YU Guo-lin,ZHANG Yan,LIU San-yang

(1.Institute of Applied Mathematics,Beifang University of Nationalities,Yinchuan 750021,China)

(2.Department of Mathematics,Xidian University,Xi’an 710071,China)

余國林1,張 燕1,劉三陽2

(1.北方民族大學(xué)應(yīng)用數(shù)學(xué)研究所, 寧夏 銀川 750021)

(2.西安電子科技大學(xué)數(shù)學(xué)系, 陜西 西安 710071)

STRONG DUALITY WITH STRICT EFFICIENCY IN VECTOR OPTIMIZATION INVOLVING NONCONVEX SET-VALUED MAPS

YU Guo-lin1,ZHANG Yan1,LIU San-yang2

(1.Institute of Applied Mathematics,Beifang University of Nationalities,Yinchuan 750021,China)

(2.Department of Mathematics,Xidian University,Xi’an 710071,China)

This paper is diverted to the study of two strong dual problems of a primal nonconvex set-valued optimization in the sense of strict effi ciency.By using the principles of Lagrange duality and Mond-Weir duality,for each dual problem,a strong duality theorem with strict effi ciency is established.The conclusions can be formulated as follows:starting from a strictly effi cient solution of the primal problem,it can be constructed a strictly effi cient solution of the dual problem such that the corresponding objective values of both problems are equal.The results generalize the strong dual theorems in which the set-valued maps are assumed to be cone-convex.

strict effi ciency;strong duality;set-valued optimization;ic-cone-convexlikeness

1 Introduction

One of the most important topics of set-valued optimization is related to proper efficiency,this is because that the range ofthe set of(weak)effi cient solutions is often too large. In order to contract the solution range,several kinds of proper effi ciency were presented.For example,Benson effi ciency[1],Henig effi ciency[2],Geoffrion effi ciency[3],Super effi ciency [4]and Strictly effi ciency[5]etc.Especially,super effi ciency,given by Borwein and Zhuang [4],was shown to have some desirable properties.However,the condition to guarantee its existence is rather strong.Later,weakening the existence condition,Professor Cheng and Fu [5]improved the concept ofsupper effi ciency and introduced the concept of strict effi ciency.

Since duality assertions allow to study a minimization problem through a maximization problem and to know what one can expect in the bestcase.At the same time,duality resulted in many applications within optimization,and it provided many unifying conceptualinsights into economics and management science.So it is not surprising that duality is one of the important topics in set-valued optimization.There were many papers dedicated to dualitytheory ofset-valued optimization(see[6–11]).Among results obtained in this field,we want to mention the strong duality.In vector optimization,it is often said that strong duality holds between primaland dualproblems,if a weakly effi cient solution ofa primalproblem is a weakly effi cient solution ofdualproblem and such that the corresponding objective values of the primaland dualproblems are equal.If in this problem “weakly effi cient solution”is replaced by “properly effi cient solution”,then it is said that strong duality with proper effi ciency holds between the primaland dualproblems.However,strong duality with proper effi ciency was considered only for the case when proper effi ciency was understood in the sense of Geoffi ron[10]and Benson[11].

On the other hand,it is wellknown that the concept of cone-convexity and its generalizations play an important role in establishing duality theorems for set-valued optimization problems.Up to now,there are many notions of generalized convexity for set-valued maps which are introduced and are proved to be usefulfor optimization theory and related topics. Among them,the notion of ic-cone-convexlikeness seemed to be more general one[12],and was successfully applied to strict effi ciency and Henig effi ciency in set-valued optimization [13–16].

Based upon the above observation,the aim of this note is to establish the strong duality theorems with strict effi ciency for set-valued optimization problems under the ic-coneconvexlikenessassumptions.Thispaperisarranged as follows:In Section 2,some well-known definitions and results used in the sequelare recalled.In Section 3,two improved dualmodels are introduced,and strong duality theorems with strict effi ciency are established under the assumption of ic-cone-convexlikeness,respectively.

2 Preliminaries

In this paper,let X,Y and Z be real topological spaces.Let D ? Y and E ? Z be pointed convex cones,and denoted

Defi nition 2.1Let M be a nonempty subset of Y, ˉy ∈ M is called a minimize (maximize)point of M,if

The set of minimize(maximize)point of M is denoted by Min[M,D](Max[M,D]).

For a set A ? Y,we write cone(A)={λ ·a: λ ≥ 0,a ∈ A}.The closure and interior of set A is denoted by cl(A)and int(A).A convex subset B of a cone D is a base of D if 0Y/∈ cl(B)and D=cone(B).

Throughout this paper,it is always assumed that the pointed convex cone D ? Y has a base B.

Defi nition 2.2[5,13]Let M be a nonempty subset of Y, ˉy ∈ M is called a strictly minimize point of M with respect to B,if there is a neighbourhood U of 0Ysuch that

The set of strictly effi cient point of M with respect to B is denoted by Strmin[M,B].

Remark 2.1[5,13](1)With respect to the defi nition of strictly minimize points, equality(2.1)is equivalent to

Moreover,if necessary,the neighbourhood U of 0Ycan be chosen to be open,convex or balanced.

(2)Strmin[M,B]? Min[M,D].

(3)Similarly, ˉy ∈ M is called a strictly maximize point of M with respect to B,ifthere is a neighbourhood V of 0Ysuch that

Remark 2.2In Defi nition 2.2,if equality(2.1)holds,then

In fact,if not,there exist λ ＞ 0,m ∈ M,d ∈ D{0Y},u ∈ U and b ∈ B,such that λ(m ? ˉy+d)=u ? b.Since B is the base of D,there exist μ ＞ 0 and b1∈ B such that d= μ ·b1.Since B is convex set,we get that

Therefore,we can get

which contradicts equality(2.1).

Defi nition 2.3[12]The set-valued map F:X → 2Yis called ic-D-convexlike if int(cone(im(F)+D))is convex and

where im(F)is the image of F,and that is

Assume that F:X → 2Yand G:X → 2Zare set-valued maps.This note considers the following set-valued optimization problem(SOP):

The set of feasible solution of(SOP)is denoted by ?,that is

Defi nition 2.4If ˉx ∈ S and ˉy ∈ F(ˉx) ∩ Strmin￡F(S),B,then we say that(ˉx,ˉy)is a strictly effi cient solution of problem(SOP).

Let L(X,Y)be the family of(single-valued)linear continuous maps from X into Y.Let

Defi nition 2.5[13]Let F:X → 2Ybe a set-valued map, ˉx ∈ X and ˉy ∈ F(ˉx).A map T ∈ L(X,Y)is said to be a strict subgradient of F at(ˉx,ˉy)if

The set of allstrict subgradients of F at(ˉx,ˉy)is denoted by ?strF(ˉx,ˉy).

Assumption(A)[12]In problem(SOP),let ˉx ∈ S, ˉy ∈ F(ˉx)and ˉz ∈ G(ˉx) ∩ (?E). It is said that Assumption(A)is satisfied if there exists β ∈ [0,1)such that the set-valued map Hβ:=(F ? ˉx)× (G ? β ·ˉz):X → 2Y×Zis ic-D × E-convexlike.

Defi nition 2.6[12]It is said that condition(CQ)holds if cl￡cone(im G+E) =Z.

Lemma 2.7[13]Let ˉx ∈ S, ˉy ∈ F(ˉx)and ˉz ∈ G(ˉx) ∩ (?E).Let Assumption(A)and condition(CQ)be satisfied.If(ˉx,ˉy)is a strictly effi cient solution of problem(SOP),then there exists ˉT ∈ L+(Z,Y)such that ˉT(ˉz)=0Yand

3 Strong Duality

3.1 Lagrange-Wolfe Strong Duality

We firstrewrite the Lagrange dualproblem in the form similar to the Wolfe dualproblem [17],which is denoted by problem(LWD)as follows:

Denote by Q1the set of allfeasible points of(LWD),i.e.,the set of points(ξ,u,v,T) ∈X × Y × Z × L(Z,Y)satisfying(3.1)–(3.3).Let S1be the set of all points u+T(v)such that there exists ξ∈ X with(ξ,u,v,T) ∈ Q1.

Defi nition 3.1If(ξ,u,v,T) ∈ Q1,and u+T(v) ∈ Strmax￡S,B,then we say that (ξ,u,v,T)is a strictly effi cient solution of problem(LWD).

Theorem 3.2(Weak Duality)If x ∈ ? and(ξ,u,v,T) ∈ Q1,then

ProofSince x ∈ ?,it holds that G(x) ∩ (?E)/= ?.So we can take a point.Hence

On the other hand,(3.2)shows that there exists a neighbourhood U of 0Ysuch that

It follows from Remark 2.2 that

So we get(3.4),as desired.

Remark 3.1In weak duality Theorem 3.2,it follows from(3.4)and Remark 2.1 that u+T(v) ∈ min￡F(x),D.This leads to

so(3.4)means that y/≤ u+T(v), ?y ∈ F(x),which is the sense of generalweak duality in literatures[6–8].

Theorem 3.3(Strong Duality)Let ˉx ∈ X, ˉy ∈ F(ˉx)and ˉz ∈ G(ˉx) ∩ (?E).Let Assumption(A)and condition(CQ)be satisfied.If(ˉx,ˉy)is a strictly effi cient solution of problem(SOP),then there exists ˉT ∈ L+(Z,Y)such that ˉT(ˉz)=0,(ˉx,ˉy,ˉz, ˉT)is a strictly effi cient solution of(LWD),and the corresponding objective values of(SOP)and(LWD)are equal.

ProofIt yields from Lemma 2.7 that there exists ˉT ∈ L+(Z,Y)such that ˉT(ˉz)=0 and (ˉx,ˉy,ˉz, ˉT) ∈ Q1.It remains to prove that ˉy= ˉy+ ˉT(ˉz) ∈ Strmax[S1,B].In fact,otherwise there exist the neighbourhood U0of 0Ysuch that

Hence,there exist b0∈ (B ? U0), λ ＞ 0 and ?u+T(?v) ∈ S1such that b0= λ(?u+T(?v) ? ˉy) or,equivalently,

This indicates that

a contradiction to the weak duality property(3.4)with x= ˉx.

3.2 Mond-Weir Strong Duality

This subsection is devoted to construct another duality problem on the basis ofthe idea of Mond-Weir[18],called the Mond-Weir duality problem(MWD),and establish a strong duality result between(SOP)and(MWD).The next problem is named the Mond-Weir dual problem of(SOP)and is denoted by(MWD):

Denote by Q2the set ofallfeasible points of(MWD),i.e.,the set ofpoints(ξ,u,v,T) ∈X × Y × Z × L(Z,Y)satisfying(3.5)–(3.8).Let S2be the set ofallpoints u such that there exists(ξ,v,T) ∈ X × Z × L(Z,Y)with(ξ,u,v,T) ∈ Q2.

Lemma 3.4It holds that Q2? Q1and S2? S1? D.

ProofAccording to the definitions of Q1and Q2,it is obviously that Q2? Q1is satisfied.So it is to prove the second one only.Let u ∈ S2.Then there exists(ξ,v,T) ∈X × Z × L(Z,Y)such that(ξ,u,v,T) ∈ Q2? Q1is satisfied.We get that

Thus,u ∈ S1? D.This completes proof.

Theorem 3.5(Weak Duality)If x ∈ ? and(ξ,u,v,T) ∈ Q2,then there exists a neighbourhood U of 0Ysuch that

ProofBy Lemma 3.4,we obtain that Q2? Q1.Again,we get from Theorem 3.2 that there exists a neighbourhood U of 0Ysuch that

Hence it follows from Remark 2.2 that

On the other hand,it yields from(3.8)that

Combing above inquality with(3.10)yields(3.9),as required.

In order to formulating the strong duality between(SOP)and(MWD),we need propose the following Lemma 3.6.

Lemma 3.6If(ˉξ,ˉu,ˉv, ˉT)is a strictly effi cient solution of(LWD)and ˉT(ˉv)=0,then ( ˉξ,ˉu,ˉv, ˉT)is a strictly effi cient solution of(MWD)and the corresponding objective values of both problems are equal.

ProofBecause(ˉξ,ˉu,ˉv, ˉT)is a strictly effi cient solution of(LWD),it follows from the definition of set S1that there exists a neighbourhood U of 0Ysuch that

Therefore,we get from Remark 2.2 that

On the other hand,according to Lemma 3.4,we have S2? S1? D.Then we derive from ˉT(ˉv)=0 that

Together(3.11)with(3.12),it is clear thatwhich is the desired result.

Theorem 3.7(Strong Duality)Let ˉx ∈ X, ˉy ∈ F(ˉx)and ˉz ∈ G(ˉx) ∩ (?E).Let Assumption(A)and condition(CQ)be satisfied.If(ˉx,ˉy)is a strictly effi cient solution of problem(SOP),then there exists ˉT ∈ L+(Z,Y)such that ˉT(ˉz)=0,(ˉx,ˉy,ˉz, ˉT)is a strictly effi cient of(MWD),and the corresponding objective values of(SOP)and(MWD)are equal.

ProofIt follows from Lemma 2.7 that there exists ˉT ∈ L+(Z,Y)such that ˉT(ˉz)=0 and(ˉx,ˉy,ˉz, ˉT) ∈ Q2? Q1.Hence,we get from the strong duality Theorem 3.3 between (SOP)and(LWD)that(ˉx,ˉy,ˉz, ˉT)is a strictly effi cient solution of(LWD)and the corresponding objective values of(SOP)and(LWD)are equal.Therefore,it yields from Lemma 3.6 that(ˉx,ˉy,ˉz, ˉT)is also a strictly effi cient of(MWD)and the corresponding objective values of(LWD)and(MWD)are equal.This can obtain the desired results.

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非凸集值優(yōu)化問題嚴(yán)有效解的強(qiáng)對偶定理

本文研究了非凸集值向量優(yōu)化的嚴(yán)有效解在兩種對偶模型的強(qiáng)對偶問題.利用Lagrange對偶和Mond-Weir對偶原理, 獲得了如下結(jié)果: 原集值優(yōu)化問題的嚴(yán)有效解, 在一些條件下是對偶問題的強(qiáng)有效解,并且原問題和對偶問題的目標(biāo)函數(shù)值相等;推廣了集值優(yōu)化對偶理論在錐-凸假設(shè)下的相應(yīng)結(jié)果.

嚴(yán)有效性;強(qiáng)對偶;集值優(yōu)化;生成錐內(nèi)部凸-錐類凸性

類 號(hào):90C29;90C46

O224

余國林1,張 燕1,劉三陽2

(1.北方民族大學(xué)應(yīng)用數(shù)學(xué)研究所, 寧夏 銀川 750021)

(2.西安電子科技大學(xué)數(shù)學(xué)系, 陜西 西安 710071)

tion:90C29;90C46

A < class="emphasis_bold">Article ID:0255-7797(2017)02-0223-08

0255-7797(2017)02-0223-08

?Received date:2015-01-27 Accepted date:2015-09-24

Foundation item:Supported by Natural Science Foundation of China(11361001);Natual Science Foundation of Ningxia(NZ14101).

Biography:Yu Guolin(1974–),male,born at Yinchuan,Ningxia,professor,major in optimization theory and applications,nonlinear analysis.