Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 570918 9 pageshttpdxdoiorg1011552013570918

Research ArticleStrict Efficiency in Vector Optimization withNearly Convexlike Set-Valued Maps

Xiaohong Hu1 Zhimiao Fang2 and Yunxuan Xiong3

1 Department of Mathematics and Physics Chongqing University of Posts and Telecommunications Chongqing 400065 China2 Chongqing Police College Chongqing 401331 China3 Basic Course Department Nanchang Institute of Science and Technology Nanchang 330108 China

Correspondence should be addressed to Xiaohong Hu huxhcqupteducn

Received 10 December 2012 Revised 6 March 2013 Accepted 11 March 2013

Academic Editor Sheng-Jie Li

Copyright copy 2013 Xiaohong Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimizationproblem involving nearly convexlike set-valued maps Two scalarization theorems and two Lagrange multiplier theorems for strictefficiency in vector optimization involving nearly convexlike set-valuedmaps are established A dual is proposed and duality resultsare obtained in terms of strictly efficient solutions A new type of saddle point called strict saddle point of an appropriate set-valuedLagrange map is introduced and is used to characterize strict efficiency

1 Introduction

One important problem in vector optimization is to findthe efficient points of a set As observed by Kuhn Tuckerand later by Geoffrion some efficient points exhibit certainabnormal properties To eliminate such abnormal efficientpoints various concepts of proper efficiency have beenintroduced The original concept was introduced by Kuhnand Tucker [1] and Geoffrion [2] and was later modifiedand formulated in a more general framework by Borwein[3] Hartley [4] Benson [5] Henig [6] and Borwein andZhuang [7] also see the references therein In particular theconcept of strict efficiencywas first introduced byBednarczakand Song [8] in order to obtain upper semicontinuity of thesection mapping 119866(119910) = 119878 cap (119910 minus 119862) at an efficient pointZaffaroni [9] used a special scalar function to characterizethe strict efficiency and obtained some properties of strictefficiency which includes well posedness

Recently several authors have turned their interests tovector optimization of set-valued maps For instance see[10ndash16] Li [17] extended the concept of Benson properefficiency to set-valued maps and presented two scalarizationtheorems and Lagrange multiplier theorems for set-valuedvector optimization problem under cone subconvexlikeness

Mehra [18] and Xia and Qiu [19] discussed the super effi-ciency in vector optimization problem involving nearly cone-convexlike set-valued maps and nearly cone-subconvexlikeset-valued maps respectively Miglierina [20] linked theproperly efficient solutions of set-valued vector optimizationwith well-posedness hypothesis of a special scalar problem

In this paper inspired by [8 17 18] we study strictefficiency for vector optimization problem involving nearlycone-convexlike set-valued maps in the framework of realnormed locally convex spaces The paper is organized as fol-lows In Section 2 we recall some basic concepts and lemmasIn Section 3 the well posedness of a special scalar problemson strict efficiency involving nearly cone-convexlike set-valued maps is discussed In Section 4 two scalarizationtheorems for strict efficiency in vector optimization prob-lems involving nearly cone-convexlike set-valued maps areobtained In Section 5 we establish two Lagrange multipliertheorems which show that strictly efficient solution of theconstrained vector optimization problem is equivalent tostrictly efficient solution of an appropriate unconstrainedvector optimization problem In Section 6 some results onstrict duality are given In Section 7 a new concept of strictsaddle point for set-valued Lagrangianmap is introduced andis then utilized to characterize strict efficiency

2 Abstract and Applied Analysis

2 Preliminaries

Throughout this paper let 119883 be a linear space and 119884 and119885 two real normed locally convex spaces with topologicaldual spaces 119884

lowast and 119885lowast For a set 119860 sub 119884 cl119860 int119860 120597119860

and119860119888 denote the closure the interior the boundary and the

complement of 119860 respectively Moreover we will denote by119861 the closed unit ball of 119884 A set 119862 sub 119884 is said to be a cone if120582119888 isin 119862 for any 119888 isin 119862 and 120582 ge 0 A cone119862 is said to be convexif 119862 + 119862 sub 119862 and it is said to be pointed if 119862 cap (minus119862) = 0The generated cone of 119862 is defined by

cone 119862 = 120582119888 | 120582 ge 0 119888 isin 119862 (1)

The dual cone of 119862 is defined as

119862+

= 120593 isin 119884lowast

| 120593 (119888) ge 0 forall119888 isin 119862 (2)

The quasi-interior of 119862+ is the set

119862+119894

= 120593 isin 119884lowast

| 120593 (119888) gt 0 forall119888 isin 119862 0119884 (3)

Recall that a base of a cone119862 is a convex subset Θ of 119862 suchthat

0119884

notin clΘ 119862 = cone Θ (4)

Of course 119862 is pointed whenever 119862 has a base Furthermoreif 119862 is a nonempty closed convex pointed cone in 119884 then119862+119894

= 0 if and only if 119862 has a base

Definition 1 Let 119878 be a nonempty subset of 119884 The set ofall strictly efficient points and the set of all efficient pointswith respect to the convex cone119862with nonempty interior aredefined as

119878119905119864 (119878 119862) = 119910 | for every 120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (120575119861 minus 119862) sube 120598119861

119864 (119878 119862) = 119910 | (119878 minus 119910) cap minus119862 = 0119884

(5)

It is easy to verify that

119878119905119864 (119878 119862) sube 119864 (119878 119862) (6)

Also in this paper we assume that 119862 sub 119884 and 119863 sub 119885

are pointed closed convex cone with nonempty interior 119865

119883 rarr 2119884 and 119866 119883 rarr 2

119885 are set-valued maps withnonempty value Let119871(119885 119884) be the space of continuous linearoperations from 119885 to 119884 and let

119871+(119885 119884) = 119879 isin 119871 (119885 119884) | 119879 (119863) sub 119862 (7)

Let (119865 119866) be the set-valued map from 119883 to 119884 times 119885 denotedby

(119865 119866) (119909) = 119865 (119909) times 119866 (119909) (8)

If 120593 isin 119884lowast 119879 isin 119871(119885 119884) we also define 120593119865 119883 rarr 2

R and119865 + 119879119866 119883 rarr 2

119884 by (120593119865)(119909) = 120593[119865(119909)] and (119865 + 119879119866)(119909) =

119865(119909) + 119879[119866(119909)] respectively

Definition 2 (see [12]) A set-valued map 119865 119883 rarr 2119884 is said

to be nearly119862-convexlike on119883 if cl (119865(119909)+119862) is convex in 119884

Lemma 3 (see [12]) Let 119865 119883 rarr 2119884 be nearly 119862-

convexlike set-valued map on 119883 Then exactly one of thefollowing statement holds

(i) exist119909 isin 119883 such that 119865(119909) cap (minus int119862) = 0(ii) exist120593 isin 119862

+

0119884 such that (120593119865)(119909) sub R

+

Lemma 4 (see [12]) If (119865 119866) is nearly119862times119863-convexlike on119883then

(i) for each 120593 isin 119862+

0119884 (120593119865 119866) is nearly R

+times 119863-

convexlike on 119883(ii) for each 119879 isin 119871

+(119885 119884) 119865 + 119879119866 is nearly 119862-convexlike

on 119883

Lemma 5 (see [21]) Let 119878 be a closed convex subset of 119884 and119870 a closed convex pointed cone Then 119878 cap (minus119870 0

119884) = 0 if

and only if there exists 120593 isin 119870+119894 such that 120593(119904) ge 0 for all 119904 isin 119878

3 Strict Efficiency and Well Posedness

Consider the following vector optimization problemwith set-valued maps

119862-min119865 (119909)

st 119866 (119909) cap (minus119863) = 0

119909 isin 119883

(VP)

Denote the feasible solution set of (VP) by

119860 = 119909 isin 119883 119866 (119909) cap (minus119863) = 0 (9)

And denote the image of 119860 under 119865 by

119865 (119860) = ⋃

119909isin119860

119865 (119909) (10)

Definition 6 Apoint 119909 is said to be a strictly efficient solutionof (VP) if there exists 119910 isin 119865(119909) such that 119910 isin 119878119905119864[119865(119860) 119862]and the point (119909 119910) is said to be a strictly efficient minimizerof (VP)

Definition 7 For a set 119878 sube 119884 let the function Δ119878

119884 rarr

R cup plusmninfin be defined as

Δ119878(119910) = 119889

119878(119910) minus 119889

119884119878(119910) (11)

where 119889119878(119910) = inf119904 minus 119910 119904 isin 119878 with 119889

0(119910) = +infin

The function Δ was first introduced in [22] and its mainproperties are gathered together in the following proposition

Proposition 8 (see [9]) If the set 119878 is nonempty and 119878 = 119884then

(1) Δ119878is real valued

(2) Δ119878is 1-Lipschitzian

Abstract and Applied Analysis 3

(3) Δ119878(119910) lt 0 for every 119910 isin int119860 Δ

119878(119910) = 0 for every

119910 isin 120597119860 and Δ119878(119910) gt 0 for every 119910 isin int 119878119888

(4) if 119878 is closed then it holds that 119878 = 119910 Δ119878(119910) le 0

(5) if 119878 is a convex then Δ119878is convex

(6) if 119878 is a cone then Δ119878is positively homogeneous

(7) if 119878 is a closed convex cone then Δ119878is nonincreasing

with respect to the ordering relation induced on 119884

We consider the following parameterized scalar problem

minΔminus119862

(119910 minus 119910)

st 119910 isin 119865 (119860)

(P119910)

The following theorem characterize the relation betweenstrictly efficient points of (VP) and the parameterized scalarproblem (P

119910)

Theorem 9 Let 119909 isin 119860 119910 isin 119865(119909) Then (119909 119910) is astrictly efficient minimizer of (VP) if and only if there existsa nondecreasing function 120601 R

+rarr R

+with 120601(0) = 0 and

120601(119905) gt 0 for all 119905 gt 0 such that Δminus119862

(119910 minus 119910) ge 120601(119910 minus 119910) forall 119910 isin 119865(119860)

Proof Since (119909 119910) is a strictly efficient minimizer of vectoroptimization problem (VP) can be rephrased as follows forevery 120598 gt 0 there exists 120575 gt 0 such that 119889

minus119862(119910 minus 119910) ge 120575 for

every 119910 isin 119865(119860) with 119910 minus 119910 gt 120598 So suppose that the point(119909 119910) is a strictly efficientminimizer of (VP) and consider thefollowing functions

1206010(120598) = inf 119889

minus119862(119910 minus 119910) | 119910 isin 119865 (119860)

1003817100381710038171003817119910 minus 1199101003817100381710038171003817 ge 120598

120601 (120598) = min (1206010(120598) 1)

(12)

It is evident that 120601 is nondecreasing null at the origin andpositive elsewhere moreover for every 119910 isin 119865(119860) it holds that

Δminus119862

(119910 minus 119910) = 119889minus119862

(119910 minus 119910) ge 120601 (1003817100381710038171003817119910 minus 119910

1003817100381710038171003817) (13)

If on the other hand there exists a nondecreasing function 120601

with the above properties and such that Δminus119862

(119910 minus119910) ge 120601(119910 minus

119910) for all 119910 isin 119865(119860) then it holds that Δminus119862

(119910 minus 119910) = 119889minus119862

(119910 minus

119910) gt 0 for all 119910 isin 119865(119860) with 119910 = 119910 To show that (119909 119910) is astrictly efficient minimizer of (VP) for every 120598 gt 0 we canlet 120575 = inf119889

minus119862(119910 minus 119910) 119910 minus 119910 gt 120598 and it implies that the

proof is completed

The scalar problem (P119910) is Tikhonov well posed if

Δminus119862

(119910 minus 119910) gt 0 for all 119910 isin 119865(119860) with 119910 = 119910 and

119910119899isin 119865 (119860) 119889

minus119862(119910119899minus 119910) 997888rarr 0 997904rArr 119910

119899997888rarr 119910 (14)

Theorem 10 Let 119909 isin 119860 119910 isin 119865(119909) The (119909 119910) is a strictlyefficient minimizer of (VP) if and only if 119910 is a solution of (P

119910)

and the scalar problem (P119910) is Tikhonov well posed

Proof If (119909 119910) is a strictly efficient minimizer of (VP) thenby Theorem 9 119910 is the unique solution of (P

119910) and there

exists a forcing function 120601 such that Δminus119862

(119910minus119910) ge 120601(119910minus119910)for all 119910 isin 119865(119860) Since 120601(119905

119899) rarr 0 implies 119905

119899rarr 0 hence for

any sequence 119910119899isin 119865(119860) such that Δ

minus119862(119910119899minus 119910) rarr 0 then it

must converge to 119910Conversely if the scalar problem (P

119910) is Tikhonov well

posed then119889minus119862

(119910minus119910) = Δminus119862

(119910minus119910)holds for every119910 isin 119865(119860)Thus we consider the function 120601(120598) = inf119889

minus119862(119910 minus 119910) 119910 minus

119910 ge 120598 it holds by the construction that 119889minus119862

(119910minus119910 ge 120601(119910minus119910)and it is to see that 120601 is nondecreasing on [0infin) with 120601(0) = 0

and 120601(119905) gt 0 for all 119905 gt 0 Hence again by theTheorem 9 weget that (119909 119910) is a strictly efficient minimizer of (VP)

4 Strict Efficiency and Linear Scalarization

In association with the vector optimization problem (VP)involving set-valuedmaps we consider the following linearlyscalar optimization problem with a set-valued map

min (120593119865) (119909)

st 119909 isin 119860

(LSP120593)

where 120593 isin 119884lowast

0119884lowast

Definition 11 If 119909 isin 119860 119910 isin 119865(119860) and

120593 (119910) le 120593 (119910) forall119910 isin 119865 (119860) (15)

then 119909 and (119909 119910) are called a minimal solution and aminimizer of (LSP

120593) respectively

Lemma 12 Let 119910 isin 119878 sub 119884 and 119862 is a closed convex cone withhaving a compact base Θ Then 119910 is a strictly efficient point of119878 if and only if cl (119878 + 119862 minus 119910) cap minus119862 = 0

119884

Proof The definition of strict efficiency of 119878 can be rephrasedas follows for every 120598 gt 0 there exists 120575 gt 0 such that 119889

minus119862(119910minus

119910) gt 120575 for every 119910 isin 119878 with 119910 minus 119910 gt 120598 Hence if thereexists sequence 119910

119899isin 119878 119888

119899isin 119862 and some 119888 isin int 119862 such

that 119910119899+ 119888119899minus 119910 rarr minus119888 then 119910

119899+ 119888119899+ 119888 minus 119910 rarr 0 hence

119889minus119862

(119910 minus 119910) rarr 0 Since 119910119899minus 119910 = minus(119888

119899+ 119888) 119888 = 0 and 119862 is

pointed 119910119899minus 119910 is outside some small ball around the origin

This shows that 119910 is not the strictly efficient point of 119878 thiscontraction shows that cl (119878 + 119862 minus 119910) cap minus119862 = 0

119884

Conversely if 119910 is not a strictly efficient point of 119878 thenthere exists 120598 gt 0 and a sequence 119910

119899isin 119878 and 119888

119899isin 119862 such that

1003817100381710038171003817119910119899 minus 1199101003817100381710038171003817 ge 120598

1003817100381710038171003817119910119899 + 119888119899minus 119910

1003817100381710038171003817 997888rarr 0 (16)

We write 119888119899

= 120582119899120579119899with 120582

119899gt 0 and 120579

119899isin Θ then by (16)

and asΘ is compact there exists 120572 gt 0 and119873 isin 119877+ such that

120572 lt 120582119899 Indeed by (16) we have 119888

119899notin (1205982)119861 furthermore

since Θ is compact thus 120582119899does not converse to 0 and it

implies that there exists a real number 120572 gt 0 and 119873 isin R+

such that 120572 lt 120582119899for all 119899 ge 119873 Now we define 1198881015840

119899= (120582119899minus120572)120579119899

119899 ge 119873Thus we obtain

119910119899+ 1198881015840

119899minus 119910 = 119910

119899+ 119888119899minus 119910 minus 120572120579

119899997888rarr minus120572120579 = 0 (17)

4 Abstract and Applied Analysis

Hence

cl (119878 + 119862 minus 119910) cap minus119862 0119884 = 0 (18)

This contradiction shows that 119910 is a strictly efficient point of119878

Theorem 13 Let 119909 isin 119860 119910 isin 119865(119909) let 119862 have a compact baseand let 120593 isin 119862

+119894 be fixed If (119909 119910) is a minimizer of (LSP120593)

then (119909 119910) is a strictly efficient minimizer of (VP)

Proof By Lemma 12 we need only to prove that

cl (119865 (119860) + 119862 minus 119910) cap minus119862 = 0119884 (19)

Indeed let 119888 isin cl (119878 + 119862 minus 119910) cap minus119862 Then there exists 119910119899 sub

119865(119860) 119888119899 sub 119862 such that

119888 = lim119899rarr+infin

(119910119899+ 119888119899minus 119910) (20)

hence

120593 (119888) = lim119899rarr+infin

[120593 (119910119899) + 120593 (119888

119899) minus 120593 (119910)] (21)

Since (119909 119910) is a minimizer of (LSP120593) and 119910

119899isin 119865(119860) we have

120593(119910119899) ge 120593(119910) while 119888

119899isin 119862 and 120593 isin 119862

+119894 imply that 120593(119888119899) ge 0

Hence 120593(119888) ge 0 On the other hand since 119888 isin minus119862 we have120593(119888) le 0 Thus 120593(119888) = 0 Again by 120593 isin 119862

+ and 119888 isin minus119862 wemust have 119888 = 0

119884

Hence we have shown that cl (119878 + 119862 minus 119910) cap minus119862 = 0119884

Therefore this proof is completed

Theorem 14 Let 119865 be nearly 119862-convexlike on 119860 119909 isin 119860 and119910 isin 119865(119909) and let 119862 have a compact base If (119909 119910) is a strictlyefficient minimizer of (VP) then there exists 120593 isin 119862

+ such that(119909 119910) is a minimizer of (LSP

120593)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP)thus by Lemma 12 we have

minus119862 cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (22)

By the definition of nearly 119862-convexlike set-valued map119865 we have that cl [119865(119860) + 119862 minus 119910] is closed convex set in 119884since 119862 is a closed convex pointed compact cone Thus byLemma 5 there exists 120593 isin 119862

+119894 such that

120593 (cl [119865 (119860) + 119862 minus 119910]) ge 0 (23)

Since

119865 (119860) minus 119910 sub 119865 (119860) + 119862 minus 119910 sub cl [119865 (119860) + 119862 minus 119910] (24)

we obtain

120593 (119910) minus 120593 (119910) ge 0 forall119910 isin 119865 (119860) (25)

Therefore (119909 119910) is a minimizer of (LSP120593)

If we denote by 119878119905119864(VP) the set of strictly efficientminimizer of (VP) and by 119872(LSP

120593) the set of minimizer of

(LSP120593) then from Theorems 13 and 14 we get immediately

the following corollary

Corollary 15 Let 119865 be nearly 119862-convexlike on 119883 Then

119878119905119864 (119881119875) = ⋃

120593isin119862+119894

119872(119871119878119875120593) (26)

5 Strict Efficiency and Lagrange Multipliers

In this sectionwe establish twoLagrangemultiplier theoremswhich show that the set of strictly efficient minimizer of theconstrained set-valued vector optimization problem (VP) itis equivalent to the set of an appropriate unconstrained vectoroptimization problem

The following concept is a generalization of Slater con-straint qualification in mathematical programming and invector optimization

Definition 16 We say that (VP) satisfies the generalized Slaterconstraint qualification if there exists 119909 isin 119883 such that 119866(119909) cap

(minus int119863) = 0

Theorem 17 Let 119865 be nearly 119862-convexlike on119883 Let (119865 119866) benearly 119862 times 119863-convexlike on 119883 and let 119862 have a compact baseFurthermore let (VP) satisfy the generalized Slater constraintqualification If (119909 119910) is a strictly efficient minimizer of (VP)then there exists 119879 isin 119871

+(119885 119884) such that 119879[119866(119909) cap (minus119863)] =

0119884and (119909 119910) is a strictly efficient minimizer of the following

unconstrained vector optimization problem

119862-min119865 (119909) + 119879 [119866 (119909)]

119904119905 119909 isin 119883

(UVP)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP) byTheorem 14 there exists 120593 isin 119862

+119894 such that

120593 [119865 (119909) minus 119910] ge 0 forall119909 isin 119860 (27)

Define 119867 119883 rarr 2Rtimes119885 by

119867(119909) = 120593 [119865 (119909) minus 119910] times 119866 (119909) = (120593119865 119866) (119909) minus (120593 (119910 0119885))

(28)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4we have that 119867 is nearlyR

+times 119863-convexlike on 119883 while (27)

implies that

119867(119909) cap [minus int (R+times 119863)] = 0 forall119909 isin 119883 (29)

has no solution and hence by Lemma 3 there exists (120582 120595) isin

R+times 119863 0

119884lowast such that

120582120593 [119865 (119909) minus 119910] + 120595 [119866 (119909)] ge 0 forall119909 isin 119883 (30)

Since 119909 isin 119860 that is 119866(119909) cap (minus119863) = 0 this implies that thereexists 119911 isin 119866(119909) such that minus119911 isin 119863 Then since 120595 isin 119863

+ we get

120595 (119911) le 0 (31)

Also let 119909 = 119909 in (30) and noting that119910 isin 119865(119909) and 119911 isin 119866(119909)we get 120595(119911) ge 0

Abstract and Applied Analysis 5

Hence

120595 [119866 (119909) cap minus119863] = 0119884 (32)

We now claim that 120582 = 0 If this is not the case then

120595 isin 119863+

0119884lowast (33)

By the generalized Slater constraint qualification there exists119909 isin 119883 such that

119866 (119909) cap minus int119863 = 0 (34)

and so there exists isin 119866(119909) such that

minus isin int119863 (35)

Hence 120595() lt 0 But substituting 120582 = 0 into (30) and bytaking 119909 = 119909 and isin 119866(119909) in (30) we have

120595 () ge 0 (36)

This contradiction shows that 120582 gt 0 From this and 120593 isin 119862+119894

we can choose 119888 isin 119862 0119884 such that 120582120593(119888) = 1 and define the

operator 119879 119885 rarr 119884 by

119879 (119911) = 120595 (119911) 119888 (37)

Obviously

119879 isin 119871+(119885 119884) 119879 [119866 (119909) cap minus119863] = 0

119884 (38)

Thus

0119884

isin 119879 [119866 (119909)] 119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (39)

From (30) and (37) we obtain

120582120593 [119865 (119909) + 119879119866 (119909)]

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)] 120582120593 (119888)

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)]

ge 120582120593 (119910) forall119909 isin 119883

(40)

Dividing the above inequality by 120582 gt 0 we obtain

120593 [119865 (119909) + 119879119866 (119909)] ge 120593 (119910) forall119909 isin 119883 (41)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4119865+119879119866 is nearly119862-convexlike on119883Therefore byTheorem 13and120593 isin 119862

+119894 we have that (119909sdot119910) is a strictly efficientminimizerof (UVP)

Theorem 18 Let 119909 119910 isin 119865(119909) and let119862 have a compact base Ifthere exists 119879 isin 119871

+(119885 119884) such that 0

119884isin 119879[119866(119909)] and (119909 119910) is

a strictly efficient minimizer of (UVP) then (119909 119910) is a strictlyefficient minimizer of (VP)

Proof From the assumption we have

119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (42)

(minus119862) cap cl[⋃

119909isin119883

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910] = 0119884 (43)

Since 119909 isin 119860 we have 119866(119909) cap (minus119863) = 0 Thus there exists 119911119909isin

119866(119909) cap (minus119863) Then

minus119879 (119911119909) isin 119862 (44)

which implies that 119862 minus 119879(119911119909) sub 119862 that is 119862 sub 119862 + 119879(119911

119909)

forall119909 isin 119860 So we have

119865 (119860) + 119862 minus 119910 sub ⋃

119909isin119860

[119865 (119909) + 119862 minus 119910]

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)] + 119862 minus 119910)

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910

(45)

This together with (43) implies

(minus119862) cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (46)

Noting that 119909 isin 119860 119910 isin 119865(119909) sub 119865(119860) and by Lemma 12 wehave that (119909 119910) is a strictly efficient minimizer of (VP)

6 Strict Efficiency and Duality

Definition 19 The Lagrange map for (VP) is the set-valuedmapL 119883 times 119871

+(119885 119884) rarr 2

119884 defined by

L (119909 119879) = 119865 (119909) + 119879 [119866 (119909)] (47)

We denote

L (119883 119879) = ⋃

119909isin119883

L (119909 119879) = ⋃

119909isin119883

(119865 + 119879119866) (119909)

L (119909 119871+(119885 119884)) = ⋃

119879isin119871+(119885119884)

L (119909 119879)

= ⋃

119879isin119871+(119885119884)

(119865 (119909) + 119879 [119866 (119909)])

(48)

Definition 20 The set-valued map Φ 119871+(119885 119884) rarr 2

119884 isdefined as

Φ (119879) = 119878119905119864 [L (119883 119879) 119862] forall119879 isin 119871+(119885 119884) (49)

which is called a strict dual map of (VP)

Using this definition we define the Lagrange dual prob-lem associated with the primal problem (VP) as follows

119862-maxΦ (119879)

st 119879 isin 119871+(119885 119884)

(VD)

Definition 21 119910 isin ⋃119879isin119871+(119885119884)

Φ(119879) is called an efficient pointof (VD) if

119910 minus 119910 notin 119862 0119884 119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (50)

We can now establish the following dual theorems

6 Abstract and Applied Analysis

Theorem 22 (weak duality) If 119909 isin 119860 and 1199100

isin

⋃119879isin119871+(119885119884)

Φ(119879) then

[1199100minus 119865 (119909)] cap (119862 0

119884) = 0 (51)

Proof From 1199100isin ⋃119879isin119871+(119885119884)

Φ(119879) there exists 119879 isin 119871+(119885 119884)

such that

1199100isin Φ (119879)

isin 119878119905119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

sub 119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

(52)

Hence

(1199100minus 119865 (119909) minus 119879 [119866 (119909)]) cap (119862 0

119884) = 0 forall119909 isin 119883 (53)

In particular

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) forall119911 isin 119866 (119909)

(54)

Noting that 119909 isin 119860 we choose 119911 isin 119866(119909)cap(minus119863)Then minus119879(119911) isin

119862 and taking 119911 = 119911 in (54) we have

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) (55)

Hence from minus119879(119911) isin 119862 and119862+1198620119884 sube 1198620

119884 we obtain

1199100minus 119910 notin 119862 0

119884 forall119910 isin 119865 (119909) (56)

This completes the proof

Theorem 23 (strong duality) Let 119865 be nearly C-convexlike on119883 Let (119865 119866) be nearly (119862times119863)-convexlike on119883 and let 119862 havea compact base Furthermore let (VP) satisfy the generalizedSlater constraint qualification If 119909 is a strictly efficient solutionof (VP) then there exists that 119910 isin 119865(119909) is an efficient point of(VD)

Proof Since119909 is a strictly efficient solution of (VP) then thereexists119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizerof (VP) According to Theorem 17 there exists 119879 isin 119871

+(119885 119884)

such that 119879[119866(119909) cap (minus119863)] = 0119884 and (119909 119910) is a strictly

efficient minimizer of (UVP) Hence 119910 isin 119878119905119864[⋃119909isin119883

(119865(119909) +

119879[119866(119909)]) 119862] = Φ(119879) sub ⋃119879isin119871+(119885119884)

Φ(119879) By Theorem 22 wehave

(119910 minus 119910) notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (57)

Therefore by Definition 20 we know that 119910 is an efficientpoint of (VD)

7 Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle pointfor a set-valued Lagrange map L and use it to characterizestrict efficiency

For a nonempty subset 119878 of 119884 we define a set

119878119905119872 (119878 119862) = 119910 isin 119878 | forall120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (minus120575119861 + 119862) sube 120598119861

(58)

It is easy to find that 119910 isin 119878119905119872(119878 119862) if and only if minus119910 isin

119878119905119864(minus119878 119862) and if 119884 is normed space and 119862 has a compactbase Then by Lemma 12 we have 119910 isin 119878119905119872(119878 119862) if and onlyif cl (119878 minus 119862 minus 119910) cap 119862 = 0

119884

Definition 24 A pair (119909 119879) isin 119883 times 119871+(119885 119884) is said to be a

strict saddle point of Lagrange mapL if

L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

= 0

(59)

We first present an important equivalent characterizationfor a strict saddle point of the Lagrange mapL

Lemma 25 Let119862 have a compact base (119909 119879) isin 119883times119871+(119885 119884)

is said to be a strict saddle point of Lagrange map L if only ifthere exist 119910 isin 119865(119909) and 119911 isin 119866(119909) such that

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(ii) 119879(119911) = 0119884

Proof (necessity) Since (119909 119879) is a strict saddle point of themap L by Definition 24 there exists 119910 isin 119865(119909) 119911 isin 119866(119909)

such that

119910 + 119879 (119911) isin 119878119905119864 [⋃

119909isin119883

L (119909 119879) 119862] (60)

119910 + 119879 (119911) isin 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(61)

From (61) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus (119910 + 119879 (119911))]

]

= 0119884

(62)

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

2 Preliminaries

Throughout this paper let 119883 be a linear space and 119884 and119885 two real normed locally convex spaces with topologicaldual spaces 119884

lowast and 119885lowast For a set 119860 sub 119884 cl119860 int119860 120597119860

and119860119888 denote the closure the interior the boundary and the

complement of 119860 respectively Moreover we will denote by119861 the closed unit ball of 119884 A set 119862 sub 119884 is said to be a cone if120582119888 isin 119862 for any 119888 isin 119862 and 120582 ge 0 A cone119862 is said to be convexif 119862 + 119862 sub 119862 and it is said to be pointed if 119862 cap (minus119862) = 0The generated cone of 119862 is defined by

cone 119862 = 120582119888 | 120582 ge 0 119888 isin 119862 (1)

The dual cone of 119862 is defined as

119862+

= 120593 isin 119884lowast

| 120593 (119888) ge 0 forall119888 isin 119862 (2)

The quasi-interior of 119862+ is the set

119862+119894

= 120593 isin 119884lowast

| 120593 (119888) gt 0 forall119888 isin 119862 0119884 (3)

Recall that a base of a cone119862 is a convex subset Θ of 119862 suchthat

0119884

notin clΘ 119862 = cone Θ (4)

Of course 119862 is pointed whenever 119862 has a base Furthermoreif 119862 is a nonempty closed convex pointed cone in 119884 then119862+119894

= 0 if and only if 119862 has a base

Definition 1 Let 119878 be a nonempty subset of 119884 The set ofall strictly efficient points and the set of all efficient pointswith respect to the convex cone119862with nonempty interior aredefined as

119878119905119864 (119878 119862) = 119910 | for every 120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (120575119861 minus 119862) sube 120598119861

119864 (119878 119862) = 119910 | (119878 minus 119910) cap minus119862 = 0119884

(5)

It is easy to verify that

119878119905119864 (119878 119862) sube 119864 (119878 119862) (6)

Also in this paper we assume that 119862 sub 119884 and 119863 sub 119885

are pointed closed convex cone with nonempty interior 119865

119883 rarr 2119884 and 119866 119883 rarr 2

119885 are set-valued maps withnonempty value Let119871(119885 119884) be the space of continuous linearoperations from 119885 to 119884 and let

119871+(119885 119884) = 119879 isin 119871 (119885 119884) | 119879 (119863) sub 119862 (7)

Let (119865 119866) be the set-valued map from 119883 to 119884 times 119885 denotedby

(119865 119866) (119909) = 119865 (119909) times 119866 (119909) (8)

If 120593 isin 119884lowast 119879 isin 119871(119885 119884) we also define 120593119865 119883 rarr 2

R and119865 + 119879119866 119883 rarr 2

119884 by (120593119865)(119909) = 120593[119865(119909)] and (119865 + 119879119866)(119909) =

119865(119909) + 119879[119866(119909)] respectively

Definition 2 (see [12]) A set-valued map 119865 119883 rarr 2119884 is said

to be nearly119862-convexlike on119883 if cl (119865(119909)+119862) is convex in 119884

Lemma 3 (see [12]) Let 119865 119883 rarr 2119884 be nearly 119862-

convexlike set-valued map on 119883 Then exactly one of thefollowing statement holds

(i) exist119909 isin 119883 such that 119865(119909) cap (minus int119862) = 0(ii) exist120593 isin 119862

+

0119884 such that (120593119865)(119909) sub R

+

Lemma 4 (see [12]) If (119865 119866) is nearly119862times119863-convexlike on119883then

(i) for each 120593 isin 119862+

0119884 (120593119865 119866) is nearly R

+times 119863-

convexlike on 119883(ii) for each 119879 isin 119871

+(119885 119884) 119865 + 119879119866 is nearly 119862-convexlike

on 119883

Lemma 5 (see [21]) Let 119878 be a closed convex subset of 119884 and119870 a closed convex pointed cone Then 119878 cap (minus119870 0

119884) = 0 if

and only if there exists 120593 isin 119870+119894 such that 120593(119904) ge 0 for all 119904 isin 119878

3 Strict Efficiency and Well Posedness

Consider the following vector optimization problemwith set-valued maps

119862-min119865 (119909)

st 119866 (119909) cap (minus119863) = 0

119909 isin 119883

(VP)

Denote the feasible solution set of (VP) by

119860 = 119909 isin 119883 119866 (119909) cap (minus119863) = 0 (9)

And denote the image of 119860 under 119865 by

119865 (119860) = ⋃

119909isin119860

119865 (119909) (10)

Definition 6 Apoint 119909 is said to be a strictly efficient solutionof (VP) if there exists 119910 isin 119865(119909) such that 119910 isin 119878119905119864[119865(119860) 119862]and the point (119909 119910) is said to be a strictly efficient minimizerof (VP)

Definition 7 For a set 119878 sube 119884 let the function Δ119878

119884 rarr

R cup plusmninfin be defined as

Δ119878(119910) = 119889

119878(119910) minus 119889

119884119878(119910) (11)

where 119889119878(119910) = inf119904 minus 119910 119904 isin 119878 with 119889

0(119910) = +infin

The function Δ was first introduced in [22] and its mainproperties are gathered together in the following proposition

Proposition 8 (see [9]) If the set 119878 is nonempty and 119878 = 119884then

(1) Δ119878is real valued

(2) Δ119878is 1-Lipschitzian

Abstract and Applied Analysis 3

(3) Δ119878(119910) lt 0 for every 119910 isin int119860 Δ

119878(119910) = 0 for every

119910 isin 120597119860 and Δ119878(119910) gt 0 for every 119910 isin int 119878119888

(4) if 119878 is closed then it holds that 119878 = 119910 Δ119878(119910) le 0

(5) if 119878 is a convex then Δ119878is convex

(6) if 119878 is a cone then Δ119878is positively homogeneous

(7) if 119878 is a closed convex cone then Δ119878is nonincreasing

with respect to the ordering relation induced on 119884

We consider the following parameterized scalar problem

minΔminus119862

(119910 minus 119910)

st 119910 isin 119865 (119860)

(P119910)

The following theorem characterize the relation betweenstrictly efficient points of (VP) and the parameterized scalarproblem (P

119910)

Theorem 9 Let 119909 isin 119860 119910 isin 119865(119909) Then (119909 119910) is astrictly efficient minimizer of (VP) if and only if there existsa nondecreasing function 120601 R

+rarr R

+with 120601(0) = 0 and

120601(119905) gt 0 for all 119905 gt 0 such that Δminus119862

(119910 minus 119910) ge 120601(119910 minus 119910) forall 119910 isin 119865(119860)

Proof Since (119909 119910) is a strictly efficient minimizer of vectoroptimization problem (VP) can be rephrased as follows forevery 120598 gt 0 there exists 120575 gt 0 such that 119889

minus119862(119910 minus 119910) ge 120575 for

every 119910 isin 119865(119860) with 119910 minus 119910 gt 120598 So suppose that the point(119909 119910) is a strictly efficientminimizer of (VP) and consider thefollowing functions

1206010(120598) = inf 119889

minus119862(119910 minus 119910) | 119910 isin 119865 (119860)

1003817100381710038171003817119910 minus 1199101003817100381710038171003817 ge 120598

120601 (120598) = min (1206010(120598) 1)

(12)

It is evident that 120601 is nondecreasing null at the origin andpositive elsewhere moreover for every 119910 isin 119865(119860) it holds that

Δminus119862

(119910 minus 119910) = 119889minus119862

(119910 minus 119910) ge 120601 (1003817100381710038171003817119910 minus 119910

1003817100381710038171003817) (13)

If on the other hand there exists a nondecreasing function 120601

with the above properties and such that Δminus119862

(119910 minus119910) ge 120601(119910 minus

119910) for all 119910 isin 119865(119860) then it holds that Δminus119862

(119910 minus 119910) = 119889minus119862

(119910 minus

119910) gt 0 for all 119910 isin 119865(119860) with 119910 = 119910 To show that (119909 119910) is astrictly efficient minimizer of (VP) for every 120598 gt 0 we canlet 120575 = inf119889

minus119862(119910 minus 119910) 119910 minus 119910 gt 120598 and it implies that the

proof is completed

The scalar problem (P119910) is Tikhonov well posed if

Δminus119862

(119910 minus 119910) gt 0 for all 119910 isin 119865(119860) with 119910 = 119910 and

119910119899isin 119865 (119860) 119889

minus119862(119910119899minus 119910) 997888rarr 0 997904rArr 119910

119899997888rarr 119910 (14)

Theorem 10 Let 119909 isin 119860 119910 isin 119865(119909) The (119909 119910) is a strictlyefficient minimizer of (VP) if and only if 119910 is a solution of (P

119910)

and the scalar problem (P119910) is Tikhonov well posed

Proof If (119909 119910) is a strictly efficient minimizer of (VP) thenby Theorem 9 119910 is the unique solution of (P

119910) and there

exists a forcing function 120601 such that Δminus119862

(119910minus119910) ge 120601(119910minus119910)for all 119910 isin 119865(119860) Since 120601(119905

119899) rarr 0 implies 119905

119899rarr 0 hence for

any sequence 119910119899isin 119865(119860) such that Δ

minus119862(119910119899minus 119910) rarr 0 then it

must converge to 119910Conversely if the scalar problem (P

119910) is Tikhonov well

posed then119889minus119862

(119910minus119910) = Δminus119862

(119910minus119910)holds for every119910 isin 119865(119860)Thus we consider the function 120601(120598) = inf119889

minus119862(119910 minus 119910) 119910 minus

119910 ge 120598 it holds by the construction that 119889minus119862

(119910minus119910 ge 120601(119910minus119910)and it is to see that 120601 is nondecreasing on [0infin) with 120601(0) = 0

and 120601(119905) gt 0 for all 119905 gt 0 Hence again by theTheorem 9 weget that (119909 119910) is a strictly efficient minimizer of (VP)

4 Strict Efficiency and Linear Scalarization

In association with the vector optimization problem (VP)involving set-valuedmaps we consider the following linearlyscalar optimization problem with a set-valued map

min (120593119865) (119909)

st 119909 isin 119860

(LSP120593)

where 120593 isin 119884lowast

0119884lowast

Definition 11 If 119909 isin 119860 119910 isin 119865(119860) and

120593 (119910) le 120593 (119910) forall119910 isin 119865 (119860) (15)

then 119909 and (119909 119910) are called a minimal solution and aminimizer of (LSP

120593) respectively

Lemma 12 Let 119910 isin 119878 sub 119884 and 119862 is a closed convex cone withhaving a compact base Θ Then 119910 is a strictly efficient point of119878 if and only if cl (119878 + 119862 minus 119910) cap minus119862 = 0

119884

Proof The definition of strict efficiency of 119878 can be rephrasedas follows for every 120598 gt 0 there exists 120575 gt 0 such that 119889

minus119862(119910minus

119910) gt 120575 for every 119910 isin 119878 with 119910 minus 119910 gt 120598 Hence if thereexists sequence 119910

119899isin 119878 119888

119899isin 119862 and some 119888 isin int 119862 such

that 119910119899+ 119888119899minus 119910 rarr minus119888 then 119910

119899+ 119888119899+ 119888 minus 119910 rarr 0 hence

119889minus119862

(119910 minus 119910) rarr 0 Since 119910119899minus 119910 = minus(119888

119899+ 119888) 119888 = 0 and 119862 is

pointed 119910119899minus 119910 is outside some small ball around the origin

This shows that 119910 is not the strictly efficient point of 119878 thiscontraction shows that cl (119878 + 119862 minus 119910) cap minus119862 = 0

119884

Conversely if 119910 is not a strictly efficient point of 119878 thenthere exists 120598 gt 0 and a sequence 119910

119899isin 119878 and 119888

119899isin 119862 such that

1003817100381710038171003817119910119899 minus 1199101003817100381710038171003817 ge 120598

1003817100381710038171003817119910119899 + 119888119899minus 119910

1003817100381710038171003817 997888rarr 0 (16)

We write 119888119899

= 120582119899120579119899with 120582

119899gt 0 and 120579

119899isin Θ then by (16)

and asΘ is compact there exists 120572 gt 0 and119873 isin 119877+ such that

120572 lt 120582119899 Indeed by (16) we have 119888

119899notin (1205982)119861 furthermore

since Θ is compact thus 120582119899does not converse to 0 and it

implies that there exists a real number 120572 gt 0 and 119873 isin R+

such that 120572 lt 120582119899for all 119899 ge 119873 Now we define 1198881015840

119899= (120582119899minus120572)120579119899

119899 ge 119873Thus we obtain

119910119899+ 1198881015840

119899minus 119910 = 119910

119899+ 119888119899minus 119910 minus 120572120579

119899997888rarr minus120572120579 = 0 (17)

4 Abstract and Applied Analysis

Hence

cl (119878 + 119862 minus 119910) cap minus119862 0119884 = 0 (18)

This contradiction shows that 119910 is a strictly efficient point of119878

Theorem 13 Let 119909 isin 119860 119910 isin 119865(119909) let 119862 have a compact baseand let 120593 isin 119862

+119894 be fixed If (119909 119910) is a minimizer of (LSP120593)

then (119909 119910) is a strictly efficient minimizer of (VP)

Proof By Lemma 12 we need only to prove that

cl (119865 (119860) + 119862 minus 119910) cap minus119862 = 0119884 (19)

Indeed let 119888 isin cl (119878 + 119862 minus 119910) cap minus119862 Then there exists 119910119899 sub

119865(119860) 119888119899 sub 119862 such that

119888 = lim119899rarr+infin

(119910119899+ 119888119899minus 119910) (20)

hence

120593 (119888) = lim119899rarr+infin

[120593 (119910119899) + 120593 (119888

119899) minus 120593 (119910)] (21)

Since (119909 119910) is a minimizer of (LSP120593) and 119910

119899isin 119865(119860) we have

120593(119910119899) ge 120593(119910) while 119888

119899isin 119862 and 120593 isin 119862

+119894 imply that 120593(119888119899) ge 0

Hence 120593(119888) ge 0 On the other hand since 119888 isin minus119862 we have120593(119888) le 0 Thus 120593(119888) = 0 Again by 120593 isin 119862

+ and 119888 isin minus119862 wemust have 119888 = 0

119884

Hence we have shown that cl (119878 + 119862 minus 119910) cap minus119862 = 0119884

Therefore this proof is completed

Theorem 14 Let 119865 be nearly 119862-convexlike on 119860 119909 isin 119860 and119910 isin 119865(119909) and let 119862 have a compact base If (119909 119910) is a strictlyefficient minimizer of (VP) then there exists 120593 isin 119862

+ such that(119909 119910) is a minimizer of (LSP

120593)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP)thus by Lemma 12 we have

minus119862 cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (22)

By the definition of nearly 119862-convexlike set-valued map119865 we have that cl [119865(119860) + 119862 minus 119910] is closed convex set in 119884since 119862 is a closed convex pointed compact cone Thus byLemma 5 there exists 120593 isin 119862

+119894 such that

120593 (cl [119865 (119860) + 119862 minus 119910]) ge 0 (23)

Since

119865 (119860) minus 119910 sub 119865 (119860) + 119862 minus 119910 sub cl [119865 (119860) + 119862 minus 119910] (24)

we obtain

120593 (119910) minus 120593 (119910) ge 0 forall119910 isin 119865 (119860) (25)

Therefore (119909 119910) is a minimizer of (LSP120593)

If we denote by 119878119905119864(VP) the set of strictly efficientminimizer of (VP) and by 119872(LSP

120593) the set of minimizer of

(LSP120593) then from Theorems 13 and 14 we get immediately

the following corollary

Corollary 15 Let 119865 be nearly 119862-convexlike on 119883 Then

119878119905119864 (119881119875) = ⋃

120593isin119862+119894

119872(119871119878119875120593) (26)

5 Strict Efficiency and Lagrange Multipliers

In this sectionwe establish twoLagrangemultiplier theoremswhich show that the set of strictly efficient minimizer of theconstrained set-valued vector optimization problem (VP) itis equivalent to the set of an appropriate unconstrained vectoroptimization problem

The following concept is a generalization of Slater con-straint qualification in mathematical programming and invector optimization

Definition 16 We say that (VP) satisfies the generalized Slaterconstraint qualification if there exists 119909 isin 119883 such that 119866(119909) cap

(minus int119863) = 0

Theorem 17 Let 119865 be nearly 119862-convexlike on119883 Let (119865 119866) benearly 119862 times 119863-convexlike on 119883 and let 119862 have a compact baseFurthermore let (VP) satisfy the generalized Slater constraintqualification If (119909 119910) is a strictly efficient minimizer of (VP)then there exists 119879 isin 119871

+(119885 119884) such that 119879[119866(119909) cap (minus119863)] =

0119884and (119909 119910) is a strictly efficient minimizer of the following

unconstrained vector optimization problem

119862-min119865 (119909) + 119879 [119866 (119909)]

119904119905 119909 isin 119883

(UVP)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP) byTheorem 14 there exists 120593 isin 119862

+119894 such that

120593 [119865 (119909) minus 119910] ge 0 forall119909 isin 119860 (27)

Define 119867 119883 rarr 2Rtimes119885 by

119867(119909) = 120593 [119865 (119909) minus 119910] times 119866 (119909) = (120593119865 119866) (119909) minus (120593 (119910 0119885))

(28)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4we have that 119867 is nearlyR

+times 119863-convexlike on 119883 while (27)

implies that

119867(119909) cap [minus int (R+times 119863)] = 0 forall119909 isin 119883 (29)

has no solution and hence by Lemma 3 there exists (120582 120595) isin

R+times 119863 0

119884lowast such that

120582120593 [119865 (119909) minus 119910] + 120595 [119866 (119909)] ge 0 forall119909 isin 119883 (30)

Since 119909 isin 119860 that is 119866(119909) cap (minus119863) = 0 this implies that thereexists 119911 isin 119866(119909) such that minus119911 isin 119863 Then since 120595 isin 119863

+ we get

120595 (119911) le 0 (31)

Also let 119909 = 119909 in (30) and noting that119910 isin 119865(119909) and 119911 isin 119866(119909)we get 120595(119911) ge 0

Abstract and Applied Analysis 5

Hence

120595 [119866 (119909) cap minus119863] = 0119884 (32)

We now claim that 120582 = 0 If this is not the case then

120595 isin 119863+

0119884lowast (33)

By the generalized Slater constraint qualification there exists119909 isin 119883 such that

119866 (119909) cap minus int119863 = 0 (34)

and so there exists isin 119866(119909) such that

minus isin int119863 (35)

Hence 120595() lt 0 But substituting 120582 = 0 into (30) and bytaking 119909 = 119909 and isin 119866(119909) in (30) we have

120595 () ge 0 (36)

This contradiction shows that 120582 gt 0 From this and 120593 isin 119862+119894

we can choose 119888 isin 119862 0119884 such that 120582120593(119888) = 1 and define the

operator 119879 119885 rarr 119884 by

119879 (119911) = 120595 (119911) 119888 (37)

Obviously

119879 isin 119871+(119885 119884) 119879 [119866 (119909) cap minus119863] = 0

119884 (38)

Thus

0119884

isin 119879 [119866 (119909)] 119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (39)

From (30) and (37) we obtain

120582120593 [119865 (119909) + 119879119866 (119909)]

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)] 120582120593 (119888)

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)]

ge 120582120593 (119910) forall119909 isin 119883

(40)

Dividing the above inequality by 120582 gt 0 we obtain

120593 [119865 (119909) + 119879119866 (119909)] ge 120593 (119910) forall119909 isin 119883 (41)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4119865+119879119866 is nearly119862-convexlike on119883Therefore byTheorem 13and120593 isin 119862

+119894 we have that (119909sdot119910) is a strictly efficientminimizerof (UVP)

Theorem 18 Let 119909 119910 isin 119865(119909) and let119862 have a compact base Ifthere exists 119879 isin 119871

+(119885 119884) such that 0

119884isin 119879[119866(119909)] and (119909 119910) is

a strictly efficient minimizer of (UVP) then (119909 119910) is a strictlyefficient minimizer of (VP)

Proof From the assumption we have

119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (42)

(minus119862) cap cl[⋃

119909isin119883

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910] = 0119884 (43)

Since 119909 isin 119860 we have 119866(119909) cap (minus119863) = 0 Thus there exists 119911119909isin

119866(119909) cap (minus119863) Then

minus119879 (119911119909) isin 119862 (44)

which implies that 119862 minus 119879(119911119909) sub 119862 that is 119862 sub 119862 + 119879(119911

119909)

forall119909 isin 119860 So we have

119865 (119860) + 119862 minus 119910 sub ⋃

119909isin119860

[119865 (119909) + 119862 minus 119910]

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)] + 119862 minus 119910)

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910

(45)

This together with (43) implies

(minus119862) cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (46)

Noting that 119909 isin 119860 119910 isin 119865(119909) sub 119865(119860) and by Lemma 12 wehave that (119909 119910) is a strictly efficient minimizer of (VP)

6 Strict Efficiency and Duality

Definition 19 The Lagrange map for (VP) is the set-valuedmapL 119883 times 119871

+(119885 119884) rarr 2

119884 defined by

L (119909 119879) = 119865 (119909) + 119879 [119866 (119909)] (47)

We denote

L (119883 119879) = ⋃

119909isin119883

L (119909 119879) = ⋃

119909isin119883

(119865 + 119879119866) (119909)

L (119909 119871+(119885 119884)) = ⋃

119879isin119871+(119885119884)

L (119909 119879)

= ⋃

119879isin119871+(119885119884)

(119865 (119909) + 119879 [119866 (119909)])

(48)

Definition 20 The set-valued map Φ 119871+(119885 119884) rarr 2

119884 isdefined as

Φ (119879) = 119878119905119864 [L (119883 119879) 119862] forall119879 isin 119871+(119885 119884) (49)

which is called a strict dual map of (VP)

Using this definition we define the Lagrange dual prob-lem associated with the primal problem (VP) as follows

119862-maxΦ (119879)

st 119879 isin 119871+(119885 119884)

(VD)

Definition 21 119910 isin ⋃119879isin119871+(119885119884)

Φ(119879) is called an efficient pointof (VD) if

119910 minus 119910 notin 119862 0119884 119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (50)

We can now establish the following dual theorems

6 Abstract and Applied Analysis

Theorem 22 (weak duality) If 119909 isin 119860 and 1199100

isin

⋃119879isin119871+(119885119884)

Φ(119879) then

[1199100minus 119865 (119909)] cap (119862 0

119884) = 0 (51)

Proof From 1199100isin ⋃119879isin119871+(119885119884)

Φ(119879) there exists 119879 isin 119871+(119885 119884)

such that

1199100isin Φ (119879)

isin 119878119905119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

sub 119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

(52)

Hence

(1199100minus 119865 (119909) minus 119879 [119866 (119909)]) cap (119862 0

119884) = 0 forall119909 isin 119883 (53)

In particular

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) forall119911 isin 119866 (119909)

(54)

Noting that 119909 isin 119860 we choose 119911 isin 119866(119909)cap(minus119863)Then minus119879(119911) isin

119862 and taking 119911 = 119911 in (54) we have

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) (55)

Hence from minus119879(119911) isin 119862 and119862+1198620119884 sube 1198620

119884 we obtain

1199100minus 119910 notin 119862 0

119884 forall119910 isin 119865 (119909) (56)

This completes the proof

Theorem 23 (strong duality) Let 119865 be nearly C-convexlike on119883 Let (119865 119866) be nearly (119862times119863)-convexlike on119883 and let 119862 havea compact base Furthermore let (VP) satisfy the generalizedSlater constraint qualification If 119909 is a strictly efficient solutionof (VP) then there exists that 119910 isin 119865(119909) is an efficient point of(VD)

Proof Since119909 is a strictly efficient solution of (VP) then thereexists119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizerof (VP) According to Theorem 17 there exists 119879 isin 119871

+(119885 119884)

such that 119879[119866(119909) cap (minus119863)] = 0119884 and (119909 119910) is a strictly

efficient minimizer of (UVP) Hence 119910 isin 119878119905119864[⋃119909isin119883

(119865(119909) +

119879[119866(119909)]) 119862] = Φ(119879) sub ⋃119879isin119871+(119885119884)

Φ(119879) By Theorem 22 wehave

(119910 minus 119910) notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (57)

Therefore by Definition 20 we know that 119910 is an efficientpoint of (VD)

7 Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle pointfor a set-valued Lagrange map L and use it to characterizestrict efficiency

For a nonempty subset 119878 of 119884 we define a set

119878119905119872 (119878 119862) = 119910 isin 119878 | forall120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (minus120575119861 + 119862) sube 120598119861

(58)

It is easy to find that 119910 isin 119878119905119872(119878 119862) if and only if minus119910 isin

119878119905119864(minus119878 119862) and if 119884 is normed space and 119862 has a compactbase Then by Lemma 12 we have 119910 isin 119878119905119872(119878 119862) if and onlyif cl (119878 minus 119862 minus 119910) cap 119862 = 0

119884

Definition 24 A pair (119909 119879) isin 119883 times 119871+(119885 119884) is said to be a

strict saddle point of Lagrange mapL if

L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

= 0

(59)

We first present an important equivalent characterizationfor a strict saddle point of the Lagrange mapL

Lemma 25 Let119862 have a compact base (119909 119879) isin 119883times119871+(119885 119884)

is said to be a strict saddle point of Lagrange map L if only ifthere exist 119910 isin 119865(119909) and 119911 isin 119866(119909) such that

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(ii) 119879(119911) = 0119884

Proof (necessity) Since (119909 119879) is a strict saddle point of themap L by Definition 24 there exists 119910 isin 119865(119909) 119911 isin 119866(119909)

such that

119910 + 119879 (119911) isin 119878119905119864 [⋃

119909isin119883

L (119909 119879) 119862] (60)

119910 + 119879 (119911) isin 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(61)

From (61) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus (119910 + 119879 (119911))]

]

= 0119884

(62)

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

(3) Δ119878(119910) lt 0 for every 119910 isin int119860 Δ

119878(119910) = 0 for every

119910 isin 120597119860 and Δ119878(119910) gt 0 for every 119910 isin int 119878119888

(4) if 119878 is closed then it holds that 119878 = 119910 Δ119878(119910) le 0

(5) if 119878 is a convex then Δ119878is convex

(6) if 119878 is a cone then Δ119878is positively homogeneous

(7) if 119878 is a closed convex cone then Δ119878is nonincreasing

with respect to the ordering relation induced on 119884

We consider the following parameterized scalar problem

minΔminus119862

(119910 minus 119910)

st 119910 isin 119865 (119860)

(P119910)

The following theorem characterize the relation betweenstrictly efficient points of (VP) and the parameterized scalarproblem (P

119910)

Theorem 9 Let 119909 isin 119860 119910 isin 119865(119909) Then (119909 119910) is astrictly efficient minimizer of (VP) if and only if there existsa nondecreasing function 120601 R

+rarr R

+with 120601(0) = 0 and

120601(119905) gt 0 for all 119905 gt 0 such that Δminus119862

(119910 minus 119910) ge 120601(119910 minus 119910) forall 119910 isin 119865(119860)

Proof Since (119909 119910) is a strictly efficient minimizer of vectoroptimization problem (VP) can be rephrased as follows forevery 120598 gt 0 there exists 120575 gt 0 such that 119889

minus119862(119910 minus 119910) ge 120575 for

every 119910 isin 119865(119860) with 119910 minus 119910 gt 120598 So suppose that the point(119909 119910) is a strictly efficientminimizer of (VP) and consider thefollowing functions

1206010(120598) = inf 119889

minus119862(119910 minus 119910) | 119910 isin 119865 (119860)

1003817100381710038171003817119910 minus 1199101003817100381710038171003817 ge 120598

120601 (120598) = min (1206010(120598) 1)

(12)

It is evident that 120601 is nondecreasing null at the origin andpositive elsewhere moreover for every 119910 isin 119865(119860) it holds that

Δminus119862

(119910 minus 119910) = 119889minus119862

(119910 minus 119910) ge 120601 (1003817100381710038171003817119910 minus 119910

1003817100381710038171003817) (13)

If on the other hand there exists a nondecreasing function 120601

with the above properties and such that Δminus119862

(119910 minus119910) ge 120601(119910 minus

119910) for all 119910 isin 119865(119860) then it holds that Δminus119862

(119910 minus 119910) = 119889minus119862

(119910 minus

119910) gt 0 for all 119910 isin 119865(119860) with 119910 = 119910 To show that (119909 119910) is astrictly efficient minimizer of (VP) for every 120598 gt 0 we canlet 120575 = inf119889

minus119862(119910 minus 119910) 119910 minus 119910 gt 120598 and it implies that the

proof is completed

The scalar problem (P119910) is Tikhonov well posed if

Δminus119862

(119910 minus 119910) gt 0 for all 119910 isin 119865(119860) with 119910 = 119910 and

119910119899isin 119865 (119860) 119889

minus119862(119910119899minus 119910) 997888rarr 0 997904rArr 119910

119899997888rarr 119910 (14)

Theorem 10 Let 119909 isin 119860 119910 isin 119865(119909) The (119909 119910) is a strictlyefficient minimizer of (VP) if and only if 119910 is a solution of (P

119910)

and the scalar problem (P119910) is Tikhonov well posed

Proof If (119909 119910) is a strictly efficient minimizer of (VP) thenby Theorem 9 119910 is the unique solution of (P

119910) and there

exists a forcing function 120601 such that Δminus119862

(119910minus119910) ge 120601(119910minus119910)for all 119910 isin 119865(119860) Since 120601(119905

119899) rarr 0 implies 119905

119899rarr 0 hence for

any sequence 119910119899isin 119865(119860) such that Δ

minus119862(119910119899minus 119910) rarr 0 then it

must converge to 119910Conversely if the scalar problem (P

119910) is Tikhonov well

posed then119889minus119862

(119910minus119910) = Δminus119862

(119910minus119910)holds for every119910 isin 119865(119860)Thus we consider the function 120601(120598) = inf119889

minus119862(119910 minus 119910) 119910 minus

119910 ge 120598 it holds by the construction that 119889minus119862

(119910minus119910 ge 120601(119910minus119910)and it is to see that 120601 is nondecreasing on [0infin) with 120601(0) = 0

and 120601(119905) gt 0 for all 119905 gt 0 Hence again by theTheorem 9 weget that (119909 119910) is a strictly efficient minimizer of (VP)

4 Strict Efficiency and Linear Scalarization

In association with the vector optimization problem (VP)involving set-valuedmaps we consider the following linearlyscalar optimization problem with a set-valued map

min (120593119865) (119909)

st 119909 isin 119860

(LSP120593)

where 120593 isin 119884lowast

0119884lowast

Definition 11 If 119909 isin 119860 119910 isin 119865(119860) and

120593 (119910) le 120593 (119910) forall119910 isin 119865 (119860) (15)

then 119909 and (119909 119910) are called a minimal solution and aminimizer of (LSP

120593) respectively

Lemma 12 Let 119910 isin 119878 sub 119884 and 119862 is a closed convex cone withhaving a compact base Θ Then 119910 is a strictly efficient point of119878 if and only if cl (119878 + 119862 minus 119910) cap minus119862 = 0

119884

Proof The definition of strict efficiency of 119878 can be rephrasedas follows for every 120598 gt 0 there exists 120575 gt 0 such that 119889

minus119862(119910minus

119910) gt 120575 for every 119910 isin 119878 with 119910 minus 119910 gt 120598 Hence if thereexists sequence 119910

119899isin 119878 119888

119899isin 119862 and some 119888 isin int 119862 such

that 119910119899+ 119888119899minus 119910 rarr minus119888 then 119910

119899+ 119888119899+ 119888 minus 119910 rarr 0 hence

119889minus119862

(119910 minus 119910) rarr 0 Since 119910119899minus 119910 = minus(119888

119899+ 119888) 119888 = 0 and 119862 is

pointed 119910119899minus 119910 is outside some small ball around the origin

This shows that 119910 is not the strictly efficient point of 119878 thiscontraction shows that cl (119878 + 119862 minus 119910) cap minus119862 = 0

119884

Conversely if 119910 is not a strictly efficient point of 119878 thenthere exists 120598 gt 0 and a sequence 119910

119899isin 119878 and 119888

119899isin 119862 such that

1003817100381710038171003817119910119899 minus 1199101003817100381710038171003817 ge 120598

1003817100381710038171003817119910119899 + 119888119899minus 119910

1003817100381710038171003817 997888rarr 0 (16)

We write 119888119899

= 120582119899120579119899with 120582

119899gt 0 and 120579

119899isin Θ then by (16)

and asΘ is compact there exists 120572 gt 0 and119873 isin 119877+ such that

120572 lt 120582119899 Indeed by (16) we have 119888

119899notin (1205982)119861 furthermore

since Θ is compact thus 120582119899does not converse to 0 and it

implies that there exists a real number 120572 gt 0 and 119873 isin R+

such that 120572 lt 120582119899for all 119899 ge 119873 Now we define 1198881015840

119899= (120582119899minus120572)120579119899

119899 ge 119873Thus we obtain

119910119899+ 1198881015840

119899minus 119910 = 119910

119899+ 119888119899minus 119910 minus 120572120579

119899997888rarr minus120572120579 = 0 (17)

4 Abstract and Applied Analysis

Hence

cl (119878 + 119862 minus 119910) cap minus119862 0119884 = 0 (18)

This contradiction shows that 119910 is a strictly efficient point of119878

Theorem 13 Let 119909 isin 119860 119910 isin 119865(119909) let 119862 have a compact baseand let 120593 isin 119862

+119894 be fixed If (119909 119910) is a minimizer of (LSP120593)

then (119909 119910) is a strictly efficient minimizer of (VP)

Proof By Lemma 12 we need only to prove that

cl (119865 (119860) + 119862 minus 119910) cap minus119862 = 0119884 (19)

Indeed let 119888 isin cl (119878 + 119862 minus 119910) cap minus119862 Then there exists 119910119899 sub

119865(119860) 119888119899 sub 119862 such that

119888 = lim119899rarr+infin

(119910119899+ 119888119899minus 119910) (20)

hence

120593 (119888) = lim119899rarr+infin

[120593 (119910119899) + 120593 (119888

119899) minus 120593 (119910)] (21)

Since (119909 119910) is a minimizer of (LSP120593) and 119910

119899isin 119865(119860) we have

120593(119910119899) ge 120593(119910) while 119888

119899isin 119862 and 120593 isin 119862

+119894 imply that 120593(119888119899) ge 0

Hence 120593(119888) ge 0 On the other hand since 119888 isin minus119862 we have120593(119888) le 0 Thus 120593(119888) = 0 Again by 120593 isin 119862

+ and 119888 isin minus119862 wemust have 119888 = 0

119884

Hence we have shown that cl (119878 + 119862 minus 119910) cap minus119862 = 0119884

Therefore this proof is completed

Theorem 14 Let 119865 be nearly 119862-convexlike on 119860 119909 isin 119860 and119910 isin 119865(119909) and let 119862 have a compact base If (119909 119910) is a strictlyefficient minimizer of (VP) then there exists 120593 isin 119862

+ such that(119909 119910) is a minimizer of (LSP

120593)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP)thus by Lemma 12 we have

minus119862 cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (22)

By the definition of nearly 119862-convexlike set-valued map119865 we have that cl [119865(119860) + 119862 minus 119910] is closed convex set in 119884since 119862 is a closed convex pointed compact cone Thus byLemma 5 there exists 120593 isin 119862

+119894 such that

120593 (cl [119865 (119860) + 119862 minus 119910]) ge 0 (23)

Since

119865 (119860) minus 119910 sub 119865 (119860) + 119862 minus 119910 sub cl [119865 (119860) + 119862 minus 119910] (24)

we obtain

120593 (119910) minus 120593 (119910) ge 0 forall119910 isin 119865 (119860) (25)

Therefore (119909 119910) is a minimizer of (LSP120593)

If we denote by 119878119905119864(VP) the set of strictly efficientminimizer of (VP) and by 119872(LSP

120593) the set of minimizer of

(LSP120593) then from Theorems 13 and 14 we get immediately

the following corollary

Corollary 15 Let 119865 be nearly 119862-convexlike on 119883 Then

119878119905119864 (119881119875) = ⋃

120593isin119862+119894

119872(119871119878119875120593) (26)

5 Strict Efficiency and Lagrange Multipliers

In this sectionwe establish twoLagrangemultiplier theoremswhich show that the set of strictly efficient minimizer of theconstrained set-valued vector optimization problem (VP) itis equivalent to the set of an appropriate unconstrained vectoroptimization problem

The following concept is a generalization of Slater con-straint qualification in mathematical programming and invector optimization

Definition 16 We say that (VP) satisfies the generalized Slaterconstraint qualification if there exists 119909 isin 119883 such that 119866(119909) cap

(minus int119863) = 0

Theorem 17 Let 119865 be nearly 119862-convexlike on119883 Let (119865 119866) benearly 119862 times 119863-convexlike on 119883 and let 119862 have a compact baseFurthermore let (VP) satisfy the generalized Slater constraintqualification If (119909 119910) is a strictly efficient minimizer of (VP)then there exists 119879 isin 119871

+(119885 119884) such that 119879[119866(119909) cap (minus119863)] =

0119884and (119909 119910) is a strictly efficient minimizer of the following

unconstrained vector optimization problem

119862-min119865 (119909) + 119879 [119866 (119909)]

119904119905 119909 isin 119883

(UVP)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP) byTheorem 14 there exists 120593 isin 119862

+119894 such that

120593 [119865 (119909) minus 119910] ge 0 forall119909 isin 119860 (27)

Define 119867 119883 rarr 2Rtimes119885 by

119867(119909) = 120593 [119865 (119909) minus 119910] times 119866 (119909) = (120593119865 119866) (119909) minus (120593 (119910 0119885))

(28)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4we have that 119867 is nearlyR

+times 119863-convexlike on 119883 while (27)

implies that

119867(119909) cap [minus int (R+times 119863)] = 0 forall119909 isin 119883 (29)

has no solution and hence by Lemma 3 there exists (120582 120595) isin

R+times 119863 0

119884lowast such that

120582120593 [119865 (119909) minus 119910] + 120595 [119866 (119909)] ge 0 forall119909 isin 119883 (30)

Since 119909 isin 119860 that is 119866(119909) cap (minus119863) = 0 this implies that thereexists 119911 isin 119866(119909) such that minus119911 isin 119863 Then since 120595 isin 119863

+ we get

120595 (119911) le 0 (31)

Also let 119909 = 119909 in (30) and noting that119910 isin 119865(119909) and 119911 isin 119866(119909)we get 120595(119911) ge 0

Abstract and Applied Analysis 5

Hence

120595 [119866 (119909) cap minus119863] = 0119884 (32)

We now claim that 120582 = 0 If this is not the case then

120595 isin 119863+

0119884lowast (33)

By the generalized Slater constraint qualification there exists119909 isin 119883 such that

119866 (119909) cap minus int119863 = 0 (34)

and so there exists isin 119866(119909) such that

minus isin int119863 (35)

Hence 120595() lt 0 But substituting 120582 = 0 into (30) and bytaking 119909 = 119909 and isin 119866(119909) in (30) we have

120595 () ge 0 (36)

This contradiction shows that 120582 gt 0 From this and 120593 isin 119862+119894

we can choose 119888 isin 119862 0119884 such that 120582120593(119888) = 1 and define the

operator 119879 119885 rarr 119884 by

119879 (119911) = 120595 (119911) 119888 (37)

Obviously

119879 isin 119871+(119885 119884) 119879 [119866 (119909) cap minus119863] = 0

119884 (38)

Thus

0119884

isin 119879 [119866 (119909)] 119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (39)

From (30) and (37) we obtain

120582120593 [119865 (119909) + 119879119866 (119909)]

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)] 120582120593 (119888)

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)]

ge 120582120593 (119910) forall119909 isin 119883

(40)

Dividing the above inequality by 120582 gt 0 we obtain

120593 [119865 (119909) + 119879119866 (119909)] ge 120593 (119910) forall119909 isin 119883 (41)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4119865+119879119866 is nearly119862-convexlike on119883Therefore byTheorem 13and120593 isin 119862

+119894 we have that (119909sdot119910) is a strictly efficientminimizerof (UVP)

Theorem 18 Let 119909 119910 isin 119865(119909) and let119862 have a compact base Ifthere exists 119879 isin 119871

+(119885 119884) such that 0

119884isin 119879[119866(119909)] and (119909 119910) is

a strictly efficient minimizer of (UVP) then (119909 119910) is a strictlyefficient minimizer of (VP)

Proof From the assumption we have

119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (42)

(minus119862) cap cl[⋃

119909isin119883

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910] = 0119884 (43)

Since 119909 isin 119860 we have 119866(119909) cap (minus119863) = 0 Thus there exists 119911119909isin

119866(119909) cap (minus119863) Then

minus119879 (119911119909) isin 119862 (44)

which implies that 119862 minus 119879(119911119909) sub 119862 that is 119862 sub 119862 + 119879(119911

119909)

forall119909 isin 119860 So we have

119865 (119860) + 119862 minus 119910 sub ⋃

119909isin119860

[119865 (119909) + 119862 minus 119910]

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)] + 119862 minus 119910)

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910

(45)

This together with (43) implies

(minus119862) cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (46)

Noting that 119909 isin 119860 119910 isin 119865(119909) sub 119865(119860) and by Lemma 12 wehave that (119909 119910) is a strictly efficient minimizer of (VP)

6 Strict Efficiency and Duality

Definition 19 The Lagrange map for (VP) is the set-valuedmapL 119883 times 119871

+(119885 119884) rarr 2

119884 defined by

L (119909 119879) = 119865 (119909) + 119879 [119866 (119909)] (47)

We denote

L (119883 119879) = ⋃

119909isin119883

L (119909 119879) = ⋃

119909isin119883

(119865 + 119879119866) (119909)

L (119909 119871+(119885 119884)) = ⋃

119879isin119871+(119885119884)

L (119909 119879)

= ⋃

119879isin119871+(119885119884)

(119865 (119909) + 119879 [119866 (119909)])

(48)

Definition 20 The set-valued map Φ 119871+(119885 119884) rarr 2

119884 isdefined as

Φ (119879) = 119878119905119864 [L (119883 119879) 119862] forall119879 isin 119871+(119885 119884) (49)

which is called a strict dual map of (VP)

Using this definition we define the Lagrange dual prob-lem associated with the primal problem (VP) as follows

119862-maxΦ (119879)

st 119879 isin 119871+(119885 119884)

(VD)

Definition 21 119910 isin ⋃119879isin119871+(119885119884)

Φ(119879) is called an efficient pointof (VD) if

119910 minus 119910 notin 119862 0119884 119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (50)

We can now establish the following dual theorems

6 Abstract and Applied Analysis

Theorem 22 (weak duality) If 119909 isin 119860 and 1199100

isin

⋃119879isin119871+(119885119884)

Φ(119879) then

[1199100minus 119865 (119909)] cap (119862 0

119884) = 0 (51)

Proof From 1199100isin ⋃119879isin119871+(119885119884)

Φ(119879) there exists 119879 isin 119871+(119885 119884)

such that

1199100isin Φ (119879)

isin 119878119905119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

sub 119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

(52)

Hence

(1199100minus 119865 (119909) minus 119879 [119866 (119909)]) cap (119862 0

119884) = 0 forall119909 isin 119883 (53)

In particular

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) forall119911 isin 119866 (119909)

(54)

Noting that 119909 isin 119860 we choose 119911 isin 119866(119909)cap(minus119863)Then minus119879(119911) isin

119862 and taking 119911 = 119911 in (54) we have

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) (55)

Hence from minus119879(119911) isin 119862 and119862+1198620119884 sube 1198620

119884 we obtain

1199100minus 119910 notin 119862 0

119884 forall119910 isin 119865 (119909) (56)

This completes the proof

Theorem 23 (strong duality) Let 119865 be nearly C-convexlike on119883 Let (119865 119866) be nearly (119862times119863)-convexlike on119883 and let 119862 havea compact base Furthermore let (VP) satisfy the generalizedSlater constraint qualification If 119909 is a strictly efficient solutionof (VP) then there exists that 119910 isin 119865(119909) is an efficient point of(VD)

Proof Since119909 is a strictly efficient solution of (VP) then thereexists119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizerof (VP) According to Theorem 17 there exists 119879 isin 119871

+(119885 119884)

such that 119879[119866(119909) cap (minus119863)] = 0119884 and (119909 119910) is a strictly

efficient minimizer of (UVP) Hence 119910 isin 119878119905119864[⋃119909isin119883

(119865(119909) +

119879[119866(119909)]) 119862] = Φ(119879) sub ⋃119879isin119871+(119885119884)

Φ(119879) By Theorem 22 wehave

(119910 minus 119910) notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (57)

Therefore by Definition 20 we know that 119910 is an efficientpoint of (VD)

7 Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle pointfor a set-valued Lagrange map L and use it to characterizestrict efficiency

For a nonempty subset 119878 of 119884 we define a set

119878119905119872 (119878 119862) = 119910 isin 119878 | forall120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (minus120575119861 + 119862) sube 120598119861

(58)

It is easy to find that 119910 isin 119878119905119872(119878 119862) if and only if minus119910 isin

119878119905119864(minus119878 119862) and if 119884 is normed space and 119862 has a compactbase Then by Lemma 12 we have 119910 isin 119878119905119872(119878 119862) if and onlyif cl (119878 minus 119862 minus 119910) cap 119862 = 0

119884

Definition 24 A pair (119909 119879) isin 119883 times 119871+(119885 119884) is said to be a

strict saddle point of Lagrange mapL if

L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

= 0

(59)

We first present an important equivalent characterizationfor a strict saddle point of the Lagrange mapL

Lemma 25 Let119862 have a compact base (119909 119879) isin 119883times119871+(119885 119884)

is said to be a strict saddle point of Lagrange map L if only ifthere exist 119910 isin 119865(119909) and 119911 isin 119866(119909) such that

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(ii) 119879(119911) = 0119884

Proof (necessity) Since (119909 119879) is a strict saddle point of themap L by Definition 24 there exists 119910 isin 119865(119909) 119911 isin 119866(119909)

such that

119910 + 119879 (119911) isin 119878119905119864 [⋃

119909isin119883

L (119909 119879) 119862] (60)

119910 + 119879 (119911) isin 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(61)

From (61) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus (119910 + 119879 (119911))]

]

= 0119884

(62)

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Hence

cl (119878 + 119862 minus 119910) cap minus119862 0119884 = 0 (18)

This contradiction shows that 119910 is a strictly efficient point of119878

Theorem 13 Let 119909 isin 119860 119910 isin 119865(119909) let 119862 have a compact baseand let 120593 isin 119862

+119894 be fixed If (119909 119910) is a minimizer of (LSP120593)

then (119909 119910) is a strictly efficient minimizer of (VP)

Proof By Lemma 12 we need only to prove that

cl (119865 (119860) + 119862 minus 119910) cap minus119862 = 0119884 (19)

Indeed let 119888 isin cl (119878 + 119862 minus 119910) cap minus119862 Then there exists 119910119899 sub

119865(119860) 119888119899 sub 119862 such that

119888 = lim119899rarr+infin

(119910119899+ 119888119899minus 119910) (20)

hence

120593 (119888) = lim119899rarr+infin

[120593 (119910119899) + 120593 (119888

119899) minus 120593 (119910)] (21)

Since (119909 119910) is a minimizer of (LSP120593) and 119910

119899isin 119865(119860) we have

120593(119910119899) ge 120593(119910) while 119888

119899isin 119862 and 120593 isin 119862

+119894 imply that 120593(119888119899) ge 0

Hence 120593(119888) ge 0 On the other hand since 119888 isin minus119862 we have120593(119888) le 0 Thus 120593(119888) = 0 Again by 120593 isin 119862

+ and 119888 isin minus119862 wemust have 119888 = 0

119884

Hence we have shown that cl (119878 + 119862 minus 119910) cap minus119862 = 0119884

Therefore this proof is completed

Theorem 14 Let 119865 be nearly 119862-convexlike on 119860 119909 isin 119860 and119910 isin 119865(119909) and let 119862 have a compact base If (119909 119910) is a strictlyefficient minimizer of (VP) then there exists 120593 isin 119862

+ such that(119909 119910) is a minimizer of (LSP

120593)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP)thus by Lemma 12 we have

minus119862 cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (22)

By the definition of nearly 119862-convexlike set-valued map119865 we have that cl [119865(119860) + 119862 minus 119910] is closed convex set in 119884since 119862 is a closed convex pointed compact cone Thus byLemma 5 there exists 120593 isin 119862

+119894 such that

120593 (cl [119865 (119860) + 119862 minus 119910]) ge 0 (23)

Since

119865 (119860) minus 119910 sub 119865 (119860) + 119862 minus 119910 sub cl [119865 (119860) + 119862 minus 119910] (24)

we obtain

120593 (119910) minus 120593 (119910) ge 0 forall119910 isin 119865 (119860) (25)

Therefore (119909 119910) is a minimizer of (LSP120593)

If we denote by 119878119905119864(VP) the set of strictly efficientminimizer of (VP) and by 119872(LSP

120593) the set of minimizer of

(LSP120593) then from Theorems 13 and 14 we get immediately

the following corollary

Corollary 15 Let 119865 be nearly 119862-convexlike on 119883 Then

119878119905119864 (119881119875) = ⋃

120593isin119862+119894

119872(119871119878119875120593) (26)

5 Strict Efficiency and Lagrange Multipliers

In this sectionwe establish twoLagrangemultiplier theoremswhich show that the set of strictly efficient minimizer of theconstrained set-valued vector optimization problem (VP) itis equivalent to the set of an appropriate unconstrained vectoroptimization problem

The following concept is a generalization of Slater con-straint qualification in mathematical programming and invector optimization

Definition 16 We say that (VP) satisfies the generalized Slaterconstraint qualification if there exists 119909 isin 119883 such that 119866(119909) cap

(minus int119863) = 0

Theorem 17 Let 119865 be nearly 119862-convexlike on119883 Let (119865 119866) benearly 119862 times 119863-convexlike on 119883 and let 119862 have a compact baseFurthermore let (VP) satisfy the generalized Slater constraintqualification If (119909 119910) is a strictly efficient minimizer of (VP)then there exists 119879 isin 119871

+(119885 119884) such that 119879[119866(119909) cap (minus119863)] =

0119884and (119909 119910) is a strictly efficient minimizer of the following

unconstrained vector optimization problem

119862-min119865 (119909) + 119879 [119866 (119909)]

119904119905 119909 isin 119883

(UVP)

Proof Since (119909 119910) is a strictly efficient minimizer of (VP) byTheorem 14 there exists 120593 isin 119862

+119894 such that

120593 [119865 (119909) minus 119910] ge 0 forall119909 isin 119860 (27)

Define 119867 119883 rarr 2Rtimes119885 by

119867(119909) = 120593 [119865 (119909) minus 119910] times 119866 (119909) = (120593119865 119866) (119909) minus (120593 (119910 0119885))

(28)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4we have that 119867 is nearlyR

+times 119863-convexlike on 119883 while (27)

implies that

119867(119909) cap [minus int (R+times 119863)] = 0 forall119909 isin 119883 (29)

has no solution and hence by Lemma 3 there exists (120582 120595) isin

R+times 119863 0

119884lowast such that

120582120593 [119865 (119909) minus 119910] + 120595 [119866 (119909)] ge 0 forall119909 isin 119883 (30)

Since 119909 isin 119860 that is 119866(119909) cap (minus119863) = 0 this implies that thereexists 119911 isin 119866(119909) such that minus119911 isin 119863 Then since 120595 isin 119863

+ we get

120595 (119911) le 0 (31)

Also let 119909 = 119909 in (30) and noting that119910 isin 119865(119909) and 119911 isin 119866(119909)we get 120595(119911) ge 0

Abstract and Applied Analysis 5

Hence

120595 [119866 (119909) cap minus119863] = 0119884 (32)

We now claim that 120582 = 0 If this is not the case then

120595 isin 119863+

0119884lowast (33)

By the generalized Slater constraint qualification there exists119909 isin 119883 such that

119866 (119909) cap minus int119863 = 0 (34)

and so there exists isin 119866(119909) such that

minus isin int119863 (35)

Hence 120595() lt 0 But substituting 120582 = 0 into (30) and bytaking 119909 = 119909 and isin 119866(119909) in (30) we have

120595 () ge 0 (36)

This contradiction shows that 120582 gt 0 From this and 120593 isin 119862+119894

we can choose 119888 isin 119862 0119884 such that 120582120593(119888) = 1 and define the

operator 119879 119885 rarr 119884 by

119879 (119911) = 120595 (119911) 119888 (37)

Obviously

119879 isin 119871+(119885 119884) 119879 [119866 (119909) cap minus119863] = 0

119884 (38)

Thus

0119884

isin 119879 [119866 (119909)] 119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (39)

From (30) and (37) we obtain

120582120593 [119865 (119909) + 119879119866 (119909)]

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)] 120582120593 (119888)

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)]

ge 120582120593 (119910) forall119909 isin 119883

(40)

Dividing the above inequality by 120582 gt 0 we obtain

120593 [119865 (119909) + 119879119866 (119909)] ge 120593 (119910) forall119909 isin 119883 (41)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4119865+119879119866 is nearly119862-convexlike on119883Therefore byTheorem 13and120593 isin 119862

+119894 we have that (119909sdot119910) is a strictly efficientminimizerof (UVP)

Theorem 18 Let 119909 119910 isin 119865(119909) and let119862 have a compact base Ifthere exists 119879 isin 119871

+(119885 119884) such that 0

119884isin 119879[119866(119909)] and (119909 119910) is

a strictly efficient minimizer of (UVP) then (119909 119910) is a strictlyefficient minimizer of (VP)

Proof From the assumption we have

119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (42)

(minus119862) cap cl[⋃

119909isin119883

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910] = 0119884 (43)

Since 119909 isin 119860 we have 119866(119909) cap (minus119863) = 0 Thus there exists 119911119909isin

119866(119909) cap (minus119863) Then

minus119879 (119911119909) isin 119862 (44)

which implies that 119862 minus 119879(119911119909) sub 119862 that is 119862 sub 119862 + 119879(119911

119909)

forall119909 isin 119860 So we have

119865 (119860) + 119862 minus 119910 sub ⋃

119909isin119860

[119865 (119909) + 119862 minus 119910]

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)] + 119862 minus 119910)

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910

(45)

This together with (43) implies

(minus119862) cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (46)

Noting that 119909 isin 119860 119910 isin 119865(119909) sub 119865(119860) and by Lemma 12 wehave that (119909 119910) is a strictly efficient minimizer of (VP)

6 Strict Efficiency and Duality

Definition 19 The Lagrange map for (VP) is the set-valuedmapL 119883 times 119871

+(119885 119884) rarr 2

119884 defined by

L (119909 119879) = 119865 (119909) + 119879 [119866 (119909)] (47)

We denote

L (119883 119879) = ⋃

119909isin119883

L (119909 119879) = ⋃

119909isin119883

(119865 + 119879119866) (119909)

L (119909 119871+(119885 119884)) = ⋃

119879isin119871+(119885119884)

L (119909 119879)

= ⋃

119879isin119871+(119885119884)

(119865 (119909) + 119879 [119866 (119909)])

(48)

Definition 20 The set-valued map Φ 119871+(119885 119884) rarr 2

119884 isdefined as

Φ (119879) = 119878119905119864 [L (119883 119879) 119862] forall119879 isin 119871+(119885 119884) (49)

which is called a strict dual map of (VP)

Using this definition we define the Lagrange dual prob-lem associated with the primal problem (VP) as follows

119862-maxΦ (119879)

st 119879 isin 119871+(119885 119884)

(VD)

Definition 21 119910 isin ⋃119879isin119871+(119885119884)

Φ(119879) is called an efficient pointof (VD) if

119910 minus 119910 notin 119862 0119884 119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (50)

We can now establish the following dual theorems

6 Abstract and Applied Analysis

Theorem 22 (weak duality) If 119909 isin 119860 and 1199100

isin

⋃119879isin119871+(119885119884)

Φ(119879) then

[1199100minus 119865 (119909)] cap (119862 0

119884) = 0 (51)

Proof From 1199100isin ⋃119879isin119871+(119885119884)

Φ(119879) there exists 119879 isin 119871+(119885 119884)

such that

1199100isin Φ (119879)

isin 119878119905119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

sub 119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

(52)

Hence

(1199100minus 119865 (119909) minus 119879 [119866 (119909)]) cap (119862 0

119884) = 0 forall119909 isin 119883 (53)

In particular

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) forall119911 isin 119866 (119909)

(54)

Noting that 119909 isin 119860 we choose 119911 isin 119866(119909)cap(minus119863)Then minus119879(119911) isin

119862 and taking 119911 = 119911 in (54) we have

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) (55)

Hence from minus119879(119911) isin 119862 and119862+1198620119884 sube 1198620

119884 we obtain

1199100minus 119910 notin 119862 0

119884 forall119910 isin 119865 (119909) (56)

This completes the proof

Theorem 23 (strong duality) Let 119865 be nearly C-convexlike on119883 Let (119865 119866) be nearly (119862times119863)-convexlike on119883 and let 119862 havea compact base Furthermore let (VP) satisfy the generalizedSlater constraint qualification If 119909 is a strictly efficient solutionof (VP) then there exists that 119910 isin 119865(119909) is an efficient point of(VD)

Proof Since119909 is a strictly efficient solution of (VP) then thereexists119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizerof (VP) According to Theorem 17 there exists 119879 isin 119871

+(119885 119884)

such that 119879[119866(119909) cap (minus119863)] = 0119884 and (119909 119910) is a strictly

efficient minimizer of (UVP) Hence 119910 isin 119878119905119864[⋃119909isin119883

(119865(119909) +

119879[119866(119909)]) 119862] = Φ(119879) sub ⋃119879isin119871+(119885119884)

Φ(119879) By Theorem 22 wehave

(119910 minus 119910) notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (57)

Therefore by Definition 20 we know that 119910 is an efficientpoint of (VD)

7 Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle pointfor a set-valued Lagrange map L and use it to characterizestrict efficiency

For a nonempty subset 119878 of 119884 we define a set

119878119905119872 (119878 119862) = 119910 isin 119878 | forall120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (minus120575119861 + 119862) sube 120598119861

(58)

It is easy to find that 119910 isin 119878119905119872(119878 119862) if and only if minus119910 isin

119878119905119864(minus119878 119862) and if 119884 is normed space and 119862 has a compactbase Then by Lemma 12 we have 119910 isin 119878119905119872(119878 119862) if and onlyif cl (119878 minus 119862 minus 119910) cap 119862 = 0

119884

Definition 24 A pair (119909 119879) isin 119883 times 119871+(119885 119884) is said to be a

strict saddle point of Lagrange mapL if

L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

= 0

(59)

We first present an important equivalent characterizationfor a strict saddle point of the Lagrange mapL

Lemma 25 Let119862 have a compact base (119909 119879) isin 119883times119871+(119885 119884)

is said to be a strict saddle point of Lagrange map L if only ifthere exist 119910 isin 119865(119909) and 119911 isin 119866(119909) such that

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(ii) 119879(119911) = 0119884

Proof (necessity) Since (119909 119879) is a strict saddle point of themap L by Definition 24 there exists 119910 isin 119865(119909) 119911 isin 119866(119909)

such that

119910 + 119879 (119911) isin 119878119905119864 [⋃

119909isin119883

L (119909 119879) 119862] (60)

119910 + 119879 (119911) isin 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(61)

From (61) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus (119910 + 119879 (119911))]

]

= 0119884

(62)

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Hence

120595 [119866 (119909) cap minus119863] = 0119884 (32)

We now claim that 120582 = 0 If this is not the case then

120595 isin 119863+

0119884lowast (33)

By the generalized Slater constraint qualification there exists119909 isin 119883 such that

119866 (119909) cap minus int119863 = 0 (34)

and so there exists isin 119866(119909) such that

minus isin int119863 (35)

Hence 120595() lt 0 But substituting 120582 = 0 into (30) and bytaking 119909 = 119909 and isin 119866(119909) in (30) we have

120595 () ge 0 (36)

This contradiction shows that 120582 gt 0 From this and 120593 isin 119862+119894

we can choose 119888 isin 119862 0119884 such that 120582120593(119888) = 1 and define the

operator 119879 119885 rarr 119884 by

119879 (119911) = 120595 (119911) 119888 (37)

Obviously

119879 isin 119871+(119885 119884) 119879 [119866 (119909) cap minus119863] = 0

119884 (38)

Thus

0119884

isin 119879 [119866 (119909)] 119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (39)

From (30) and (37) we obtain

120582120593 [119865 (119909) + 119879119866 (119909)]

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)] 120582120593 (119888)

= 120582120593 [119865 (119909)] + 120595 [119866 (119909)]

ge 120582120593 (119910) forall119909 isin 119883

(40)

Dividing the above inequality by 120582 gt 0 we obtain

120593 [119865 (119909) + 119879119866 (119909)] ge 120593 (119910) forall119909 isin 119883 (41)

Since (119865 119866) is nearly 119862 times 119863-convexlike on 119883 by Lemma 4119865+119879119866 is nearly119862-convexlike on119883Therefore byTheorem 13and120593 isin 119862

+119894 we have that (119909sdot119910) is a strictly efficientminimizerof (UVP)

Theorem 18 Let 119909 119910 isin 119865(119909) and let119862 have a compact base Ifthere exists 119879 isin 119871

+(119885 119884) such that 0

119884isin 119879[119866(119909)] and (119909 119910) is

a strictly efficient minimizer of (UVP) then (119909 119910) is a strictlyefficient minimizer of (VP)

Proof From the assumption we have

119910 isin 119865 (119909) sub 119865 (119909) + 119879 [119866 (119909)] (42)

(minus119862) cap cl[⋃

119909isin119883

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910] = 0119884 (43)

Since 119909 isin 119860 we have 119866(119909) cap (minus119863) = 0 Thus there exists 119911119909isin

119866(119909) cap (minus119863) Then

minus119879 (119911119909) isin 119862 (44)

which implies that 119862 minus 119879(119911119909) sub 119862 that is 119862 sub 119862 + 119879(119911

119909)

forall119909 isin 119860 So we have

119865 (119860) + 119862 minus 119910 sub ⋃

119909isin119860

[119865 (119909) + 119862 minus 119910]

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)] + 119862 minus 119910)

sub ⋃

119909isin119860

(119865 (119909) + 119879 [119866 (119909)]) + 119862 minus 119910

(45)

This together with (43) implies

(minus119862) cap cl [119865 (119860) + 119862 minus 119910] = 0119884 (46)

Noting that 119909 isin 119860 119910 isin 119865(119909) sub 119865(119860) and by Lemma 12 wehave that (119909 119910) is a strictly efficient minimizer of (VP)

6 Strict Efficiency and Duality

Definition 19 The Lagrange map for (VP) is the set-valuedmapL 119883 times 119871

+(119885 119884) rarr 2

119884 defined by

L (119909 119879) = 119865 (119909) + 119879 [119866 (119909)] (47)

We denote

L (119883 119879) = ⋃

119909isin119883

L (119909 119879) = ⋃

119909isin119883

(119865 + 119879119866) (119909)

L (119909 119871+(119885 119884)) = ⋃

119879isin119871+(119885119884)

L (119909 119879)

= ⋃

119879isin119871+(119885119884)

(119865 (119909) + 119879 [119866 (119909)])

(48)

Definition 20 The set-valued map Φ 119871+(119885 119884) rarr 2

119884 isdefined as

Φ (119879) = 119878119905119864 [L (119883 119879) 119862] forall119879 isin 119871+(119885 119884) (49)

which is called a strict dual map of (VP)

Using this definition we define the Lagrange dual prob-lem associated with the primal problem (VP) as follows

119862-maxΦ (119879)

st 119879 isin 119871+(119885 119884)

(VD)

Definition 21 119910 isin ⋃119879isin119871+(119885119884)

Φ(119879) is called an efficient pointof (VD) if

119910 minus 119910 notin 119862 0119884 119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (50)

We can now establish the following dual theorems

6 Abstract and Applied Analysis

Theorem 22 (weak duality) If 119909 isin 119860 and 1199100

isin

⋃119879isin119871+(119885119884)

Φ(119879) then

[1199100minus 119865 (119909)] cap (119862 0

119884) = 0 (51)

Proof From 1199100isin ⋃119879isin119871+(119885119884)

Φ(119879) there exists 119879 isin 119871+(119885 119884)

such that

1199100isin Φ (119879)

isin 119878119905119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

sub 119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

(52)

Hence

(1199100minus 119865 (119909) minus 119879 [119866 (119909)]) cap (119862 0

119884) = 0 forall119909 isin 119883 (53)

In particular

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) forall119911 isin 119866 (119909)

(54)

Noting that 119909 isin 119860 we choose 119911 isin 119866(119909)cap(minus119863)Then minus119879(119911) isin

119862 and taking 119911 = 119911 in (54) we have

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) (55)

Hence from minus119879(119911) isin 119862 and119862+1198620119884 sube 1198620

119884 we obtain

1199100minus 119910 notin 119862 0

119884 forall119910 isin 119865 (119909) (56)

This completes the proof

Theorem 23 (strong duality) Let 119865 be nearly C-convexlike on119883 Let (119865 119866) be nearly (119862times119863)-convexlike on119883 and let 119862 havea compact base Furthermore let (VP) satisfy the generalizedSlater constraint qualification If 119909 is a strictly efficient solutionof (VP) then there exists that 119910 isin 119865(119909) is an efficient point of(VD)

Proof Since119909 is a strictly efficient solution of (VP) then thereexists119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizerof (VP) According to Theorem 17 there exists 119879 isin 119871

+(119885 119884)

such that 119879[119866(119909) cap (minus119863)] = 0119884 and (119909 119910) is a strictly

efficient minimizer of (UVP) Hence 119910 isin 119878119905119864[⋃119909isin119883

(119865(119909) +

119879[119866(119909)]) 119862] = Φ(119879) sub ⋃119879isin119871+(119885119884)

Φ(119879) By Theorem 22 wehave

(119910 minus 119910) notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (57)

Therefore by Definition 20 we know that 119910 is an efficientpoint of (VD)

7 Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle pointfor a set-valued Lagrange map L and use it to characterizestrict efficiency

For a nonempty subset 119878 of 119884 we define a set

119878119905119872 (119878 119862) = 119910 isin 119878 | forall120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (minus120575119861 + 119862) sube 120598119861

(58)

It is easy to find that 119910 isin 119878119905119872(119878 119862) if and only if minus119910 isin

119878119905119864(minus119878 119862) and if 119884 is normed space and 119862 has a compactbase Then by Lemma 12 we have 119910 isin 119878119905119872(119878 119862) if and onlyif cl (119878 minus 119862 minus 119910) cap 119862 = 0

119884

Definition 24 A pair (119909 119879) isin 119883 times 119871+(119885 119884) is said to be a

strict saddle point of Lagrange mapL if

L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

= 0

(59)

We first present an important equivalent characterizationfor a strict saddle point of the Lagrange mapL

Lemma 25 Let119862 have a compact base (119909 119879) isin 119883times119871+(119885 119884)

is said to be a strict saddle point of Lagrange map L if only ifthere exist 119910 isin 119865(119909) and 119911 isin 119866(119909) such that

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(ii) 119879(119911) = 0119884

Proof (necessity) Since (119909 119879) is a strict saddle point of themap L by Definition 24 there exists 119910 isin 119865(119909) 119911 isin 119866(119909)

such that

119910 + 119879 (119911) isin 119878119905119864 [⋃

119909isin119883

L (119909 119879) 119862] (60)

119910 + 119879 (119911) isin 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(61)

From (61) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus (119910 + 119879 (119911))]

]

= 0119884

(62)

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Theorem 22 (weak duality) If 119909 isin 119860 and 1199100

isin

⋃119879isin119871+(119885119884)

Φ(119879) then

[1199100minus 119865 (119909)] cap (119862 0

119884) = 0 (51)

Proof From 1199100isin ⋃119879isin119871+(119885119884)

Φ(119879) there exists 119879 isin 119871+(119885 119884)

such that

1199100isin Φ (119879)

isin 119878119905119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

sub 119864(⋃

119909isin119883

[119865 (119909) + 119879119866 (119909)] 119862)

(52)

Hence

(1199100minus 119865 (119909) minus 119879 [119866 (119909)]) cap (119862 0

119884) = 0 forall119909 isin 119883 (53)

In particular

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) forall119911 isin 119866 (119909)

(54)

Noting that 119909 isin 119860 we choose 119911 isin 119866(119909)cap(minus119863)Then minus119879(119911) isin

119862 and taking 119911 = 119911 in (54) we have

1199100minus 119910 minus 119879 (119911) notin 119862 0

119884 forall119910 isin 119865 (119909) (55)

Hence from minus119879(119911) isin 119862 and119862+1198620119884 sube 1198620

119884 we obtain

1199100minus 119910 notin 119862 0

119884 forall119910 isin 119865 (119909) (56)

This completes the proof

Theorem 23 (strong duality) Let 119865 be nearly C-convexlike on119883 Let (119865 119866) be nearly (119862times119863)-convexlike on119883 and let 119862 havea compact base Furthermore let (VP) satisfy the generalizedSlater constraint qualification If 119909 is a strictly efficient solutionof (VP) then there exists that 119910 isin 119865(119909) is an efficient point of(VD)

Proof Since119909 is a strictly efficient solution of (VP) then thereexists119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizerof (VP) According to Theorem 17 there exists 119879 isin 119871

+(119885 119884)

such that 119879[119866(119909) cap (minus119863)] = 0119884 and (119909 119910) is a strictly

efficient minimizer of (UVP) Hence 119910 isin 119878119905119864[⋃119909isin119883

(119865(119909) +

119879[119866(119909)]) 119862] = Φ(119879) sub ⋃119879isin119871+(119885119884)

Φ(119879) By Theorem 22 wehave

(119910 minus 119910) notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

Φ (119879) (57)

Therefore by Definition 20 we know that 119910 is an efficientpoint of (VD)

7 Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle pointfor a set-valued Lagrange map L and use it to characterizestrict efficiency

For a nonempty subset 119878 of 119884 we define a set

119878119905119872 (119878 119862) = 119910 isin 119878 | forall120598 gt 0 exist120575 gt 0

such that (119878 minus 119910) cap (minus120575119861 + 119862) sube 120598119861

(58)

It is easy to find that 119910 isin 119878119905119872(119878 119862) if and only if minus119910 isin

119878119905119864(minus119878 119862) and if 119884 is normed space and 119862 has a compactbase Then by Lemma 12 we have 119910 isin 119878119905119872(119878 119862) if and onlyif cl (119878 minus 119862 minus 119910) cap 119862 = 0

119884

Definition 24 A pair (119909 119879) isin 119883 times 119871+(119885 119884) is said to be a

strict saddle point of Lagrange mapL if

L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

= 0

(59)

We first present an important equivalent characterizationfor a strict saddle point of the Lagrange mapL

Lemma 25 Let119862 have a compact base (119909 119879) isin 119883times119871+(119885 119884)

is said to be a strict saddle point of Lagrange map L if only ifthere exist 119910 isin 119865(119909) and 119911 isin 119866(119909) such that

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(ii) 119879(119911) = 0119884

Proof (necessity) Since (119909 119879) is a strict saddle point of themap L by Definition 24 there exists 119910 isin 119865(119909) 119911 isin 119866(119909)

such that

119910 + 119879 (119911) isin 119878119905119864 [⋃

119909isin119883

L (119909 119879) 119862] (60)

119910 + 119879 (119911) isin 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(61)

From (61) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus (119910 + 119879 (119911))]

]

= 0119884

(62)

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Since every 119879 isin 119871+(119885 119884) we have

119879 (119911) minus 119879 (119911)

= [119910 + 119879 (119911)] minus [119910 + 119879 (119911)]

isin 119865 (119909) + 119879 [119866 (119909)] minus [119910 + 119879 (119911)]

= L (119909 119879) minus [119910 + 119879 (119911)]

(63)

We have

119879 (119911) 119879 isin 119871+(119885 119884) minus 119862 minus 119879 (119911)

sub ⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)] (64)

Hence

cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

sub cl[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) minus 119862 minus [119910 + 119879 (119911)]]

]

(65)

Thus from (62) we have

119862 cap cl[

[

⋃

119879isin119871+(119885119884)

119879 (119911) minus 119862 minus 119879 (119911)]

]

= 0119884 (66)

Let 119891 119871(119885 119884) rarr 119884 be defined by

119891 (119879) = minus119879 (119911) (67)

Then (66) can be written as

(minus119862) cap cl [119891 (119871+(119885 119884)) + 119862 minus 119891 (119879)] = 0

119884 (68)

This together with Lemma 12 shows that 119879 isin 119871+(119885 119884) is a

strictly efficient point of the vector optimization problem

119862-min119891 (119879)

st 119879 isin 119871+(119885 119884)

(69)

Since 119891 is a linear map then of course minus119891 is nearly 119862-convexlike on 119871

+(119885 119884) Hence by Theorem 14 there exists

120593 isin 119862+119894 such that

120593 [minus119879 (119911)] = 120593 [119891 (119879)] le 120593 [119891 (119879)]

= 120593 [minus119879 (119911)] forall119879 isin 119871+(119885 119884)

(70)

Now we claim that

minus119911 isin 119863 (71)

If this is not true then since 119863 is a closed convex cone setby the strong separation theorem in topological vector space[21] there exists 120583 isin 119885

lowast

0119885lowast such that

120583 (minus119911) lt 120583 (120582119889) forall119889 isin 119863 forall120582 gt 0 (72)

In the above expression taking 119889 = 0119911isin 119863 gets

120583 (119911) gt 0 (73)

while letting 120582 rarr +infin leads to

120583 (119889) ge 0 forall119889 isin 119863 (74)

Hence

120583 isin 119863+

0119885lowast (75)

Let 119888lowast isin int119862 be fixed and define 119879lowast

119885 rarr 119884 as

119879lowast

(119911) = [120583 (119911)

120583 (119911)] 119888lowast

+ 119879 (119911) (76)

It is evident that 119879lowast isin 119871(119885 119884) and that

119879lowast

(119889) = [120583 (119889)

120583 (119911119888lowast)] + 119879 (119889) isin 119862 + 119862 sub 119862 forall119889 isin 119863 (77)

Hence 119879lowast isin 119871+(119885 119884) Taking 119911 = 119911 in (66) we obtain

119879lowast

(119911) minus 119879 (119911) = 119888lowast

(78)

Hence

120593 [119879lowast

(119911)] minus 120593 [119879 (119911)] = 120593 (119888lowast

) gt 0 (79)

which contradicts (70) Therefore

minus119911 isin 119863 (80)

Thus minus119879(119911) isin 119862 since 119879 isin 119871+(119885 119884) If 119879(119911) = 0

119884 then

minus119879 (119911) isin 119862 0119884 (81)

hence 120593[119879(119911)] lt 0 by 120593 isin 119862+119894 while taking 119879 = 0 isin

119871+(119885 119884) leads to

120593 (119879 (119911)) ge 0 (82)

This contradiction shows that 119879(119911) = 0119884 that is condition

(ii) holdsTherefore by (60) and (61) we know

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(83)

that is condition (i) holds

Sufficiency From 119910 isin 119865(119909) 119911 isin 119866(119909) and condition (ii) weget

119910 = 119910 + 119879 (119911) isin 119865 (119909) + 119879 [119866 (119909)] = L (119909 119879) (84)

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

And by condition (i) we obtain

119910 isin L (119909 119879) cap 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

cap 119878119905119872[

[

⋃

119879isin119871+(119885119884)

L (119909 119879) 119862]

]

(85)

Therefore (119909 119879) is a strict saddle point of the set-valuedLagrange mapL and the proof is complete

The following saddle-point theorems allow us to expressa strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange mapL

Theorem 26 Let 119865 be nearly 119862-convexlike on 119860 let (119865 119866) benearly (119862 times 119863)-convexlike on 119860 and let 119862 have a compactbase Moreover (VP) satisfies generalized Slater constrainedqualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119910 isin 119878119905119872[⋃

119879isin119871+(119885119884)

L(119909 119879) 119862] then there exists 119879 isin

119871+(119885 119884) such that (119909 119879) is a strict saddle point of the

mapL

Proof (i) By the necessity of Lemma 25 we have

0119884

isin 119879 [119866 (119909)] (86)

and there exists 119910 isin 119865(119909) such that (119909 119910) is a strictly efficientminimizer of the problem

119862-min119865 (119909) + 119879 [119866 (119909)]

st 119909 isin 119883

(UVP)

According to Theorem 18 (119909 119910) is a strictly efficientminimizer of (VP) Therefore 119909 is strictly efficient solutionof (VP)

(ii) From the assumption and by the Theorem 17 thereexists 119879 isin 119871

+(119885 119884) such that

119910 isin 119878119905119864[⋃

119909isin119883

L (119909 119879) 119862]

119879 [119866 (119909) cap (minus119863)] = 0119884

(87)

Therefore there exists 119911 isin 119866(119909) such that 119879(119911) = 0119884 Hence

from Lemma 25 it follows that (119909 119879) is a strict saddle pointof the mapL

Lemma 27 Let (119909 119879) isin 119883 times 119871+(119885 119884) 119910 isin 119865(119909) and 119911 isin

119866(119909) Then the following conditions

(a) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap

119878119905119872[⋃119879isin119871+(119885119884)

L(119909 119879) 119862]

(b) 119879(119911) = 0119884

are equivalent to the following conditions

(i) 119910 isin 119878119905119864[⋃119909isin119883

L(119909 119879) 119862] cap 119878119905119872[119865(119909) 119862](ii) 119866(119909) sub minus119863(iii) 119879(119911) = 0

119884

Proof By conditions (a) and (b) it is easy to verify thatconditions (i) and (iii) hold Now we show that 119866(119909) sub minus119863If this is not true then there would exist 119911

0isin 119866(119909) such that

minus1199110notin 119863 (88)

Then since119863 is a closed convex set by the strong separationtheorem in topological vector space (see [21]) there exists1205950isin 119885lowast

0119885lowast such that

1205950(minus1199110) lt 120595 (120582119889) forall119889 isin 119863 forall120582 gt 0 (89)

In the above expression taking 119889 = 0119911isin 119863 gives

1205950(1199110) gt 0 (90)

and taking 120582 rarr +infin leads to

1205950(119889) ge 0 forall119889 isin 119863 (91)

Hence

1205950isin 119863+

0119911lowast (92)

Take 1198880isin int119862 and define 119879

0 119885 rarr 119884 as

1198790(119911) = 120595

0(119911) 1198880 (93)

Then 1198790isin Ł+(119885 119884) and

1198790(1199110) = 1205950(1199110) 1198880isin int119862 sub 119862 0

119884 (94)

And by condition (a) we have

119910 minus 119910 notin 119862 0119884 forall119910 isin ⋃

119879isin119871+(119885119884)

L (119909 119879) (95)

From

119910 + 1198790(1199110) isin 119865 (119909) + 119879

0[119866 (119909)]

= 119871 (119909 1198790) sub ⋃

119879isin119871+(119885119884)

L (119909 119879) (96)

and by (95) we have

1198790(1199110) = [119910 + 119879

0(1199110)] minus 119910 notin 119862 0

119884 (97)

This conflicts with (99) Therefore 119866(119909) sub minus119863Conversely by (i) we have

119910 isin 119878119905119872 [119865 (119909) 119862] (98)

that is

cl [119865 (119909) minus 119862 minus 119910] cap 119862 = 0119884 (99)

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

By condition (ii) for every 119879 isin 119871+(119885 119884) we have

119879 [119866 (119909)] sub 119879 (minus119863) sub minus119862 (100)

Thus

⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910

= ⋃

119909isin119883

[119865 (119909) + 119879 [119866 (119909)]] minus 119862 minus 119910

= ⋃

119909isin119883

119879 [119866 (119909)] + 119865 (119909) minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119862 minus 119910

sub 119865 (119909) minus 119862 minus 119910

(101)

Therefore by (99) we have

cl[⋃

119909isin119883

L (119909 119879) minus 119862 minus 119910] cap 119862 = 0119884 (102)

that is 119910 isin 119878119905119872[⋃119909isin119883

L(119909 119879)] Therefore this proof iscompleted

By Lemmas 25 and 27 and Theorem 26 we can obtainimmediately the following corollary

Corollary 28 Let 119865 be nearly 119862-convexlike on119860 let (119865 119866) benearly (119862times119863)-convexlike on119860 and let (VP) satisfy generalizedSlater constrained qualification

(i) If (119909 119879) is a strict saddle point of the mapL then 119909 isstrictly efficient solution of (VP)

(ii) If (119909 119910) is a strictly efficient minimizer of (VP) and119866(119909) sub minus119863 119910 isin 119878119905119872[119865(119909) 119862] then there exists119879 isin 119871

+(119885 119884) such that (119909 119879) is a strict saddle point

of the mapL

Acknowledgments

Zhimiao Fangrsquos research was supported by the Natu-ral Science Foundation Project of CQ CSTC (Grant nocstc2012jjA00033) The authors would like to thank twoanonymous referees for their valuable comments and sugges-tions which helped to improve the paper

References

[1] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Los Angeles Calif USA 1951

[2] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 pp 618ndash630 1968

[3] J Borwein ldquoProper efficient points for maximizations withrespect to conesrdquo SIAM Journal on Control and Optimizationvol 15 no 1 pp 57ndash63 1977

[4] R Hartley ldquoOn cone-efficiency cone-convexity and cone-com-pactnessrdquo SIAM Journal on Applied Mathematics vol 34 no 2pp 211ndash222 1978

[5] H P Benson ldquoAn improved definition of proper efficiency forvector maximization with respect to conesrdquo Journal of Mathe-matical Analysis and Applications vol 71 no 1 pp 232ndash2411979

[6] M I Henig ldquoProper efficiency with respect to conesrdquo Journalof OptimizationTheory and Applications vol 36 no 3 pp 387ndash407 1982

[7] J M Borwein and D Zhuang ldquoSuper efficiency in vector opti-mizationrdquo Transactions of the American Mathematical Societyvol 338 no 1 pp 105ndash122 1993

[8] E Bednarczuk andW Song ldquoPC points and their application tovector optimizationrdquo Pliska Studia Mathematica Bulgarica vol12 pp 21ndash30 1998

[9] A Zaffaroni ldquoDegrees of efficiency and degrees of minimalityrdquoSIAM Journal on Control and Optimization vol 42 no 3 pp1071ndash1086 2003

[10] H W Corley ldquoExistence and Lagrangian duality for maximiza-tions of set-valued functionsrdquo Journal of Optimization Theoryand Applications vol 54 no 3 pp 489ndash500 1987

[11] Z-F Li and G-Y Chen ldquoLagrangian multipliers saddle pointsand duality in vector optimization of set-valued mapsrdquo Journalof Mathematical Analysis and Applications vol 215 no 2 pp297ndash316 1997

[12] W Song ldquoLagrangian duality for minimization of nonconvexmultifunctionsrdquo Journal of Optimization Theory and Applica-tions vol 93 no 1 pp 167ndash182 1997

[13] G Y Chen and J Jahn ldquoOptimality conditions for set-valuedoptimization problemsrdquo Mathematical Methods of OperationsResearch vol 48 no 2 pp 187ndash200 1998

[14] W D Rong and Y NWu ldquoCharacterizations of super efficiencyin cone-convexlike vector optimization with set-valued mapsrdquoMathematical Methods of Operations Research vol 48 no 2 pp247ndash258 1998

[15] X M Yang D Li and S Y Wang ldquoNear-subconvexlikeness invector optimization with set-valued functionsrdquo Journal of Opti-mization Theory and Applications vol 110 no 2 pp 413ndash4272001

[16] S J Li X Q Yang and G Y Chen ldquoNonconvex vector opti-mization of set-valuedmappingsrdquo Journal ofMathematical Ana-lysis and Applications vol 283 no 2 pp 337ndash350 2003

[17] Z F Li ldquoBenson proper efficiency in the vector optimization ofset-valued mapsrdquo Journal of Optimization Theory and Applica-tions vol 98 no 3 pp 623ndash649 1998

[18] A Mehra ldquoSuper efficiency in vector optimization with nearlyconvexlike set-valued mapsrdquo Journal of Mathematical Analysisand Applications vol 276 no 2 pp 815ndash832 2002

[19] L Y Xia and J H Qiu ldquoSuperefficiency in vector optimizationwith nearly subconvexlike set-valued mapsrdquo Journal of Opti-mization Theory and Applications vol 136 no 1 pp 125ndash1372008

[20] E Miglierina ldquoCharacterization of solutions of multiobjectiveoptimization problemrdquo Rendiconti del Circolo Matematico diPalermo vol 50 no 1 pp 153ndash164 2001

[21] J Jahn Mathematical Vector Optimization in Partially OrderedLinear Spaces vol 31 ofMethods andProcedures inMathematicalPhysics Peter D Lang Frankfurt Germany 1986

[22] J-B Hiriart-Urruty ldquoTangent cones generalized gradients andmathematical programming in Banach spacesrdquoMathematics ofOperations Research vol 4 no 1 pp 79ndash97 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of