Research Article Optimality Conditions for Nondifferentiable Multiobjective …downloads.hindawi.com/journals/aaa/2016/5367190.pdf · 2019-07-30 · Research Article Optimality Conditions

Research ArticleOptimality Conditions for Nondifferentiable MultiobjectiveSemi-Infinite Programming Problems

D Barilla G Caristi and A Puglisi

Department of Economics University of Messina Via dei Verdi 75 Messina Italy

Correspondence should be addressed to G Caristi gcaristiunimeit

Received 2 May 2016 Accepted 18 August 2016

Academic Editor Jozef Banas

Copyright copy 2016 D Barilla et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We have considered a multiobjective semi-infinite programming problem with a feasible set defined by inequality constraintsFirst we studied a Fritz-John type necessary condition Then we introduced two constraint qualifications and derive the weak andstrong Karush-Kuhn-Tucker (KKT in brief) types necessary conditions for an efficient solution of the considered problem Finallyan extension of a Caristi-Ferrara-Stefanescu result for the (Φ 120588)-invexity is proved and some sufficient conditions are presentedunder this weak assumption All results are given in terms of Clark subdifferential

1 Preliminaries and Introduction

First we briefly overview some notions of convex analysisand nonsmooth analysis widely used in the formulations andproofs of the main results of the paper For more detailsdiscussion and applications see [1ndash3]

Given a nonempty set 119860 sube R119899 we denote with 119860 ri(119860)conv(119860) and cone(119860) the closure of 119860 the relative interiorof 119860 convex hull and convex cone (containing the origin)generated by 119860 respectively The polar cone and strict polarcone of 119860 are defined respectively by

119860minus fl 119909 isin R

119899

| ⟨119909 119886⟩ le 0 forall119886 isin 119860

119860119904 fl 119909 isin R

119899

| ⟨119909 119886⟩ lt 0 forall119886 isin 119860

(1)

It is easy to show that if119860119904 = 120601 then119860119904 = 119860minus The bipolar

theorem states that

119860minusminus

= cone (119860) fl cone (119860) (2)

The cone of feasible direction of119860 at isin 119860 is the cone definedby

119863 (119860 ) fl V isin R119899

| exist120575 gt 0 + 120576V isin 119860 forall120576 isin (0 120575) (3)

It is worth observing that if is a minimizer of convexfunction 120601 on a convex set 119862 then

0 isin 120597120601 () + 119873 (119862 ) (4)

where 119873(119862 ) and 120597120601() denote respectively the normalcone of 119862 at and the convex subdifferential of 120601 at thatis

119873(119862 ) fl 119910 isin R119899

| ⟨119910 119909 minus ⟩ le 0 forall119909 isin 119862

120597120601 ()

fl 120585 isin R119899

| 120601 (119909) ge 120601 () + ⟨120585 119909 minus ⟩ forall119909 isin R119899

(5)

We observe that if 119870 sube R119899 is an arbitrary set and isin 119870then

119873(119870minus

) = 119870minusminus

(6)

If 119860120572

| 120572 isin Λ is a collection of convex sets in R119899 and119861 fl ⋃

120572isinΛ119860120572 then it is easy to see that

conv (119861) =

119896

sum

119895=1

120582120572119895119886120572119895

| 119886120572119895

isin 119860120572119895 119896 isin N 120582

120572119895

ge 0

119896

sum

119895=1

120582120572119895

= 1

(7)

cone (119861) =

119896

sum

119895=1

120582120572119895119886120572119895

| 119886120572119895

isin 119860120572119895 119896 isin N 120582

120572119895ge 0

(8)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 5367190 6 pageshttpdxdoiorg10115520165367190

2 Abstract and Applied Analysis

Let isin R119899 and let120593 R119899 rarr R be a locally Lipschitz func-tion The Clarke directional derivative of 120593 at in thedirection V isin R119899 and the Clarke subdifferential of 120593 at arerespectively given by

1205930

( V) fl lim sup119910rarr119905darr0

120593 (119910 + 119905V) minus 120593 (119910)

119905

120597119888120593 () fl 120585 isin R

119899

| ⟨120585 V⟩ le 1205930

( V) forallV isin R119899

(9)

The Clarke subdifferential is a natural generalization of theclassical derivative since it is known that when function 120593 iscontinuously differentiable at 120597

119888120593() = nabla120593() Moreover

when a function 120593 is convex the Clarke subdifferential coin-cides with the subdifferential in the sense of convex analysis

In the following theorem we summarize some importantproperties of the Clarke directional derivative and the Clarkesubdifferential from [1] which are widely used in what fol-lows

Theorem 1 Let 120593 and 120601 be functions from R119899 to R which areLipschitz near Then

(i) the function V rarr 1205930

( V) is finite positively homoge-neous and subadditive on R119899 and

1205930

( V) = max ⟨120585 V⟩ | 120585 isin 120597119888120593 () (10)

120597 (1205930

( sdot)) (0) = 120597119888120593 () (11)

(ii) 120597119888120593() is a nonempty convex and compact subset of

R119899(iii) 120593

0

(119909 V) is upper semicontinuous as a function of (119909 V)

In this paper we have considered the following multiob-jective semi-infinite programming problem

(MOSIP) inf (1198911(119909) 119891

2(119909) 119891

119901(119909))

st 119892119905(119909) le 0 119905 isin 119879

119909 isin R119899

(12)

where 119891119894 119894 isin 119868 fl 1 2 119901 and 119892

119905 119905 isin 119879 are locally

Lipschitz functions fromR119899 toRcup +infin and the index set 119879is arbitrary not necessarily finite (but nonempty)

For differentiable MOSIP where 119879 is finite necessaryconditions of KKT type have been established under vari-ous constraint qualifications in [4] The Abadie constraintqualification and related constraint qualification for semi-infinite systems of convex inequalities and linear inequalitiesare also studied in [5] There the characterizations of variousconstraint qualifications in terms of upper semicontinuity ofcertain multifunctions are given

There are only a few works available that deal with opti-mality conditions for MOSIP For instance for differentiableMOSIPs some optimality conditions have been presented byCaristi et al in [6] Glover et al in [7] considered a nondiffer-entiable convex MOSIP and presented optimality theorems

for it For a nonsmooth MOSIP the ldquobasic constraint qualifi-cationrdquo has been studied by Chuong andKim in [8] who havegiven optimality and duality conditions of Karush-Kuhn-Tucker (KKT briefly) type for the problemwhich involves thenotion ofMordukhovich subdifferential Also Gao presentedsome sufficient and duality results for MOSIPs under thevarious generalized convexity assumptions in [9 10]

This paper is structured as follows In Section 2 we pro-pose a Fritz-John type necessary condition after we derive aKKT type necessary condition for optimality of the consid-ered problem under a suitable qualification condition andweestablish the strong KKT necessary conditions for an efficientsolution of the considered problem In Section 3 we obtain anextension of a Caristi-Ferrara-Stefanescu result for the (Φ 120588)-invexity

2 Necessary Conditions

As a starting point of this section we denote with 119872 thefeasible region of (MOSIP) that is

119872 fl 119909 isin R119899

| 119892119905(119909) le 0 forall119905 isin 119879 (13)

For a given isin 119872 let 119879() denote the index set of all activeconstraints at

119879 () fl 119905 isin 119879 | 119892119905() = 0 (14)

A feasible point is said to be an efficient solution [respweakly efficient solution] to problem (MOSIP) iff there is no119909 isin 119872 satisfying 119891

119894(119909) le 119891

119894() 119894 isin 119868 and (119891

1(119909)

119891119901(119909)) = (119891

1() 119891

119901()) [resp 119891

119894(119909) lt 119891

119894() 119894 isin 119868]

For each isin 119872 set

119865fl ⋃

119894isin119868

120597119888119891119894()

119866fl ⋃

119905isin119879()

120597119888119892119905()

(15)

For each 119909 isin 119872 set

Ψ (119909) fl sup119905isin119879

119892119905(119909) (16)

Recall the following definition from [11]

Definition 2 We say that MOSIP has the Pshenichnyi-Levin-Valadire (PLV) property at 119909 isin 119872 if Ψ(sdot) is Lipschitz around119909 and

120597119888Ψ (119909) sube conv( ⋃

119905isin119879(119909)

120597119888119892119905(119909)) = conv (119866

119909) (17)

The following condition is well known even in differen-tiable cases (see eg [5 6])

Assumption A The index set119879 is a nonempty compact subsetofR119897 the function (119909 119905) rarr 119892

119905(119909) is upper semicontinuous on

R119899 times 119879 and 120597119888119892119905(119909) is an upper semicontinuous mapping in

119905 for each 119909

Abstract and Applied Analysis 3

The following lemma from [5Theorem 5] will be used insequel

Lemma 3 Suppose that Assumption A holds Then

(1) 119866is a compact set for each isin 119872

(2) the PLV property holds at each isin 119872

The following result is an extension of [6 Theorem 4]

Theorem 4 (FJ necessary condition) Let be a weaklyefficient solution ofMOSIP If condition A holds at then thereexist 120572

119894ge 0 (for 119894 = 1 2 119901) and 120573

119905ge 0 for 119905 isin 119879() with

120573119905

= 0 for finitely many indexes such that

0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905()

119901

sum

119894=1

120572119894+ sum

119905isin119879()

120573119905= 1

(18)

Proof Weknow fromLemma 3 that119866is a compact setThus

119865cup 119866is also a compact set and hence conv(119865

cup 119866) is

closedIf 0 notin conv(119865

cup119866) by strict separation theoremwe find

119902 isin R119899 such that ⟨119902 119906⟩ lt 0 for all 119906 isin conv(119865cup 119866) This

implies that

119902 isin (conv (119865cup 119866))119904

= (119865cup 119866)119904

= 119865119904

cap 119866119904

(19)

Since 119889 isin 119866119904

and the PLV property is satisfied at (by

Lemma 3) we have

119889 = (conv (119866))119904

sube (120597119888Ψ ())

119904

997904rArr

Ψ0

( 119889) lt 0

(20)

Thus there exists 120575 gt 0 such that Ψ( + 120576119889) minus Ψ() lt 0 forall 120576 isin (0 120575) The last inequality and the fact that Ψ() le 0

(since isin 119872) conclude that Ψ( + 120576119889) lt 0 and hence

119892119905( + 120576119889) lt 0 forall120576 isin (0 120575) (21)

Moreover we have

119889 isin 119865119904

= (

119901

⋃

119894=1

120597119888119891119894())

119904

=

119901

⋂

119894=1

(120597119888119891119894())119904

997904rArr

1198910

119894( 119889) lt 0 forall119894 = 1 119901

(22)

For each 119894 = 1 119901 we find 120575119894gt 0 such that

119891119894( + 120576119889) minus 119891

119894() lt 0 forall120576 isin (0 120575

119894) (23)

Take fl min120575 1205751 120575

119901 By (21) and (23) for each 120576 isin

(0 ) we have

(1198911( + 120576119889) 119891

119901( + 120576119889))

lt (1198911() 119891

119901()) + 120576119889 isin 119872

(24)

which contradicts the weak efficiency of This contradictionimplies that

0 isin conv (119865cup 119866) (25)

Now (7) proves the result

Thenecessary conditions of Fritz-John type can be viewedas being degenerate when the multiplier corresponding tothe objective function vanishes because the function beingminimized is not involved In the next theorem we derive aKarush-Kuhn-Tuker type necessary condition for optimalityof MOSIP under a suitable qualification condition

Definition 5 Let isin 119872 We say that MOSIP satisfies theZangwill CQ (ZCQ briefly) at if

119866minus

sube 119863 (119872 ) (26)

Theorem 6 Let be a weakly efficient solution of MOSIPZCQ hold at and 119888119900119899119890(119866()) be a closed cone Then thereexist 120572

119894ge 0 (for 119894 = 1 2 119901) and 120573

119905ge 0 for 119905 isin 119879() with

120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905()

119901

sum

119894=1

120572119894= 1

(27)

Proof We first claim that

max1le119894le119901

1198910

119894( 119889) ge 0 forall119889 isin 119863 (119872 ) (28)

On the contrary suppose that there exists 119889 isin 119863(119872 ) suchthat 1198910

119894( 119889) lt 0 for all 119894 = 1 119901 Thus there exist positive

scalars 120575 1205751 120575

119901such that

+ 120576119889 isin 119872 forall120576 isin (0 120575)

119891119894( + 120576119889) minus 119891

119894() lt 0 forall120576 isin (0 120575

119894)

(29)

Take fl min120575 1205751 120575

119901 By above inequalities for each

120576 isin (0 ) we have

(1198911( + 120576119889) 119891

119901( + 120576119889))

lt (1198911() 119891

119901()) + 120576119889 isin 119872

(30)

which contradicts the weak efficiency of Thus (28) is trueIf isin 119863(119872 ) there exists a sequence 119889

119896infin

119896=1in119863(119872 )

converging to Owing to (28) and continuity of function120593(119889) fl max

1le119894le1199011198910

119894( 119889) we deduce that

120593 () = lim119896rarrinfin

120593 (119889119896) ge 0 (31)

We thus proved that (by assumption of ZCQ at )

120593 (119889) = max1le119894le119901

1198910

119894( 119889) ge 0 forall119889 isin 119866

(32)


Since 0 isin 119866minus

and 120593(0) = 0 the last relation implies that the

following convex problem has a minimum at 119889 fl 0

min 120593 (119889)

subject to 119889 isin 119866

(33)

Hence by (4) (6) and (11) we obtain that

0 isin 120597120593 (0) + 119873 (119866minus

0)

= conv(

119901

⋃

119894=1

1205971198910

119894( sdot) (0)) + 119866

minusminus

= conv (119865) + cone (119866

)

(34)

Now the closeness of cone(119866()) (2) (7) and (8) prove theresults

In almost all examples we were not able to obtain positiveKKT multipliers associated with the vector-valued objectivefunction that is to say some of the multipliers may beequal to zero This means that the components of the vector-valued objective function do not have a role in the necessaryconditions for weak efficiency In order to avoid the casewhere some of the KKT multipliers associated with theobjective function vanish for the MOSIP an approach hasbeen developed in [5] and strong KKT necessary optimalityconditions have been obtained We say that strong KKTcondition holds for a multiobjective optimization problemwhen the KKTmultipliers are positive for all the componentsof the objective function Below we establish the necessarystrong KKT conditions for an efficient solution (not a weakefficient solution) of MOSIP under a suitable qualificationcondition

Let isin 119878 On the lines of [4] for each 119894 isin 119868 define the set

119876119894

() fl 119909 isin 119872 | 119891119897(119909) le 119891

119897() forall119897 isin 119868 119894

119876119894

() fl 119872 if 119901 = 1

(35)

For the sake of simplicity we denote119876119894() by119876119894 in this paper

Definition 7 Let isin 119872 We say that MOSIP satisfies thestrong Zangwill CQ (SZCQ briefly) at if

119866minus

sube

119901

⋂

119894=1

119863(119876119894 ) (36)

Theorem 8 (strong KKT necessary condition) Let be anefficient solution of MOSIP If in addition SZCQ and thecondition

(A) (

119901

⋃

119894=1

120597119888119891119894())

minus

0 sube

119901

⋃

119894=119894

(120597119888119891119894())119904 (37)

hold at then 120572119894gt 0 exist (for 119894 = 1 2 119901) and 120573

119905ge 0 for

119905 isin 119879() with 120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905() (38)

Proof We present the proof in four steps

Step 1 We claim that

(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894

)) = 0 (39)

It suffices only to prove that

(120597119888119891119897())119904

cap 119863(119876119897

) = 0 forall119897 isin 119868 (40)

On the contrary suppose that for some 119897 isin 119868 there is a vector119889 such that

119889 isin (120597119888119891119897())119904

cap 119863(119876119897

) (41)

By the definition of 119863(119876119897

) there exists 120575 gt 0 such that +

120576119889 isin 119876119897 for each 120576 isin (0 120575) Thus owing to the definition of

119876119897 we obtain that

119891119894( + 120576119889) le 119891

119894() forall119894 isin 119868 119897 forall120576 isin (0 120575)

+ 120576119889 isin 119872 forall120576 isin (0 120575)

(42)

On the other hand (41) leads to1198910

119897( 119889) lt 0This means that

120575119897gt 0 exists satisfying

119891119897( + 120576119889) minus 119891

119897() lt 0 forall120576 isin (0 120575

119897) (43)

The above inequality with (42) implies that for each 120576 isin (0 )

with fl min120575 120575119897 we have

119891119894( + 120576119889) le 119891

119894() forall119894 isin 119868 119897

119891119897( + 120576119889) lt 119891

119897()

+ 120576119889 isin 119872

(44)

This contradicts the efficiency of Therefore our claimholds

Step 2 Let isin 119863(119876119897 ) for some 119897 isin 119868 Then there existsa sequence 119889

119896infin

119896=1in 119863(119876

119897

) converging to By (40) andcontinuity of 1198910

119897( sdot) we concluded that

1198910

119897( ) = lim

119896rarrinfin

1198910

119897( 119889119896) ge 0 (45)

Therefore

(120597119888119891119897())119904

cap 119863 (119876119897 ) = 0 forall119897 isin 119868 (46)

and hence

(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) = 0 (47)


0 isin ri (conv (119865)) + cone (119866

) (48)


On the contrary we suppose that (48) does not hold Then

ri (conv (119865)) cap (minuscone (119866

)) = 0 (49)

Thus by the strong convex separation theorem [3 Theorem113] and noting that (minuscone(119866

)) is a convex cone it follows

that there is a hyperplane

119867 fl 119909 | exist119889 isin R119899

0 st ⟨119909 119889⟩ = 0 (50)

separating conv(119865) and (minuscone(119866

)) properly Hence there

exists a vector 119889 isin R119899 satisfying

0 = 119889 isin (conv (119865))minus

cap (cone (119866))minus

= 119865minus

cap 119866minus

(51)

Thus owing to SZCQ andA we conclude that

119889 isin (

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) (52)

which contradicts (47)

Step 4 The result is immediate from (48) (8) and the factthat (see [3 Theorem 69])

ri (conv (119865))

sube

119901

sum

119894=1

120572119894120585119894| 120585119894isin 120597119888119891119894() 120572

119894gt 0

119901

sum

119894=1

120572119894= 1

(53)

3 Sufficient Conditions

Similar to [6] letΦ R119899timesR119899timesR119899+1 rarr R and 120588 R119899timesR119899 rarr

R be given functions satisfying

Φ(119909 119909lowast

(0 119903)) ge 0 forall119909 isin R119899

119903 ge 0 (54)

Observe that an element of R119899+1 is represented as the orderpair (119910 119903) with 119910 isin R119899 and 119903 isin R

In [6] a differentiable function ℎ R119899 rarr R was named(Φ 120588)-invex at 119909lowast with respect to A sube R119899 if for each 119909 isin A

Φ(119909 119909lowast

(nablaℎ (119909lowast

) 120588 (119909 119909lowast

))) le ℎ (119909) minus ℎ (119909lowast

)

Φ (119909 119909lowast

sdot) is convex on R119899+1

(55)

We extend this result as follows

Definition 9 The locally Lipschitz function ℏ R119899 rarr R at119909lowast

isin R119899 is called generalized (Φ 120588)-invex at 119909lowast with respectto A sube R119899 if for each 119909 isin A it satisfies

Φ(119909 119909lowast

(120585 120588 (119909 119909lowast

))) le ℏ (119909) minus ℏ (119909lowast

)

forall120585 isin 120597119888ℏ (119909lowast

)

Φ (119909 119909lowast


(56)

In the rest of this paper we will always assume A to be equalto the set 119872 of the feasible solution of MOSIP

Theorem 10 (sufficient KKT condition) Suppose that thereexist a feasible solution isin 119872 and scalars 120572

119894ge 0 (for 119894 isin 119868)

with sum119901

119894=1120572119894= 1 and a finite set 119879lowast fl 119905

1 1199052 119905

119898 sube 119879()

and scalars 120573119895119904

ge 0 (for 119904 isin 1 2 119898) such that

0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (57)

Moreover if the 119891119894functions and the 119892

119905functions (for (119894 119905) isin

119868 times 119879()) are (Φ 120588)-invex at and sum119901

119894=1120572119894120588119894(119909 ) +

sum119898

119904=1120573119905119904120588119905119904(119909 ) ge 0 for all 119909 isin 119872 then is a weak efficient

solution for MOSIP

Proof Inclusion (57) implies that some 120585119894isin 120597119888119891119894() (for 119894 isin 119868)

and 120577119905119904

isin 120597119888119892119905119904() (for 119904 isin 1 119898) exist satisfying119901

sum

119894=1

120572119894120585119894+

119898

sum

119904=1

120573119905119904120577119905119904

= 0

119901

sum

119894=1

120572119894+

119898

sum

119904=1

120573119905119904

= 119887 gt 0

(58)

Taking 119894fl 120572119894119887 and

119905119904fl 120573119905119904119887 we conclude that

119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

= 0

119901

sum

119894=1

119894+

119898

sum

119904=1

119905119904

= 1

(59)

Owing to these equalities (54)sum119901119894=1

120572119894120588119894(119909 )+sum

119898

119904=1120573119905119904120588119905119904(119909

) ge 0 and convexity of Φ(119909 119909lowast

sdot) we obtain that

0 le Φ(119909

119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

119901

sum

119894=1

120572119894120588119894(119909 )

+

119898

sum

119904=1

120573119905119904120588119905119904(119909 )) = Φ(119909

119901

sum

119894=1

119894(120585119894 120588119894(119909 ))

+

119898

sum

119904=1

119905119904(120577119905119904 120588119905119904(119909 )))

le

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 ))

(60)

Now if is not a weak efficient of MOSIP there exists a point119909 isin 119872 such that 119891

119894(119909) lt 119891

119894() for all 119894 isin 119868 Hence by

generalized (Φ 120588)-invexity of 119891119894functions we have

Φ(119909 120585119894 120588119894(119909 )) le 119891

119894(119909) minus 119891

119894() lt 0 (61)

Similarly for each 119904 = 1 119898 we have

Φ(119909 120577119905119904 120588119905119904(119909 )) le 119892

119905119904(119909) minus 119892

119905119904() = 119892

119905119904(119909)

le 0

(62)


From (61) (62) and sum119901

119894=1119894gt 0 we conclude that

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(63)

which contradicts (60) This contradiction shows that is anweak efficient for MOSIP

Strengthening the assumptions concerning 120572119894s we obtain

sufficient conditions for efficiency

Theorem 11 (strong sufficient KKT condition) Suppose thatthere exist a feasible solution isin 119872 and scalars 120572

119894gt 0 (for

119894 isin 119868) and a finite set 119879lowast fl 1199051 1199052 119905

119898 sube 119879() and scalars

120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (64)




119894=1120572119894120588119894(119909 ) +

sum119898

119904=1120573119905119904120588119905119904(119909 ) ge 0 for all 119909 isin 119872 then is an efficient

solution for MOSIP

Proof Similar to (61) and (62) if is not efficient we find119909 isin 119872 and 119897 isin 119868 such that

Φ(119909 120585119894 120588119894(119909 )) le 0 forall119894 isin 119868 119897

Φ (119909 120585119897 120588119897(119909 )) lt 0

Φ (119909 120577119905119904 120588119905119904(119909 )) le 0 forall119904 isin 1 119898

(65)

From these and 119894gt 0 (for all 119894 isin 119868) and

119905119904ge 0 (for all

119904 isin 1 119898) we obtain

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(66)


Remark 12 Similar to [6] we can define some weaker (Φ 120588)-invexiy assumption for the function ℏ and then we can provesome weaker sufficient conditions for optimality of MOSIPSince the proof of these extensions is similar to previoustheorems we omit them

Competing Interests

The authors declare that they have no competing interests

References

[1] F H Clarke Optimization and Nonsmooth Analysis CanadianMathematical Society Series of Monographs and AdvancedTexts John Wiley amp Sons 1983

[2] J B Hiriart-Urruty and C Lemarechal Convex Analysis andMinimization Algorithms I amp II Springer Berlin Germany1991

[3] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[4] T Maeda ldquoConstraint qualifications in multiobjective opti-mization problems differentiable caserdquo Journal of OptimizationTheory and Applications vol 80 no 3 pp 483ndash500 1994

[5] N Kanzi ldquoOn strong KKT optimality conditions for multiob-jective semi-infinite programming problems with lipschitziandatardquo Optimization Letters vol 9 no 6 pp 1121ndash1129 2015

[6] G Caristi M Ferrara and A Stefanescu ldquoSemi-infinite multi-objective programming with grneralized invexityrdquo Mathemati-cal Reports vol 62 pp 217ndash233 2010

[7] B M Glover V Jeyakumar and A M Rubinov ldquoDual con-ditions characterizing optimality for convex multi-objectiveprogramsrdquo Mathematical Programming vol 84 no 1 pp 201ndash217 1999

[8] T D Chuong and D S Kim ldquoNonsmooth semi-infinite multi-objective optimization problemsrdquo Journal of Optimization The-ory and Applications vol 160 no 3 pp 748ndash762 2014

[9] X Y Gao ldquoNecessary optimality and duality for multiobjectivesemi-infinite programmingrdquo Journal of Theoretical and AppliedInformation Technology vol 46 no 1 pp 347ndash354 2012

[10] X Y Gao ldquoOptimality and duality for non-smooth multipleobjective semi-infinite programmingrdquo Journal of Networks vol8 no 2 pp 413ndash420 2013

[11] N Kanzi ldquoConstraint qualifications in semi-infinite systemsand their applications in nonsmooth semi-infinite problemswith mixed constraintsrdquo SIAM Journal on Optimization vol 24no 2 pp 559ndash572 2014

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Let isin R119899 and let120593 R119899 rarr R be a locally Lipschitz func-tion The Clarke directional derivative of 120593 at in thedirection V isin R119899 and the Clarke subdifferential of 120593 at arerespectively given by

1205930

( V) fl lim sup119910rarr119905darr0

120593 (119910 + 119905V) minus 120593 (119910)

119905

120597119888120593 () fl 120585 isin R

119899

| ⟨120585 V⟩ le 1205930

( V) forallV isin R119899

(9)

The Clarke subdifferential is a natural generalization of theclassical derivative since it is known that when function 120593 iscontinuously differentiable at 120597

119888120593() = nabla120593() Moreover

when a function 120593 is convex the Clarke subdifferential coin-cides with the subdifferential in the sense of convex analysis

In the following theorem we summarize some importantproperties of the Clarke directional derivative and the Clarkesubdifferential from [1] which are widely used in what fol-lows

Theorem 1 Let 120593 and 120601 be functions from R119899 to R which areLipschitz near Then

(i) the function V rarr 1205930

( V) is finite positively homoge-neous and subadditive on R119899 and

1205930

( V) = max ⟨120585 V⟩ | 120585 isin 120597119888120593 () (10)

120597 (1205930

( sdot)) (0) = 120597119888120593 () (11)

(ii) 120597119888120593() is a nonempty convex and compact subset of

R119899(iii) 120593

0

(119909 V) is upper semicontinuous as a function of (119909 V)

In this paper we have considered the following multiob-jective semi-infinite programming problem

(MOSIP) inf (1198911(119909) 119891

2(119909) 119891

119901(119909))

st 119892119905(119909) le 0 119905 isin 119879

119909 isin R119899

(12)

where 119891119894 119894 isin 119868 fl 1 2 119901 and 119892

119905 119905 isin 119879 are locally

Lipschitz functions fromR119899 toRcup +infin and the index set 119879is arbitrary not necessarily finite (but nonempty)

For differentiable MOSIP where 119879 is finite necessaryconditions of KKT type have been established under vari-ous constraint qualifications in [4] The Abadie constraintqualification and related constraint qualification for semi-infinite systems of convex inequalities and linear inequalitiesare also studied in [5] There the characterizations of variousconstraint qualifications in terms of upper semicontinuity ofcertain multifunctions are given

There are only a few works available that deal with opti-mality conditions for MOSIP For instance for differentiableMOSIPs some optimality conditions have been presented byCaristi et al in [6] Glover et al in [7] considered a nondiffer-entiable convex MOSIP and presented optimality theorems

for it For a nonsmooth MOSIP the ldquobasic constraint qualifi-cationrdquo has been studied by Chuong andKim in [8] who havegiven optimality and duality conditions of Karush-Kuhn-Tucker (KKT briefly) type for the problemwhich involves thenotion ofMordukhovich subdifferential Also Gao presentedsome sufficient and duality results for MOSIPs under thevarious generalized convexity assumptions in [9 10]

This paper is structured as follows In Section 2 we pro-pose a Fritz-John type necessary condition after we derive aKKT type necessary condition for optimality of the consid-ered problem under a suitable qualification condition andweestablish the strong KKT necessary conditions for an efficientsolution of the considered problem In Section 3 we obtain anextension of a Caristi-Ferrara-Stefanescu result for the (Φ 120588)-invexity

2 Necessary Conditions

As a starting point of this section we denote with 119872 thefeasible region of (MOSIP) that is

119872 fl 119909 isin R119899

| 119892119905(119909) le 0 forall119905 isin 119879 (13)

For a given isin 119872 let 119879() denote the index set of all activeconstraints at

119879 () fl 119905 isin 119879 | 119892119905() = 0 (14)

A feasible point is said to be an efficient solution [respweakly efficient solution] to problem (MOSIP) iff there is no119909 isin 119872 satisfying 119891

119894(119909) le 119891

119894() 119894 isin 119868 and (119891

1(119909)

119891119901(119909)) = (119891

1() 119891

119901()) [resp 119891

119894(119909) lt 119891

119894() 119894 isin 119868]

For each isin 119872 set

119865fl ⋃

119894isin119868

120597119888119891119894()

119866fl ⋃

119905isin119879()

120597119888119892119905()

(15)

For each 119909 isin 119872 set

Ψ (119909) fl sup119905isin119879

119892119905(119909) (16)

Recall the following definition from [11]

Definition 2 We say that MOSIP has the Pshenichnyi-Levin-Valadire (PLV) property at 119909 isin 119872 if Ψ(sdot) is Lipschitz around119909 and

120597119888Ψ (119909) sube conv( ⋃

119905isin119879(119909)

120597119888119892119905(119909)) = conv (119866

119909) (17)

The following condition is well known even in differen-tiable cases (see eg [5 6])

Assumption A The index set119879 is a nonempty compact subsetofR119897 the function (119909 119905) rarr 119892

119905(119909) is upper semicontinuous on

R119899 times 119879 and 120597119888119892119905(119909) is an upper semicontinuous mapping in

119905 for each 119909








119894ge 0 (for 119894 = 1 2 119901) and 120573

119905ge 0 for 119905 isin 119879() with

120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905()

119901

sum

119894=1

120572119894+ sum

119905isin119879()

120573119905= 1

(18)



cup 119866) is




implies that

119902 isin (conv (119865cup 119866))119904

= (119865cup 119866)119904

= 119865119904

cap 119866119904

(19)

Since 119889 isin 119866119904


Lemma 3) we have

119889 = (conv (119866))119904

sube (120597119888Ψ ())

119904

997904rArr

Ψ0

( 119889) lt 0

(20)



119892119905( + 120576119889) lt 0 forall120576 isin (0 120575) (21)

Moreover we have

119889 isin 119865119904

= (

119901

⋃

119894=1

120597119888119891119894())

119904

=

119901

⋂

119894=1

(120597119888119891119894())119904

997904rArr

1198910

119894( 119889) lt 0 forall119894 = 1 119901

(22)


119891119894( + 120576119889) minus 119891

119894() lt 0 forall120576 isin (0 120575

119894) (23)

Take fl min120575 1205751 120575


(0 ) we have

(1198911( + 120576119889) 119891

119901( + 120576119889))

lt (1198911() 119891

119901()) + 120576119889 isin 119872

(24)


0 isin conv (119865cup 119866) (25)




119866minus

sube 119863 (119872 ) (26)


119894ge 0 (for 119894 = 1 2 119901) and 120573

119905ge 0 for 119905 isin 119879() with

120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905()

119901

sum

119894=1

120572119894= 1

(27)


max1le119894le119901

1198910

119894( 119889) ge 0 forall119889 isin 119863 (119872 ) (28)



scalars 120575 1205751 120575

119901such that

+ 120576119889 isin 119872 forall120576 isin (0 120575)

119891119894( + 120576119889) minus 119891

119894() lt 0 forall120576 isin (0 120575

119894)

(29)

Take fl min120575 1205751 120575



(1198911( + 120576119889) 119891

119901( + 120576119889))

lt (1198911() 119891

119901()) + 120576119889 isin 119872

(30)


119896infin

119896=1in119863(119872 )


1le119894le1199011198910



120593 (119889119896) ge 0 (31)


120593 (119889) = max1le119894le119901

1198910

119894( 119889) ge 0 forall119889 isin 119866

(32)





min 120593 (119889)


(33)


0 isin 120597120593 (0) + 119873 (119866minus

0)

= conv(

119901

⋃

119894=1

1205971198910

119894( sdot) (0)) + 119866

minusminus

= conv (119865) + cone (119866

)

(34)




119876119894

() fl 119909 isin 119872 | 119891119897(119909) le 119891

119897() forall119897 isin 119868 119894

119876119894

() fl 119872 if 119901 = 1

(35)



119866minus

sube

119901

⋂

119894=1

119863(119876119894 ) (36)


(A) (

119901

⋃

119894=1

120597119888119891119894())

minus

0 sube

119901

⋃

119894=119894

(120597119888119891119894())119904 (37)


119905ge 0 for

119905 isin 119879() with 120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905() (38)



(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894

)) = 0 (39)


(120597119888119891119897())119904

cap 119863(119876119897

) = 0 forall119897 isin 119868 (40)


119889 isin (120597119888119891119897())119904

cap 119863(119876119897

) (41)





119891119894( + 120576119889) le 119891


+ 120576119889 isin 119872 forall120576 isin (0 120575)

(42)




119891119897( + 120576119889) minus 119891

119897() lt 0 forall120576 isin (0 120575

119897) (43)



119891119894( + 120576119889) le 119891

119894() forall119894 isin 119868 119897

119891119897( + 120576119889) lt 119891

119897()

+ 120576119889 isin 119872

(44)



119896infin

119896=1in 119863(119876

119897



1198910

119897( ) = lim

119896rarrinfin

1198910

119897( 119889119896) ge 0 (45)

Therefore

(120597119888119891119897())119904

cap 119863 (119876119897 ) = 0 forall119897 isin 119868 (46)

and hence

(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) = 0 (47)


0 isin ri (conv (119865)) + cone (119866

) (48)




)) = 0 (49)




119867 fl 119909 | exist119889 isin R119899

0 st ⟨119909 119889⟩ = 0 (50)




0 = 119889 isin (conv (119865))minus


= 119865minus

cap 119866minus

(51)


119889 isin (

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) (52)



ri (conv (119865))

sube

119901

sum

119894=1

120572119894120585119894| 120585119894isin 120597119888119891119894() 120572

119894gt 0

119901

sum

119894=1

120572119894= 1

(53)




Φ(119909 119909lowast

(0 119903)) ge 0 forall119909 isin R119899

119903 ge 0 (54)



Φ(119909 119909lowast


) 120588 (119909 119909lowast


)

Φ (119909 119909lowast


(55)




Φ(119909 119909lowast

(120585 120588 (119909 119909lowast


)


)

Φ (119909 119909lowast


(56)



119894ge 0 (for 119894 isin 119868)

with sum119901


1 1199052 119905

119898 sube 119879()

and scalars 120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (57)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


and 120577119905119904


sum

119894=1

120572119894120585119894+

119898

sum

119904=1

120573119905119904120577119905119904

= 0

119901

sum

119894=1

120572119894+

119898

sum

119904=1

120573119905119904

= 119887 gt 0

(58)

Taking 119894fl 120572119894119887 and


119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

= 0

119901

sum

119894=1

119894+

119898

sum

119904=1

119905119904

= 1

(59)


120572119894120588119894(119909 )+sum

119898

119904=1120573119905119904120588119905119904(119909



0 le Φ(119909

119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

119901

sum

119894=1

120572119894120588119894(119909 )

+

119898

sum

119904=1

120573119905119904120588119905119904(119909 )) = Φ(119909

119901

sum

119894=1

119894(120585119894 120588119894(119909 ))

+

119898

sum

119904=1

119905119904(120577119905119904 120588119905119904(119909 )))

le

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 ))

(60)


119894(119909) lt 119891



Φ(119909 120585119894 120588119894(119909 )) le 119891

119894(119909) minus 119891

119894() lt 0 (61)


Φ(119909 120577119905119904 120588119905119904(119909 )) le 119892

119905119904(119909) minus 119892

119905119904() = 119892

119905119904(119909)

le 0

(62)


From (61) (62) and sum119901


119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(63)





119894gt 0 (for



120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (64)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


Φ(119909 120585119894 120588119894(119909 )) le 0 forall119894 isin 119868 119897

Φ (119909 120585119897 120588119897(119909 )) lt 0

Φ (119909 120577119905119904 120588119905119904(119909 )) le 0 forall119904 isin 1 119898

(65)


119905119904ge 0 (for all

119904 isin 1 119898) we obtain

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(66)



Competing Interests


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119894ge 0 (for 119894 = 1 2 119901) and 120573

119905ge 0 for 119905 isin 119879() with

120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905()

119901

sum

119894=1

120572119894+ sum

119905isin119879()

120573119905= 1

(18)



cup 119866) is




implies that

119902 isin (conv (119865cup 119866))119904

= (119865cup 119866)119904

= 119865119904

cap 119866119904

(19)

Since 119889 isin 119866119904


Lemma 3) we have

119889 = (conv (119866))119904

sube (120597119888Ψ ())

119904

997904rArr

Ψ0

( 119889) lt 0

(20)



119892119905( + 120576119889) lt 0 forall120576 isin (0 120575) (21)

Moreover we have

119889 isin 119865119904

= (

119901

⋃

119894=1

120597119888119891119894())

119904

=

119901

⋂

119894=1

(120597119888119891119894())119904

997904rArr

1198910

119894( 119889) lt 0 forall119894 = 1 119901

(22)


119891119894( + 120576119889) minus 119891

119894() lt 0 forall120576 isin (0 120575

119894) (23)

Take fl min120575 1205751 120575


(0 ) we have

(1198911( + 120576119889) 119891

119901( + 120576119889))

lt (1198911() 119891

119901()) + 120576119889 isin 119872

(24)


0 isin conv (119865cup 119866) (25)




119866minus

sube 119863 (119872 ) (26)


119894ge 0 (for 119894 = 1 2 119901) and 120573

119905ge 0 for 119905 isin 119879() with

120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905()

119901

sum

119894=1

120572119894= 1

(27)


max1le119894le119901

1198910

119894( 119889) ge 0 forall119889 isin 119863 (119872 ) (28)



scalars 120575 1205751 120575

119901such that

+ 120576119889 isin 119872 forall120576 isin (0 120575)

119891119894( + 120576119889) minus 119891

119894() lt 0 forall120576 isin (0 120575

119894)

(29)

Take fl min120575 1205751 120575



(1198911( + 120576119889) 119891

119901( + 120576119889))

lt (1198911() 119891

119901()) + 120576119889 isin 119872

(30)


119896infin

119896=1in119863(119872 )


1le119894le1199011198910



120593 (119889119896) ge 0 (31)


120593 (119889) = max1le119894le119901

1198910

119894( 119889) ge 0 forall119889 isin 119866

(32)





min 120593 (119889)


(33)


0 isin 120597120593 (0) + 119873 (119866minus

0)

= conv(

119901

⋃

119894=1

1205971198910

119894( sdot) (0)) + 119866

minusminus

= conv (119865) + cone (119866

)

(34)




119876119894

() fl 119909 isin 119872 | 119891119897(119909) le 119891

119897() forall119897 isin 119868 119894

119876119894

() fl 119872 if 119901 = 1

(35)



119866minus

sube

119901

⋂

119894=1

119863(119876119894 ) (36)


(A) (

119901

⋃

119894=1

120597119888119891119894())

minus

0 sube

119901

⋃

119894=119894

(120597119888119891119894())119904 (37)


119905ge 0 for

119905 isin 119879() with 120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905() (38)



(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894

)) = 0 (39)


(120597119888119891119897())119904

cap 119863(119876119897

) = 0 forall119897 isin 119868 (40)


119889 isin (120597119888119891119897())119904

cap 119863(119876119897

) (41)





119891119894( + 120576119889) le 119891


+ 120576119889 isin 119872 forall120576 isin (0 120575)

(42)




119891119897( + 120576119889) minus 119891

119897() lt 0 forall120576 isin (0 120575

119897) (43)



119891119894( + 120576119889) le 119891

119894() forall119894 isin 119868 119897

119891119897( + 120576119889) lt 119891

119897()

+ 120576119889 isin 119872

(44)



119896infin

119896=1in 119863(119876

119897



1198910

119897( ) = lim

119896rarrinfin

1198910

119897( 119889119896) ge 0 (45)

Therefore

(120597119888119891119897())119904

cap 119863 (119876119897 ) = 0 forall119897 isin 119868 (46)

and hence

(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) = 0 (47)


0 isin ri (conv (119865)) + cone (119866

) (48)




)) = 0 (49)




119867 fl 119909 | exist119889 isin R119899

0 st ⟨119909 119889⟩ = 0 (50)




0 = 119889 isin (conv (119865))minus


= 119865minus

cap 119866minus

(51)


119889 isin (

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) (52)



ri (conv (119865))

sube

119901

sum

119894=1

120572119894120585119894| 120585119894isin 120597119888119891119894() 120572

119894gt 0

119901

sum

119894=1

120572119894= 1

(53)




Φ(119909 119909lowast

(0 119903)) ge 0 forall119909 isin R119899

119903 ge 0 (54)



Φ(119909 119909lowast


) 120588 (119909 119909lowast


)

Φ (119909 119909lowast


(55)




Φ(119909 119909lowast

(120585 120588 (119909 119909lowast


)


)

Φ (119909 119909lowast


(56)



119894ge 0 (for 119894 isin 119868)

with sum119901


1 1199052 119905

119898 sube 119879()

and scalars 120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (57)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


and 120577119905119904


sum

119894=1

120572119894120585119894+

119898

sum

119904=1

120573119905119904120577119905119904

= 0

119901

sum

119894=1

120572119894+

119898

sum

119904=1

120573119905119904

= 119887 gt 0

(58)

Taking 119894fl 120572119894119887 and


119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

= 0

119901

sum

119894=1

119894+

119898

sum

119904=1

119905119904

= 1

(59)


120572119894120588119894(119909 )+sum

119898

119904=1120573119905119904120588119905119904(119909



0 le Φ(119909

119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

119901

sum

119894=1

120572119894120588119894(119909 )

+

119898

sum

119904=1

120573119905119904120588119905119904(119909 )) = Φ(119909

119901

sum

119894=1

119894(120585119894 120588119894(119909 ))

+

119898

sum

119904=1

119905119904(120577119905119904 120588119905119904(119909 )))

le

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 ))

(60)


119894(119909) lt 119891



Φ(119909 120585119894 120588119894(119909 )) le 119891

119894(119909) minus 119891

119894() lt 0 (61)


Φ(119909 120577119905119904 120588119905119904(119909 )) le 119892

119905119904(119909) minus 119892

119905119904() = 119892

119905119904(119909)

le 0

(62)


From (61) (62) and sum119901


119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(63)





119894gt 0 (for



120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (64)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


Φ(119909 120585119894 120588119894(119909 )) le 0 forall119894 isin 119868 119897

Φ (119909 120585119897 120588119897(119909 )) lt 0

Φ (119909 120577119905119904 120588119905119904(119909 )) le 0 forall119904 isin 1 119898

(65)


119905119904ge 0 (for all

119904 isin 1 119898) we obtain

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(66)



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References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













min 120593 (119889)


(33)


0 isin 120597120593 (0) + 119873 (119866minus

0)

= conv(

119901

⋃

119894=1

1205971198910

119894( sdot) (0)) + 119866

minusminus

= conv (119865) + cone (119866

)

(34)




119876119894

() fl 119909 isin 119872 | 119891119897(119909) le 119891

119897() forall119897 isin 119868 119894

119876119894

() fl 119872 if 119901 = 1

(35)



119866minus

sube

119901

⋂

119894=1

119863(119876119894 ) (36)


(A) (

119901

⋃

119894=1

120597119888119891119894())

minus

0 sube

119901

⋃

119894=119894

(120597119888119891119894())119904 (37)


119905ge 0 for

119905 isin 119879() with 120573119905


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() + sum

119905isin119879()

120573119905120597119888119892119905() (38)



(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894

)) = 0 (39)


(120597119888119891119897())119904

cap 119863(119876119897

) = 0 forall119897 isin 119868 (40)


119889 isin (120597119888119891119897())119904

cap 119863(119876119897

) (41)





119891119894( + 120576119889) le 119891


+ 120576119889 isin 119872 forall120576 isin (0 120575)

(42)




119891119897( + 120576119889) minus 119891

119897() lt 0 forall120576 isin (0 120575

119897) (43)



119891119894( + 120576119889) le 119891

119894() forall119894 isin 119868 119897

119891119897( + 120576119889) lt 119891

119897()

+ 120576119889 isin 119872

(44)



119896infin

119896=1in 119863(119876

119897



1198910

119897( ) = lim

119896rarrinfin

1198910

119897( 119889119896) ge 0 (45)

Therefore

(120597119888119891119897())119904

cap 119863 (119876119897 ) = 0 forall119897 isin 119868 (46)

and hence

(

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) = 0 (47)


0 isin ri (conv (119865)) + cone (119866

) (48)




)) = 0 (49)




119867 fl 119909 | exist119889 isin R119899

0 st ⟨119909 119889⟩ = 0 (50)




0 = 119889 isin (conv (119865))minus


= 119865minus

cap 119866minus

(51)


119889 isin (

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) (52)



ri (conv (119865))

sube

119901

sum

119894=1

120572119894120585119894| 120585119894isin 120597119888119891119894() 120572

119894gt 0

119901

sum

119894=1

120572119894= 1

(53)




Φ(119909 119909lowast

(0 119903)) ge 0 forall119909 isin R119899

119903 ge 0 (54)



Φ(119909 119909lowast


) 120588 (119909 119909lowast


)

Φ (119909 119909lowast


(55)




Φ(119909 119909lowast

(120585 120588 (119909 119909lowast


)


)

Φ (119909 119909lowast


(56)



119894ge 0 (for 119894 isin 119868)

with sum119901


1 1199052 119905

119898 sube 119879()

and scalars 120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (57)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


and 120577119905119904


sum

119894=1

120572119894120585119894+

119898

sum

119904=1

120573119905119904120577119905119904

= 0

119901

sum

119894=1

120572119894+

119898

sum

119904=1

120573119905119904

= 119887 gt 0

(58)

Taking 119894fl 120572119894119887 and


119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

= 0

119901

sum

119894=1

119894+

119898

sum

119904=1

119905119904

= 1

(59)


120572119894120588119894(119909 )+sum

119898

119904=1120573119905119904120588119905119904(119909



0 le Φ(119909

119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

119901

sum

119894=1

120572119894120588119894(119909 )

+

119898

sum

119904=1

120573119905119904120588119905119904(119909 )) = Φ(119909

119901

sum

119894=1

119894(120585119894 120588119894(119909 ))

+

119898

sum

119904=1

119905119904(120577119905119904 120588119905119904(119909 )))

le

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 ))

(60)


119894(119909) lt 119891



Φ(119909 120585119894 120588119894(119909 )) le 119891

119894(119909) minus 119891

119894() lt 0 (61)


Φ(119909 120577119905119904 120588119905119904(119909 )) le 119892

119905119904(119909) minus 119892

119905119904() = 119892

119905119904(119909)

le 0

(62)


From (61) (62) and sum119901


119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(63)





119894gt 0 (for



120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (64)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


Φ(119909 120585119894 120588119894(119909 )) le 0 forall119894 isin 119868 119897

Φ (119909 120585119897 120588119897(119909 )) lt 0

Φ (119909 120577119905119904 120588119905119904(119909 )) le 0 forall119904 isin 1 119898

(65)


119905119904ge 0 (for all

119904 isin 1 119898) we obtain

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(66)



Competing Interests


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












)) = 0 (49)




119867 fl 119909 | exist119889 isin R119899

0 st ⟨119909 119889⟩ = 0 (50)




0 = 119889 isin (conv (119865))minus


= 119865minus

cap 119866minus

(51)


119889 isin (

119901

⋃

119894=1

(120597119888119891119894())119904

) cap (

119901

⋂

119894=1

119863(119876119894 )) (52)



ri (conv (119865))

sube

119901

sum

119894=1

120572119894120585119894| 120585119894isin 120597119888119891119894() 120572

119894gt 0

119901

sum

119894=1

120572119894= 1

(53)




Φ(119909 119909lowast

(0 119903)) ge 0 forall119909 isin R119899

119903 ge 0 (54)



Φ(119909 119909lowast


) 120588 (119909 119909lowast


)

Φ (119909 119909lowast


(55)




Φ(119909 119909lowast

(120585 120588 (119909 119909lowast


)


)

Φ (119909 119909lowast


(56)



119894ge 0 (for 119894 isin 119868)

with sum119901


1 1199052 119905

119898 sube 119879()

and scalars 120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (57)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


and 120577119905119904


sum

119894=1

120572119894120585119894+

119898

sum

119904=1

120573119905119904120577119905119904

= 0

119901

sum

119894=1

120572119894+

119898

sum

119904=1

120573119905119904

= 119887 gt 0

(58)

Taking 119894fl 120572119894119887 and


119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

= 0

119901

sum

119894=1

119894+

119898

sum

119904=1

119905119904

= 1

(59)


120572119894120588119894(119909 )+sum

119898

119904=1120573119905119904120588119905119904(119909



0 le Φ(119909

119901

sum

119894=1

119894120585119894+

119898

sum

119904=1

119905119904120577119905119904

119901

sum

119894=1

120572119894120588119894(119909 )

+

119898

sum

119904=1

120573119905119904120588119905119904(119909 )) = Φ(119909

119901

sum

119894=1

119894(120585119894 120588119894(119909 ))

+

119898

sum

119904=1

119905119904(120577119905119904 120588119905119904(119909 )))

le

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 ))

(60)


119894(119909) lt 119891



Φ(119909 120585119894 120588119894(119909 )) le 119891

119894(119909) minus 119891

119894() lt 0 (61)


Φ(119909 120577119905119904 120588119905119904(119909 )) le 119892

119905119904(119909) minus 119892

119905119904() = 119892

119905119904(119909)

le 0

(62)


From (61) (62) and sum119901


119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(63)





119894gt 0 (for



120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (64)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


Φ(119909 120585119894 120588119894(119909 )) le 0 forall119894 isin 119868 119897

Φ (119909 120585119897 120588119897(119909 )) lt 0

Φ (119909 120577119905119904 120588119905119904(119909 )) le 0 forall119904 isin 1 119898

(65)


119905119904ge 0 (for all

119904 isin 1 119898) we obtain

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(66)



Competing Interests


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










From (61) (62) and sum119901


119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(63)





119894gt 0 (for



120573119895119904


0 isin

119901

sum

119894=1

120572119894120597119888119891119894() +

119898

sum

119904=1

120573119905119904120597119888119892119905119904() (64)




119894=1120572119894120588119894(119909 ) +

sum119898


solution for MOSIP


Φ(119909 120585119894 120588119894(119909 )) le 0 forall119894 isin 119868 119897

Φ (119909 120585119897 120588119897(119909 )) lt 0

Φ (119909 120577119905119904 120588119905119904(119909 )) le 0 forall119904 isin 1 119898

(65)


119905119904ge 0 (for all

119904 isin 1 119898) we obtain

119901

sum

119894=1

119894Φ(119909 120585

119894 120588 (119909 ))

+

119898

sum

119904=1

119905119904Φ(119909 120577

119905119904 120588119905119904(119909 )) lt 0

(66)



Competing Interests


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Optimality Conditions for Nondifferentiable Multiobjective …downloads.hindawi.com/journals/aaa/2016/5367190.pdf · 2019-07-30 · Research Article Optimality Conditions