B)=Φ(T¯)⊆⋃T∈L+(Z,Y)‍Φ(T).
Sincey¯∈F(x¯), it follows from Theorem 19 that y¯ is ϵ-efficient point of (VD).


## 5. Conclusions

Based on [7], we introduce the concept of ϵ-Henig saddle point of the set-valued map in linear spaces. The relationships between the ϵ-Henig saddle point of the set-valued map and the ϵ-Henig properly efficient element of the set-valued optimization problem are established. Some duality theorems are obtained in the sense of ϵ-Henig proper efficiency. When ϵ-Henig proper efficiency is replaced by ϵ-super efficiency in linear spaces, whether the conclusions of this paper hold is an interesting topic.



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