# A Generalized Alternative Theorem of Partial and Generalized Cone Subconvexlike Set-Valued Maps and Its Applications in Linear Spaces

Authors: Zhi-Ang Zhou; Jian-Wen Peng
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We first introduce a new notion of the partial and generalized cone subconvexlike set-valued map and give an equivalent characterization of the partial and generalized cone subconvexlike set-valued map in linear spaces. Secondly, a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map was presented. Finally, Kuhn-Tucker conditions of set-valued optimization problems were established in the sense of globally proper efficiency.


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## Body

## 1. Introduction

Generalized convexity plays an important role in set-valued optimization. The generalization of convexity from vector-valued maps to set-valued maps happened in the 1970s. Borwein [1] and Giannessi [2] introduced and studied the cone convexity of set-valued maps. Based on Borwein and Giannessi’s work, some authors [3–7] established a series of optimality conditions of set-valued optimization problems under different types of generalized convexity of set-valued maps in topological spaces. Since linear spaces are wider than topological spaces, generalizing some results of the above mentioned references from topological spaces to linear spaces is an interesting topic. Li [8] introduced a cone subconvexlike set-valued map involving the algebraic interior and established Kuhn-Tucker conditions. Huang and Li [9] studied Lagrangian multiplier rules of set-valued optimization problems with generalized cone subconvexlike set-valued maps in linear spaces. When the algebraic interior of the convex cone is empty, Hernández et al. [10] used the relative algebraic interior of the convex cone to introduce cone subconvexlikeness of set-valued maps and investigated Benson proper efficiency of set-valued optimization problems in linear spaces.The aim of this paper is to study globally proper efficiency of set-valued optimization problems in linear spaces. This paper is organized as follows. In Section2, we recalled some basic notions and gave some lemmas. In Section 3, we presented a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map and established Kuhn-Tucker conditions of set-valued optimization problems in the sense of globally proper efficiency.


## 2. Preliminaries

In this paper, letY and Z be two real-ordered linear spaces, and let 0 denote the zero element of every space. Let K be a nonempty subset in Y. The cone hull of K is defined as coneK:={λk∣k∈K,λ≥0}.K  is called a convex cone if and only if
(2.1)λ1k1+λ2k2∈K,∀λ1,λ2≥0,∀k1,k2∈K.
A cone K is said to be pointed if and only if K∩(-K)={0}. A cone K is said to be nontrivial if and only if  K≠{0} and K≠Y.LetY* and Z* stand for the algebraic dual spaces of Y and Z, respectively. Let C and D be nontrivial, pointed, and convex cones in Y and Z, respectively. The algebraic dual cone C+ of C is defined as C+:={y*∈Y*∣〈y,y*〉⩾0,∀y∈C}, and the strictly algebraic dual cone C+i of C is defined as 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+.LetK be a nonempty subset of Y. The linear hull spanK of K is defined as spanK:={k∣k=∑i=1nλiki,λi∈ℝ,ki∈K,i=1,…,n}, and the affine hull affK of K is defined as affK:={k∣k=∑i=1nλiki,∑i=1nλi=1,λi∈ℝ,ki∈K,i=1,…,n}. The generated linear subspace L(K) of K is defined as L(K):=span(K-K).Definition 2.1 (see [11]).
LetK be a nonempty subset of Y. The algebraic interior of K is the set
(2.2)corK:={k∈K∣∀k′∈Y,∃λ′>0,∀λ∈[0,λ′],k+λk′∈K}.Definition 2.2 (see [12]).
LetK be a nonempty subset of Y. The relative algebraic interior of K is the set
(2.3)icrK={k∈K∣∀v∈affK-k,∃λ0>0,∀λ∈[0,λ0],k+λv∈K}.
Clearly,affK-k=L(K),forallk∈K. Therefore, Definition 2.2 is consistent with the definition of the relative algebraic interior of K in [13, 14]. However, Definition 2.2 seems to be more convenient than the ones in [13, 14].It is worth noting that ifK is a nontrivial and pointed cone in Y, then 0∉icrK, and if K is a convex cone, then icrK is a convex set, and  icrK∪{0} is a convex cone.Lemma 2.3 (see [13]).
IfK is a convex cone in Y, then K+icrK=icrK.Lemma 2.4 (see [10, 12, 14]).
IfK is a nonempty subset in Y, then(a)
affK-k=affK-K,forallk∈K;
ifK is convex in Y and icrK≠∅, then(b)
icr(icrK)=icrK;(c)
aff(icrK)=affK.Lemma 2.5 (see [12]).
LetK be a convex set with icr(K)≠∅ in Y. If 0∉icrK, then there exists y*∈Y*∖{0} such that
(2.4)〈k,y*〉≥0,∀k∈K.


## 3. Main Results

LetA be a nonempty set, and let F:A⇉Y and G:A⇉Z be two set-valued maps on A. Write F(A):=⋃x∈AF(x)  and 〈F(x),y*〉:={〈y,y*〉∣y∈F(x)}. The meanings of G(A) and 〈G(x),z*〉 are similar to those of F(A) and 〈F(x),y*〉.Now, we introduce a new notion of the partial and generalized cone subconvexlike set-valued map.Definition 3.1.
A set-valued mapJ=(F,G):A⇉Y×Z is called partial and generalized C×D-subconvexlike on A if and only if cone(J(A))+icrC×D is a convex set in Y×Z.The following theorem will give some equivalent characterizations of the partial and generalizedC×D-subconvexlike set-valued map in linear spaces.Theorem 3.2.
LeticrC≠∅. Then the following statements are equivalent:(a)
the set-valued mapJ:A⇉Y×Z is partial and generalized C×D-subconvexlike on A,(b)
Forall(c,d)∈icrC×D,∀x1,x2∈A,∀λ∈]0,1[,
(3.1)(c,d)+λJ(x1)+(1-λ)J(x2)⊆cone(J(A))+icrC×D,(c)
∃c'∈icrC,∀x1,x2∈A,∀λ∈]0,1[,∀ε>0,
(3.2)ε(c′,0)+λJ(x1)+(1-λ)J(x2)⊆cone(J(A))+C×D,(d)
∃c′′∈C,∀x1,x2∈A,∀λ∈]0,1[,∀ε>0,
(3.3)ε(c′′,0)+λJ(x1)+(1-λ)J(x2)⊆cone(J(A))+C×D.Proof.
(
a
)
⇒
(
b
). Let (c,d)∈icrC×D,x1,x2∈A,λ∈]0,1[,(y1,z1)∈J(x1), and (y2,z2)∈J(x2). Clearly,
(3.4)(y1,z1)+(c,d)∈cone(J(A))+icrC×D,(y2,z2)+(c,d)∈cone(J(A))+icrC×D.
Since J is partial and generalized C×D-subconvexlike on A, it follows from (3.4) that
(3.5)(c,d)+λ(y1,z1)+(1-λ)(y2,z2)=λ((y1,z1)+(c,d))+(1-λ)((y2,z2)+(c,d))∈cone(J(A))+icrC×D,
which implies that (3.1) holds.
The implications(b)⇒(c)⇒(d) are clear.
(
d
)
⇒
(
a
). Let (mi,ni)∈cone(J(A))+icrC×D(i=1,2),λ∈]0,1[. Then there exist ρi≥0,xi∈A,(yi,zi)∈J(xi), and (ci,di)∈icrC×D(i=1,2) such that (mi,ni)=ρi(yi,zi)+(ci,di).
Case one: ifρ1=0 or ρ2=0, we have λ(m1,n1)+(1-λ)(m2,n2)∈cone(J(A))+icrC×D.
Case two: ifρ1>0 and ρ2>0, we have
(3.6)λ(m1,n1)+(1-λ)(m2,n2)=λ(ρ1(y1,z1)+(c1,d1))+(1-λ)(ρ2(y2,z2)+(c2,d2))=[λ(c1,d1)+(1-λ)(c2,d2)]+[λρ1(y1,z1)+(1-λ)ρ2(y2,z2)]=β{1β[λ(c1,d1)+(1-λ)(c2,d2)]+[λρ1β(y1,z1)+(1-λ)ρ2β(y2,z2)]},
where β=λρ1+(1-λ)ρ2.
By Lemma2.4, we obtain
(3.7)-c′′∈C-C⊆affC-C=affC-1β[λc1+(1-λ)c2]=aff(icrC)-1β[λc1+(1-λ)c2].
Since (1/β)[λc1+(1-λ)c2]∈icrC=icr(icrC), there exists λ0>0 such that
(3.8)1β[λc1+(1-λ)c2]+λ0(-c′′)∈icrC.
By (3.3), (3.6), (3.8), and Lemma 2.3, we have
(3.9)λ(m1,n1)+(1-λ)(m2,n2)=β{{λρ1β}{λρ1β}1β[λ(c1,d1)+(1-λ)(c2,d2)]+λ0(-c′′,0)+[λ0(c′′,0)+λρ1β(y1,z1)+(1-λ)ρ2β(y2,z2)]}=β{(1β[λc1+(1-λ)c2]+λ0(-c′′),1β[λd1+(1-λ)d2])+[λ0(c′′,0)+λρ1β(y1,z1)+(1-λ)ρ2β(y2,z2)]}∈β(icrC×D)+cone(J(A))+C×D⊆cone(J(A))+icrC×D.
Cases one and two imply thatcone(J(A))+icrC×D is a convex set in Y×Z. Therefore, (a) holds.Remark 3.3.
Theorem3.2 generalizes the sixth item of Proposition  2.4 in [14], Lemma  2.1 in [15], and Lemma  2 in [16].
Now, we will give a generalized alternative theorem of the partial and generalizedC×D-subconvexlike map. We consider the following two systems.System1. There exists x0∈A such that -J(x0)∩(icrC×D)≠∅.System2. There exists (y*,z*)∈(C+×D+)∖{(0,0)} such that
(3.10)〈y,y*〉+〈z,z*〉≥0,∀(y,z)∈J(A).Theorem 3.4 (generalized alternative theorem).
Leticr(cone(J(A))+icrC×D)≠∅, and let the set-valued map J:A⇉Y×Z be partial and generalized C×D-subconvexlike on A. Then, (i)
if System 1 has no solutions, then System 2 has a solution;(ii)
if(y*,z*)∈C+i×D+ is a solution of System 2, then System 1 has no solutions.Proof.
(i) Firstly, we assert that(0,0)∉cone(J(A))+icrC×D. Otherwise, there exist x0∈A and α≥0 such that (0,0)∈αJ(x0)+icrC×D.
Case one: ifα=0, then 0∈icrC. Since C is a nontrivial, pointed, and convex cone, 0∉icrC. Thus, we obtain a contradiction.
Case two: ifα>0, then there exists (y0,z0)∈J(x0) such that
(3.11)-(y0,z0)∈1α(icrC×D)⊆icrC×D,
which contradicts that System 1 has no solutions.
Cases one and two show that our assertion is true. Since the set-valued mapJ is partial and generalized C×D-subconvexlike on A,cone(J(A))+icrC×D is a convex set in Y×Z. Note that icr(cone(J(A))+icrC×D)≠∅. Thus, all conditions of Lemma 2.5 are satisfied. Therefore, there exists (y*,z*)∈(Y*×Z*)∖{(0,0)} such that
(3.12)〈ry+c,y*〉+〈rz+d,z*〉≥0,∀r≥0,x∈A,y∈F(x),z∈G(x),c∈icrC,d∈D.
Letting r=1 in (3.12), we have
(3.13)〈y+c,y*〉+〈z+d,z*〉≥0,∀x∈A,y∈F(x),z∈G(x),c∈icrC,d∈D.
We again assert thaty*∈C+. Otherwise, there exists y'∈C such that 〈y',y*〉<0. Let x¯∈A,y¯∈F(x¯),z¯∈G(x¯),c¯∈icrC, and d¯∈D be fixed. Then there exists sufficiently large positive number λ such that λ〈y',y*〉+〈y¯+c¯,y*〉+〈z¯+d¯,z*〉<0, that is,
(3.14)〈y¯+(c¯+λy′),y*〉+〈z¯+d¯,z*〉<0.
By Lemma 2.3, c¯+λy'∈icrC. Thus, (3.14) contradicts (3.13). Therefore, y*∈C+. Similarly, we can prove that z*∈D+.
Letc∈icrC be fixed in (3.13). Then, βc∈icrC,∀β>0. Letting d=0 in (3.13), we have
(3.15)〈y,y*〉+β〈c,y*〉+〈z,z*〉≥0,∀x∈A,y∈F(x),z∈G(x).
Letting β→0 in (3.15), we obtain
(3.16)〈y,y*〉+〈z,z*〉≥0,∀x∈A,y∈F(x),z∈G(x),
which implies that System 2 has a solution.
(ii) If(y*,z*)∈C+i×D+ is a solution of System 2, then
(3.17)〈y,y*〉+〈z,z*〉≥0,∀x∈A,y∈F(x),z∈G(x).
We assert that System 1 has no solutions. Otherwise, there exist p∈F(x0) and q∈G(x0) such that -p∈icrC⊆C∖{0} and -q∈D. Therefore, we have 〈p,y*〉+〈q,z*〉<0, which contradicts (3.17). Therefore, our assertion is true.Remark 3.5.
IfY×Z is a finite-dimensional space, then the partial and generalized C×D-subconvexlikeness of J:A⇉Y×Z implies that cone(J(A))+icrC×D is a nonempty convex in Y×Z, which in turn implies that the condition icr(cone(J(A))+icrC×D)≠∅ holds trivially.Remark 3.6.
Theorem3.4 generalizes Theorem  3.7 in [14], Theorem  2.1 in [15], and Theorem  1 in [16].
From now on, we suppose thaticrC≠∅.Definition 3.7 (see [17]).
LetB⊆Y.y¯∈B be called a global properly efficient point with respect to C (denoted by y¯∈GPE(B,C)) if and only if there exists a nontrivial, pointed, and convex cone C' with C∖{0}⊆icrC' such that (B-y¯)∩(-C'∖{0})=∅.
Now, we consider the following set-valued optimization problem:(3.18)MinF(x)subjectto-G(x)∩D≠∅.
The feasible set of (3.18) is defined by S:={x∈A∣-G(x)∩D≠∅}.Definition 3.8.
Letx¯∈S be called a global properly efficient solution of (3.18) if and only if there exists y¯∈F(x¯) such that y¯∈GPE(F(S),C). The pair (x¯,y¯) is called a global properly efficient element of (3.18).
Now, we will establish Kuhn-Tucker conditions of set-valued optimization problem (3.18) in the sense of globally proper efficiency.Theorem 3.9.
Suppose that the following conditions hold:(i)
(x0,y0) is a global properly efficient element of (3.18);(ii)
the set-valued mapI:A⇉Y×Z is partial and generalized C×D-subconvexlike on A, where I(x)=(F(x)-y0,G(x)),forallx∈A.Then, there exists(y*,z*)∈(C+×D+)∖{(0,0)} such that
(3.19)infx∈A(〈F(x),y*〉+〈G(x),z*〉)=〈y0,y*〉,inf〈G(x0),z*〉=0.Proof.
Since(x0,y0) is a global properly efficient element of (3.18), there exists a nontrivial, pointed, and convex cone C' with C∖{0}⊆icrC' such that
(3.20)-(F(x)-y0)∩(C′∖{0})=∅,∀x∈A.
It follows from (3.20) that
(3.21)-(F(x)-y0)∩icrC=∅,∀x∈A.
By (3.21), we obtain
(3.22)-I(x)∩(icrC×D)=∅,∀x∈A.
Since I is partial and generalized C×D-subconvexlike on A, it follows from (3.22) and Theorem 3.4 that there exists (y*,z*)∈(C+×D+)∖{(0,0)} such that
(3.23)〈F(x)-y0,y*〉+〈G(x),z*〉≥0,∀x∈A,
that is
(3.24)〈F(x),y*〉+〈G(x),z*〉≥〈y0,y*〉,∀x∈A.
Because x0∈S, there exists p∈G(x0) such that -p∈D. Since z*∈D+, we have
(3.25)〈p,z*〉≤0.
Letting x=x0 in (3.24), we obtain
(3.26)〈p,z*〉≥0.
It follows from (3.25) and (3.26) that
(3.27)〈p,z*〉=0.
Therefore, we have
(3.28)〈y0,y*〉∈〈F(x0),y*〉+〈G(x0),z*〉.
By (3.24) and (3.28), we have infx∈A(〈F(x),y*〉+〈G(x),z*〉)=〈y0,y*〉. Letting x=x0 in (3.24), we have
(3.29)〈G(x0),z*〉≥0.
It follows from (3.27) and (3.29) that inf〈G(x0),z*〉=0.The following theorem, which can be found in [17], is a sufficient condition of global properly efficient elements of (3.18).Theorem 3.10.
Suppose that the following conditions hold:(i)
x0∈S,(ii)
there existy0∈F(x0) and (y*,z*)∈C+i×D+ such that
(3.30)infx∈A(〈F(x),y*〉+〈G(x),z*〉)≥〈y0,y*〉.Then, (x0,y0) is a global properly efficient element of (3.18).



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