Research Article Exact Penalization and Necessary Optimality...

Research ArticleExact Penalization and Necessary OptimalityConditions for Multiobjective Optimization Problemswith Equilibrium Constraints

Shengkun Zhu12 and Shengjie Li2

1 Department of Economic Mathematics Southwestern University of Finance and Economics Chengdu 611130 China2 College of Mathematics and Statistics Chongqing University Chongqing 401331 China

Correspondence should be addressed to Shengkun Zhu zskcqu163com

Received 1 December 2013 Accepted 16 March 2014 Published 15 May 2014

Academic Editor Geraldo Botelho

Copyright copy 2014 S Zhu and S Li 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

A calmness condition for a general multiobjective optimization problem with equilibrium constraints is proposed Some exactpenalization properties for two classes of multiobjective penalty problems are established and shown to be equivalent to thecalmness condition Subsequently a Mordukhovich stationary necessary optimality condition based on the exact penalizationresults is obtained Moreover some applications to a multiobjective optimization problem with complementarity constraints and amultiobjective optimization problem with weak vector variational inequality constraints are given

1 Introduction

In this paper we consider a general multiobjective optimiza-tion problem with equilibrium constraints as follows

(MOPEC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

0R119898 isin 119902 (119909) + 119876 (119909) 119909 isin Θ

(1)

where 119891 R119899 rarr R119901 119891(119909) = (1198911(119909) 119891

2(119909) 119891

119901(119909))

119892 R119899 rarr R119903 119892(119909) = (1198921(119909) 119892

2(119909) 119892

119903(119909)) ℎ R119899 rarr

R119904 ℎ(119909) = (ℎ1(119909) ℎ

2(119909) ℎ

119904(119909)) 119902 R119899 rarr R119898 and

119902(119909) = (1199021(119909) 119902

2(119909) 119902

119898(119909)) are vector-valued maps 119876

R119899 999490999473 R119898 is a set-valued map and Θ is a nonempty andclosed subset ofR119899 As usual we denote by int Θ the interiorof Θ and by gph119876 = (119909 119910) isin R119899 times R119898 | 119910 isin 119876(119909) thegraph of 119876 Moreover R119903

+denotes the nonnegative quadrant

inR119903 and 0R119904 and 0R119898 denote respectively the origins of R

119904

and R119898 Throughout this paper we assume that 119891 is locallyLipschitz 119892 ℎ and 119902 are continuously Frechet differentiable

119876 is closed (ie gph119876 is closed inR119899 timesR119898) and the feasibleset 119878 = 119909 isin R119899 | 119892(119909) isin minusR119903

+ ℎ(119909) = 0R119904 0R119898 isin

119902(119909) + 119876(119909) 119909 isin Θ of (MOPEC) is nonempty Obviously119878 is a closed subset of R119899 Recall that a point 119909 isin 119878 is said tobe an efficient (resp weak efficient) solution for (MOPEC) ifand only if

119891 (119909) minus 119891 (119909) notin minusR119901

+ 0R119901 (resp minus intR119901

+) forall119909 isin 119878

(2)

A point 119909 isin 119878 is said to be a local efficient (resp local weakefficient) solution for (MOPEC) if and only if there exists aneighborhood 119880 of 119909 such that

119891 (119909) minus 119891 (119909) notin minusR119901


+) forall119909 isin 119878 cap 119880

(3)

During the past few decades there have been a lot ofpapers devoted to study the scalar optimization problem (iethe case 119901 = 1) with equilibrium constraints which plays animportant role in engineering design economic equilibriaoperations research and so on It is well recognized thatthe scalar optimization problemwith equilibrium constraintscovers various classes of optimization-related problems and

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 630547 13 pageshttpdxdoiorg1011552014630547

2 Abstract and Applied Analysis

models arisen in practical applications such as mathematicalprograms with geometric constraints mathematical pro-grams with complementarity constraints and mathematicalprograms with variational inequality constraints For moredetails we refer to [1ndash4] It is worth noting that when 119876 isa general closed set-valued map even if119876(119909) is a fixed closedsubset of R119898 for all 119909 isin R119899 the general constraint system(1) fails to satisfy the standard linear independence constraintqualification and Mangasarian-Fromovitz constraint qualifi-cation at any feasible point [5]Thus it is a hardwork to estab-lish Karush-Kuhn-Tucker (in short KKT) necessary optimal-ity conditions for (MOPEC) Recently by virtue of advancedtools of variational analysis and various coderivatives forset-valued maps developed in [6ndash8] and references thereinsome necessary optimality conditions including the strongMordukhovich Clarke and Bouligand stationary conditionsare obtained by using different reformulations under somegeneralized constraint qualifications Simultaneously Ye andZhu [3] claimed that the Mordukhovich stationary (in shortM-stationary) condition is the strongest stationary conditionexcept the strong stationary condition which is equivalentto the classical KKT condition and proposed some newconstraint qualifications for M-stationary conditions to hold

It is well known that the penalization method is a veryimportant and effective tool for dealing with optimizationtheories and numerical algorithms of constrained extremumproblems In scalar optimization with equality and inequalityconstraints the classical exact penalty function with order 1was extensively used to investigate optimality conditions andconvergence analysis see [6 9 10] and references thereinClarke [6] derived some Fritz-John necessary optimalityconditions for a constrained mathematical programmingproblem on a Banach space by virtue of exact penaltyfunctions with order 1 Moreover Burke [9] showed that theexistence of an exact penalization function is equivalent to acalmness condition involving with the objective function andthe equality and inequality constrained system SubsequentlyFlegel and Kanzow [4] demonstrated that the correspondingrelationships still held in a generalized bilevel programmingproblem and a mathematical programming problem withcomplementarity constraints respectively Simultaneouslythey obtained some KKT necessary optimality conditionsby using exact penalty formulations and nonsmooth anal-ysis Recently the classical penalization theory has beenwidely generalized by various kinds of Lagrangian functionsespecially the augmented Lagrangian function introducedby Rockafellar and Wets [7] and the nonlinear Lagrangianfunction proposed by Rubinov et al [11] It has also beenproved that the exactness of these types of penalty functionsis equivalent to some generalized calmness conditions seemore details in [11 12]

However to the best of our knowledge there are onlya few papers devoted to study the penalty method for con-strainedmultiobjective optimization problems especially for(MOPEC) Huang and Yang [13] first introduced a vector-valued nonlinear Lagrangian and penalty functions formulti-objective optimization problems with equality and inequalityconstraints and obtained some relationships between theexact penalization property and a generalized calmness-type

condition Moreover Mordukhovich [8] and Bao et al [14]investigated some more general optimization problems withequilibrium constraints by methods of modern variationalanalysis It is worth noting that the standard Mangasarian-Fromovitz constraint qualification and error bound conditionfor a nonlinear programming problem with equality andinequality constraints implies the calmness condition see[6 15] for details Taking into account this fact it is neces-sary to further investigate the calmness condition and thepenalty method for constrained multiobjective optimizationproblems

The main motivation of this work is that there hasbeen no study on the penalization method and M-stationarycondition for (MOPEC) by using an appropriate calmnesscondition associated with the objective function and theconstraint system Although there have been many papersdealing with constrained multiobjective optimization prob-lems for example [3 8] and references therein the KKTnecessary optimality conditions are obtained under somegeneralized qualification conditions only involved with theconstraint system Inspired by the ideas reported in [3 4 6 813] we introduce a so-called (MOPEC-) calmness conditionwith order 120590 gt 0 at a local efficient (weak efficient) solutionassociated with the objective function and the constraintsystem for (MOPEC) and show that the (MOPEC-) calmnesscondition can be implied by an error bound condition of theconstraint system Moreover we establish some equivalentrelationships between the exact penalization property withorder 120590 and the (MOPEC-) calmness condition Simultane-ously we apply a nonlinear scalar technical to obtain a KKTnecessary optimality condition for (MOPEC) by using Mor-dukhovich generalized differentiation and the (MOPEC-)calmness condition with order 1

The organization of this paper is as follows In Section 2we recall some basic concepts and tools generally used invariational analysis and set-valued analysis In Section 3 weintroduce a (MOPEC-) calmness condition for (MOPEC)and establish some relationships between the exact penal-ization property and the (MOPEC-) calmness conditionMoreover we obtain a KKT necessary optimality conditionunder the (MOPEC-) calmness condition with order 1 InSection 4 we apply the obtained results to a multiobjectiveoptimization problem with complementarity constraints anda multiobjective optimization problem with weak vectorvariational inequality constraints respectively

2 Notations and Preliminaries

Throughout this paper all vectors are viewed as columnvectors Since all the norms on finite dimensional spaces areequivalent we take specially the sum norm on R119899 and theproduct space R119899 times R119898 for simplicity that is for all 119909 =

(1199091 119909

2 119909

119899)119879

isin R119899 119909 = |1199091| + |119909

2| + sdot sdot sdot + |119909

119899| and

for all (119909 119910) isin R119899 times R119898 (119909 119910) = 119909 + 119910 As usual wedenote by 119909119879 the transposition of 119909 and by ⟨119909 119910⟩ = 119909119879119910 theinner product of vectors 119909 and 119910 respectively For a givenmap 119891 R119899 rarr R119901 and a vector 120582 isin R119901 the function⟨120582 119891⟩ R119899 rarr R is defined by ⟨120582 119891⟩(119909) = ⟨120582 119891(119909)⟩ for

Abstract and Applied Analysis 3

all 119909 isin R119899 In general we denote byBR119899 the closed unit ballinR119899 and by B(119909 119903) the open ball with center at 119909 and radius119903 gt 0 for any 119909 isin R119899

The main tools for our study in this paper are theMordukhovich generalized differentiation notions which aregenerally used in variational analysis and set-valued analysissee more details in [6ndash8 16] and references therein Recallthat 119891 R119899

rarr R119901 is said to be Frechet differentiable at 119909if and only if there exists a matrix 119860 isin R119901times119899 such that

lim119909rarr119909

1003817100381710038171003817119891 (119909) minus 119891 (119909) minus 119860 (119909 minus 119909)1003817100381710038171003817

119909 minus 119909= 0 (4)

Obviously 119860 is uniquely determined by 119909 As usual 119860 iscalled the Frechet derivative of 119891 at 119909 and denoted by nabla119891(119909)If 119891 is Frechet differentiable at every 119909 isin R119899 then 119891 is said tobe Frechet differentiable on R119899 119891 is said to be continuouslyFrechet differentiable at 119909 if and only if the map nabla119891(∙) R119899 rarr R119901times119899 is continuous at 119909 Specially we denote by(nabla119891(119909))

lowast

R119901 rarr R119899 the adjoint operator of nabla119891(119909) that is⟨nabla119891(119909)(119909) 119910⟩ = ⟨119909 (nabla119891(119909))

lowast

(119910)⟩ for all 119909 isin R119899 and 119910 isin R119901Moreover119891 is said to be strictly differentiable at 119909 if and onlyif

lim119909rarr119909119906rarr119909119909 = 119906

1003817100381710038171003817119891 (119909) minus 119891 (119906) minus nabla119891 (119909) (119909 minus 119906)1003817100381710038171003817

119909 minus 119906= 0 (5)

Obviously if 119891 is continuously Frechet differentiable at 119909then 119891 is strictly differentiable at 119909

For a nonempty subset 119878 sub R119899 the indicator function120595(∙ 119878) R119899 rarr R cup +infin is defined by 120595(119909 119878) = 0forall119909 isin 119878 and 120595(119909 119878) = +infin forall119909 notin 119878 and the distance function119889(∙ 119878) R119899 rarr R is defined by 119889(119909 119878) = inf

119910isin119878119909 minus 119910

for all 119909 isin R119899 respectively Given a point 119909 isin 119878 recall thatthe Frechet normal cone (119878 119909) of 119878 at 119909 which is a convexclosed subset of R119899 and consisted of all the Frechet normalshas the form

(119878 119909) =

119909lowast

isin R119899

| lim sup119909

119878

997888rarr119909

⟨119909lowast

119909 minus 119909⟩

119909 minus 119909le 0

(6)

where 119909 119878

997888rarr 119909 means 119909 isin 119878 and 119909 rarr 119909 The Mordukhovich(or basic limiting) normal cone of 119878 at 119909 is

119873(119878 119909) = 119909lowast

isin R119899

| exist119909119899

119904

997888rarr 119909 exist119909lowast

119899997888rarr 119909

lowast

with 119909lowast119899isin (119878 119909

119899) forall119899 isin N

(7)

Specially if 119878 is convex then we have

(119878 119909) = 119873 (119878 119909) = 119909lowast

isin R119899

| ⟨119909lowast

119909 minus 119909⟩ le 0 forall119909 isin 119878

(8)

Let ℎ R119899 rarr Rcup+infin be an extended real-valued functionand let 119909 isin dom ℎ where dom ℎ = 119909 isin R119899 | ℎ(119909) lt +infin

denotes the domain of ℎ The Frechet subdifferential ℎ(119909) ofℎ at 119909 is defined in the geometric form by ℎ(119909) = 119909lowast isin R119899 |

(119909lowast minus1) isin (epi ℎ (119909 ℎ(119909))) or equivalently is defined in

the analytical form by

ℎ (119909)

= 119909lowast

isin R119899

| lim inf119909rarr119909 119909 = 119909

ℎ (119909) minus ℎ (119909) minus ⟨119909lowast 119909 minus 119909⟩

119909 minus 119909ge 0

(9)

The Mordukhovich (or basic limiting) subdifferential 120597ℎ(119909)and singular subdifferential 120597infinℎ(119909) of ℎ at 119909 are definedrespectively by 120597ℎ(119909) = 119909lowast isin R119899 | (119909lowast minus1) isin

119873(epi ℎ (119909 ℎ(119909))) and 120597infinℎ(119909) = 119909lowast isin R119899 | (119909lowast 0) isin

119873(epi ℎ (119909 ℎ(119909))) Clearly we have ℎ(119909) sub 120597ℎ(119909) and

120597ℎ (119909) = 119909lowast

isin R119899

| exist119909119899

ℎ


119899997888rarr 119909

lowast

with 119909lowast119899isin ℎ (119909

119899)

(10)

where 119909119899

ℎ

997888rarr 119909means 119909119899rarr 119909 and ℎ(119909

119899) rarr ℎ(119909) Specially

for any119909 isin 119878 it follows that 120595(119909 119878) = (119878 119909) and 120597120595(119909 119878) =120597infin120595(119909 119878) = 119873(119878 119909) Furthermore if ℎ is a convex functionthen we have

ℎ (119909) = 120597ℎ (119909)

= 119909lowast

isin R119899

| ⟨119909lowast

119909 minus 119909⟩ le ℎ (119909) minus ℎ (119909) forall119909 isin R119899

120597infin

ℎ (119909) sub 119909lowast

isin R119899

| ⟨119909lowast

119909 minus 119909⟩ le 0 forall119909 isin dom ℎ

= 119873 (dom ℎ 119909) (11)

Recall that the Frechet coderivative 119863lowast119865(119909 119910) and the Mor-dukhovich (or basic limiting) coderivative 119863lowast119865(119909 119910) of theset-valued map 119865 R119899 999490999473 R119901 at (119909 119910) isin gph119865 are the set-valued maps from R119901 to R119899 defined respectively by

119863lowast

119865 (119909 119910) (119910lowast

)

= 119909lowast

isin R119899

| (119909lowast

minus119910lowast

) isin (gph119865 (119909 119910))

forall119910lowast

isin R119901

119863lowast

119865 (119909 119910) (119910lowast

)

= 119909lowast

isin R119899

| (119909lowast

minus119910lowast

) isin 119873 (gph119865 (119909 119910))

forall119910lowast

isin R119901

(12)

Next we collect some useful and important propositionsand definitions for this paper

Proposition 1 (see [8]) For every nonempty subset Ω sub R119899

and every 119909 isin Ω we have

(i) 119889(∙ Ω)(119909) = BR119899 cap (Ω 119909) and (Ω 119909) =

⋃120582gt0120582119889(∙ Ω)(119909)


In addition if Ω is closed then we get

(ii) 120597119889(∙ Ω)(119909) sub BR119899 cap 119873(Ω 119909) and 119873(Ω 119909) =

⋃120582gt0120582120597119889(∙ Ω)(119909)

The following necessary optimality condition calledgeneralized Fermat rule for a function to attain its localminimum is useful for our analysis

Proposition 2 (see [7 8]) Let 120593 R119899 rarr R cup +infin bea proper lower semicontinuous function If 119891 attains a localminimum at 119909 isin R119899 then 0R119899 isin 119891(119909) and 0R119899 isin 120597119891(119909)

We recall the following sum rule for the Mordukhovichsubdifferential which is important in the sequel

Proposition 3 (see [8]) Let 1205931 120593

2 R119899 rarr R cup +infin

be proper lower semicontinuous functions and 119909 isin dom1205931cap

dom1205932 Suppose that the qualification condition

120597infin

1205931(119909) cap (minus120597

infin

1205932(119909)) = 0R119899 (13)

is fulfilled Then one has

120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (14)

Specially if either 1205931or 120593

2is locally Lipschitz around 119909 then

one always has

120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (15)

The following propositions of the scalarization of Mor-dukhovich coderivatives and the chain rule ofMordukhovichsubdifferentials are important for this paper

Proposition 4 (see [8 16]) Let 120593 R119899 rarr R119901 be continuousaround 119909 Then

120597 ⟨119910lowast

120593⟩ (119909) sub 119863lowast

120593 (119909) (119910lowast

) forall119910lowast

isin R119901

(16)

If in addition 120593 is locally Lipschitz around 119909 then

119863lowast

120593 (119909) (119910lowast

) = 120597 ⟨119910lowast

120593⟩ (119909) forall119910lowast

isin R119901

(17)

Proposition 5 (see [8 16]) Let the vector-valued map 119867

R119899 rarr Rℓ be locally Lipschitz and let ℎ Rℓ rarr R be lowersemicontinuous If

119910lowast

isin 120597infin

ℎ (119867 (119909)) 0R119899 isin 119863lowast

119867(119909) (119910lowast

)

119894119898119901119897119894119890119904 119910lowast

= 0Rℓ (18)

then

120597ℎ ∘ 119867 (119909) sub 119863lowast

119867(119909) (119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (19)

Moreover if119867 is strictly differentiable and ℎ is locally Lipschitzthen one always has

120597ℎ ∘ 119867 (119909) sub (nabla119867 (119909))lowast

(119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (20)

Finally in this section we recall the following usefulconcept called nonlinear scalar function and some of itsproperties For more details we refer to [17ndash20]

Lemma 6 Given 119890 = (1 1 1) isin intR119901

+ the nonlinear

scalar function 120585119890 R119901 rarr R defined by

120585119890(119910) = inf 120572 isin R | 119910 isin 120572119890 minusR

119901

+ forall119910 isin R

119901

(21)

is convex strictly intR119901

+-monotone R

119901

+-monotone

nonnegative homogeneous globally Lipschitz with modulus119889(119890 bd R

119901

+)minus1 Simultaneously for every 120572 isin R it follows that

120585119890(120572119890) = 120572

119910 isin R119901

| 120585119890(119910) le 120572 = 120572119890 minusR

119901

+

119910 isin R119901

| 120585119890(119910) lt 120572 = 120572119890 minus intR119901

+

(22)

Furthermore for every 119910 isin R119901

120597120585119890(119910) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 ⟨120582 119910⟩ = 120585

119890(119910) (23)

Specially one has

120597120585119890(0R119901) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 (24)

3 Exact Penalization CalmnessCondition and Necessary OptimalityCondition for (MOPEC)

In this section we focus our attention on establishingsome equivalent properties between a multiobjective exactpenalization and a calmness condition called (MOPEC-)calmness for (MOPEC) Simultaneously we show that a localerror bound condition associated merely with the constraintsystem equivalently a calmness condition of the parametricconstraint system implies the (MOPEC-) calmness condi-tion Subsequently we apply a nonlinear scalar method toobtain a M-stationary necessary optimality condition underthe (MOPEC-) calmness condition

Consider the following parametric form of the feasible set119878 with parameter (119906 V 119910 119911) isin R119903+119904+119899+119898

119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ(25)

Denote the corresponding feasible set by

119878 (119906 V 119910 119911) = 119909 isin R119899

| 119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ

(26)

Obviously for the set-valued map 119878 R119903+119904+119899+119898 999490999473 R119899 wehave 119878 = 119878(0R119903+119904+119899+119898)

We are now in the position to introduce a (MOPEC-)calmness concept for (MOPEC)

Definition 7 Given 120590 gt 0 and 119909 isin 119878 being a localefficient (resp local weak efficient) solution for (MOPEC)


then (MOPEC) is said to be (MOPEC-) calm with order 120590 at119909 if and only if there exist 120575 gt 0 and119872 gt 0 such that for all(119906 V 119910 119911) isin B(0R119903+119904+119899+119898 120575) and all 119909 isin 119878(119906 V 119910 119911) cap B(119909 120575)one has

119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (27)

Remark 8 Given 120590 gt 0 and 119909 isin 119878 being a local efficient(resp local weak efficient) solution for (MOPEC) we can alsocharacterize the (MOPEC-) calmness condition by means ofsequences It is easy to verify that (MOPEC) is (MOPEC-)calm with order 120590 at 119909 if and only if there exists 119872 gt 0

such that for every sequence (119906119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898

with (119906119896 V119896 119910

119896 119911119896) rarr 0R119903+119904+119899+119898 and every sequence 119909119896 sub Θ

satisfying 119892(119909119896) + 119906

119896isin R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin 119892(119909119896) +

119876(119909119896+ 119910

119896) and 119909

119896rarr 119909 it holds that

119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (28)

Note that the (MOPEC-) calmness condition depends onnot only the objective function but also the constraint systemIn order to make up this deficiency we propose the followinglocal error bound notion for (MOPEC) associated merelywith the constraint system

Definition 9 Given 120590 gt 0 and 119909 isin 119878 then the constraintsystem of (MOPEC) is said to have a local error bound withorder 120590 at 119909 if and only if there exist 120575 gt 0 and119872 gt 0 suchthat for all (119906 V 119910 119911) isin B(0R119903+119904+119899+119898 120575) 0R119903+119904+119899+119898 and all 119909 isin119878(119906 V 119910 119911) cap B(119909 120575) one has

119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (29)

Next we show that the local error bound implies the(MOPEC-) calmness

Theorem 10 Let 119909 isin 119878 be a local efficient (resp local weakefficient) solution for (MOPEC) If the constraint system of(MOPEC) has a local error bound with order 120590 at 119909 then(MOPEC) is (MOPEC-) calm with order 120590 at 119909

Proof Since 119909 isin 119878 is a local efficient (resp local weakefficient) solution for (MOPEC) and 119878 = 119878(0R119903+119904+119899+119898) itimmediately follows that

119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+(30)

holds for all 119909 isin 119878(119906 V 119910 119911) cap B(119909 120575) with (119906 V 119910 119911) =0R119903+119904+119899+119898 and sufficiently small 120575 gt 0 Thus we only need toprove the case (119906 V 119910 119911) = 0R119903+119904+119899+119898 Assume that (MOPEC) isnot (MOPEC-) calm with order 120590 at 119909 Then for every 119896 isin Nthere exist (119906

119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) 0R119903+119904+119899+119898 and

119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) cap B(119909 1119896) such that

119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (31)

Since 119878 is nonempty and closed there exists a projection119875(119909

119896 119878) of 119909

119896onto 119878 such that 119889(119909

119896 119878) = 119909

119896minus 119875(119909

119896 119878) for

all 119896 isin N Note that 119909119896rarr 119909 and 119909 isin 119878 Then it follows that

1003817100381710038171003817119875 (119909119896 119878) minus 1199091003817100381710038171003817 le

1003817100381710038171003817119875 (119909119896 119878) minus 1199091198961003817100381710038171003817 +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817

= 119889 (119909119896 119878) +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817 997888rarr 0

(32)

Together with 119909 isin 119878 being a local efficient (resp local weakefficient) solution for (MOPEC) there exists some 119873

1isin N

such that

119891 (119875 (119909119896 119878)) minus 119891 (119909) notin minusR

119901


+)

forall119896 ge 1198731

(33)

Moreover since 119891 is locally Lipschitz there exist 119871 gt 0 and1198732isin N such that1003817100381710038171003817119891 (119909119896) minus 119891 (119875 (119909119896 119878))

1003817100381710038171003817 le 1198711003817100381710038171003817119909119896 minus 119875 (119909119896 119878)

1003817100381710038171003817 forall119896 ge 1198732

(34)

By (31) and (33) we have for all 119896 ge 1198731

119891 (119875 (119909119896 119878)) minus 119891 (119909

119896)

= 119891 (119875 (119909119896 119878)) minus 119891 (119909)

+ (119891 (119909) minus 119891 (119909119896)) notin 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

(35)

Together with 119889(119909119896 119878) = 119909

119896minus 119875(119909

119896 119878) and (34) we can

conclude that

119889 (119909119896 119878)BR119901 sub

119896

119871

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

forall119896 ge max 1198731 119873

2

(36)

This is a contradiction to the assumption that (MOPEC) hasa local error bound with order 120590 at 119909 since 119896119871 rarr +infin(119906119896 V119896 119910

119896 119911119896) = 0R119903+119904+119899+119898 (119906119896 V119896 119910119896 119911119896) rarr 0R119903+119904+119899+119898 119909119896 isin

119878(119906119896 V119896 119910

119896 119911119896) and 119909

119896rarr 119909

Remark 11 Specially if we consider the case 119901 = 1 for everygiven 120590 gt 0 and 119909 isin 119878 then Definition 9 reduces to the factthat there exist 120575 gt 0 and119872 gt 0 such that for all (119906 V 119910 119911) isinB(0R119903+119904+119899+119898 120575) 0R119903+119904+119899+119898 and all 119909 isin 119878(119906 V 119910 119911) cap B(119909 120575)one has

119889 (119909 119878) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(37)

It is worth noting that this condition is essentially sufficientand necessary for the situation 119901 gt 1 Clearly the necessityholds In fact since (1119901)119890 isinBR119901 it follows that

119889 (119909 119878)1

119901119890 isin 119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

119890 minus intR119901

+ (38)

for all (119906 V 119910 119911) isin B(0R119903+119904+119899+119898 120575) 0R119903+119904+119899+119898 and all119909 isin 119878(119906 V 119910 119911) cap B(119909 120575) Moreover 120585

119890is nonnegative

homogeneous and 120585119890((1119901)119890) = 1119901 By Lemma 6 we have

120585119890(119889(119909 119878)(1119901)119890) lt 119872(119906 V 119910 119911)120590 which implies 119889(119909 119878) lt

119901119872(119906 V 119910 119911)120590 For the sufficiency since 120585119890is continuous

andBR119901 is compact there exists some119898 isin R such that

119898 = max119908isinBR119901

120585119890(119908) (39)

Obviously (1119901)119890 isin BR119901 and 120585119890((1119901)119890) = 1119901 gt 0 then wehave119898 gt 0 Thus we get from the nonnegative homogeneity


of 120585119890that for all (119906 V 119910 119911) isin B(0R119903+119904+119899+119898 120575) 0R119903+119904+119899+119898 all

119909 isin 119878(119906 V 119910 119911) cap B(119909 120575) and all 119908 isinBR119901

120585119890(119889 (119909 119878) 119908) = 119889 (119909 119878) 120585

119890(119908) lt 119898119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

(40)

By Lemma 6 we have

119889 (119909 119878) 119908 isin 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (41)

which implies

119889 (119909 119878)BR119901 sub 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (42)

Furthermore recall that a set-valued map Ψ R119905 999490999473 R119899 issaid to be calm with order 120590 gt 0 at (119909 119910) isin gphΨ if andonly if there exist neighborhoods119880 of 119909 and119881 of 119910 and a realnumber ℓ gt 0 such that

Ψ (119909) cap 119881 sub Ψ (119909) + ℓ119909 minus 119909120590

BR119899 forall119909 isin 119880 (43)

Then we can immediately obtain the following character-ization of local error bounds for the constraint system of(MOPEC) based on the arguments in Remark 11

Proposition 12 Given 120590 gt 0 and 119909 isin 119878 then the followingassertions are equivalent

(i) The constraint system of (MOPEC) has a local errorbound with order 120590 at 119909

(ii) There exist 120575 gt 0 and 119872 gt 0 such that for all(119906 V 119910 119911) isin B(0R119903+119904+119899+119898 120575) 0R119903+119904+119899+119898 and all 119909 isin

119878(119906 V 119910 119911) cap B(119909 120575)

119889 (119909 119878 (0R119903+119904+119899+119898)) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(44)

(iii) The set-valued map 119878 R119903+119904+119899+119898 999490999473 R119899 defined by(26) is calm with order 120590 at (0R119903+119904+119899+119898 119909)

Proof As discussed in Remark 11 (i) is equivalent to (ii) Weonly need to prove the equivalence of (ii) and (iii) In fact itfollows from the definition of calmness for a set-valued mapthat (ii) is obviously equivalent to the calmness with order 120590at (0R119903+119904+119899+119898 119909) of the set-valued map 119878

As we know there have been many papers devoted toinvestigate the calmness of a set-valued map Ψ (which isequivalent to the metric subregularity of its converse Ψminus1)For more details we refer to [21ndash24] and references thereinIt has been shown in Remark 11 and Proposition 12 that therehave been no differences between the scalar (119901 = 1) and themultiobjective (119901 gt 1) settings when we only consider thecalmness or the local error bound for the constraint systemof (MOPEC) However if we pay attention to the weaker(MOPEC-) calmness we cannot negative the differencesbetween them

We now give the following equivalent characterizationsof two classes of multiobjective penalty problems and the(MOPEC-) calmness condition

Theorem 13 Let 119909 isin 119878 be a local efficient (resp local weakefficient) solution for (MOPEC) Then the following assertionsare equivalent

(i) (MOPEC) is (MOPEC-) calm with order 120590 gt 0 at 119909

(ii) There exists some 120588 gt 0 such that for any 120588 ge 120588(119909 0R119899+119898) is a local efficient (resp local weak efficient)solution for the following multiobjective penalty prob-lem with order 120590

(119872119875119875)119868

min 119891 (119909) + 120588 (1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) +1003817100381710038171003817(119910 119911)

1003817100381710038171003817)120590

119890

119904119905 119911 isin 119902 (119909) + 119876 (119909 + 119910)

119909 isin Θ (119910 119911) isin R119899+119898

(45)

where 119892+(119909) = (max119892

1(119909) 0 max119892

2(119909) 0

max119892119903(119909) 0)

(iii) There exists some 120583 gt 0 such that for any 120583 ge 120583 119909 is alocal efficient (resp local weak efficient) solution for thefollowing multiobjective penalty problem with order 120590

(119872119875119875)119868119868

min 119891 (119909)

+ 120583[1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)]120590

119890

119904119905 119909 isin Θ

(46)

Proof We only prove the case for 119909 being a local weakefficient solution since the proof of the case for 119909 being a localefficient solution is similar

(i)rArr(ii) Suppose to the contrary that for every 119896 isin Nthere exists (119909

119896 119910

119896 119911119896) isin B((119909 0R119899+119898) 1119896) with 119909119896 isin Θ and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) such that

119891 (119909119896) + 119896 (

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

119890 isin 119891 (119909) minus intR119901

+

(47)

Take 119906119896= minus119892

+(119909119896) and V

119896= minusℎ(119909

119896) Then it follows that

119892(119909119896) + 119906

119896isin minusR119903

+and ℎ(119909

119896) + V

119896= 0R119904 Together with 119911119896 isin

119902(119909119896) + 119876(119909

119896+ 119910

119896) and 119909

119896isin Θ we get 119909

119896isin 119878(119906

119896 V119896 119910

119896 119911119896)

for all 119896 isin N Moreover by (47) we have

119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (48)

Note that 119909119896rarr 119909 119892(119909) isin minusR119903

+ ℎ(119909) = 0R119904 and 119892 and ℎ

are continuously Frechet differentiable Then it follows that119906119896= 119892

+(119909119896) rarr 0R119903 and V

119896= ℎ(119909

119896) rarr 0R119904 Together

with (119910119896 119911119896) rarr 0R119899+119898 and (48) this is a contradiction to the

(MOPEC-) calmness with order 120590 of (MOPEC) at 119909(ii)rArr(i) Suppose that (MOPEC) is not (MOPEC-) calm

with order 120590 gt 0 at 119909 Then for every 119896 isin N there exist(119906119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin 119878(119906119896 V119896 119910119896 119911119896) cap

B(119909 1119896) such that (31) holds Since 119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) it

follows that 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876

for all 119896 isin N Thus we have1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910

119896 119911119896)120590 forall119896 isin N Together with (31) we get

119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901

+ forall119896 isin N

(50)

This shows that the multiobjective penalty problem (MPP)Iwith order 120590 does not admit a local exact penalization at(119909 0R119899+119898) since 119909119896 isin Θ 119911

119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)

(i)rArr(iii) Assume that for every 119896 isin N there exists 119909119896isin

Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890

minus 119891 (119909) isin minus intR119901

+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)

Thus for every 119896 isin N there exists (120572119896 120573

119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573

119896 Then it follows that 119892(119909

119896) + 119906


+ ℎ(119909

119896) +

V119896= 0R119904 and 119911119896 isin 119902(119909119896) + 119876(119909119896 + 119910119896) which implies 119909

119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)

Connecting 119890 isin int R119901

+ (51) and (54) we have for any 119896 isin N

119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890

isin minus intR119901

+minus intR119901

+= minus intR119901

+

(55)

Moreover it follows from [25 Lemma 321] and (51) that forany 120582 isin R

119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)

Note that 119891 is locally Lipschitz and 119909119896rarr 119909 Then it follows

that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910

119896 119911119896) rarr 0R119903+119904+119899+119898 from (54) Together

with 119896120590+1(119896 + 1)120590 rarr +infin 119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909

and (55) this is a contradiction to the (MOPEC-) calmnesswith order 120590 of (MOPEC) at 119909

(iii)rArr(i) Assume that (MOPEC) is not (MOPEC-) calmwith order 120590 gt 0 at 119909 Then by the same argument to theproof of (ii)rArr(i) it follows that for every 119896 isin N thereexist (119906

119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876

such that (31) holds Thus we have

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

+1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (119909119896 + 119910119896 119911119896 minus 119902 (119909119896))

1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)


Together with (31) and 119890 isin intR119901

+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(58)which implies that the multiobjective penalty problem(MPP)II with order 120590 does not admit a local exact penaliza-tion at 119909 since the sequence 119909

119896 sub Θ and 119909

119896rarr 119909

It is well known that a calmness condition with order1 for standard nonlinear programming can lead to a KKTcondition In fact we can also obtain a M-stationary condi-tion for (MOPEC) under the (MOPEC-) calmness conditionwith order 1 To this end we need the following generalizedFermat rule for a multiobjective optimization problem withan abstract constraint which is established by applying thenonlinear scalar function in Lemma 6

Lemma 14 Let 120601 R119899 rarr R119901 be a locally Lipschitz vector-valued map and let Ω sub R119899 be a nonempty and closed subsetIf 119909 isin Ω is a local weak efficient solution for the multiobjectiveoptimization problem

min 120601 (119909)

119904119905 119909 isin Ω(59)

then there exists some 120582 isin R119901

+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)

Proof Define the functionΦ R119899 rarr R byΦ (119909) = 120585

119890(120601 (119909) minus 120601 (119909)) + 120595 (Ω 119909) forall119909 isin R

119899

(61)Since 119909 isin Ω is a local weak efficient solutionΦ attains a localminimum at 119909 Otherwise there exists a sequence 119909

119899 sub R119899

converging to 119909 such thatΦ(119909119899) lt 0 sinceΦ(119909) = 0 Then we

have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909

119899) minus 120601(119909)) lt 0 Together

with Lemma 6 we get120601 (119909

119899) minus 120601 (119909) isin minus intR119901

+ forall119899 isin N (62)

This is a contradiction to 119909 isin Ω being a local weak efficientsolution since 119909

119899 sub Ω and 119909

119899rarr 119909 Note that 120601 is locally

Lipschitz and Ω is closed It follows from Propositions 2 3and 5 and Lemma 6 that0R119899 isin 120597Φ (119909) sub 120597120585119890 (120601 (∙) minus 120601 (119909)) (119909) + 120597120595 (Ω ∙) (119909)

sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)

Therefore there exists some 120582 isin R119901

+with sum119901

119894=1120582119894= 1 such

that

0R119899 isin 119863lowast

120601 (119909) (120582) + 119873 (Ω 119909) (64)

By Proposition 4 it follows that

0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)

This completes the proof

Next we show that the (MOPEC-) calmness conditionwith order 1 is sufficient to establish aM-stationary conditionfor (MOPEC)

Theorem 15 Suppose that 119909 isin 119878 is a local weak efficientsolution for (MOPEC) and (MOPEC) is (MOPEC-) calm withorder 1 at 119909 Then 119909 is a M-stationary point for (MOPEC) thatis there exist 120582 isin R

119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0

and (119909lowast 119910lowast) isin R119899+119898 with 119909lowast isin 119863lowast119876(119909 minus119902(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)

Proof Since 119909 isin 119878 is a local weak efficient solution for(MOPEC) and (MOPEC) is (MOPEC-) calm with order 1at 119909 it follows from Theorem 13 (i)hArr(iii) that there existssome 120583 gt 0 such that 119909 is a local weak efficient solution forthe multiobjective penalty problem (MPP)II with order 1 Forsimplicity let the real-valued functionT R119899

rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)

forall119909 isin R119899

(67)

Note that 119891 is locally Lipschitz 119892 ℎ and 119902 are continuouslyFrechet differentiable and 119876 is closed Then T is locallyLipschitz and the penalty function119891(∙)+120583T(∙)119890 R119899 rarr R119901

is also locally Lipschitz Together with the closedness of Θand Lemma 14 it follows that there exists some 120582 isin R

119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)

Moreover by using ⟨120582 119890⟩ = sum119901

119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)

Note that 119892 ℎ and 119902 are continuously Frechet differentiableand119876 is closedThen it follows fromPropositions 1 (ii) 4 and5 that for all 119894 isin 1 2 119903

120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)

sub (IR119899 minusnabla119902 (119909))lowast

(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)

where IR119899 denotes the identity map from R119899 to itself LetJ(119909) = 119894 isin 1 2 119903 | 119892

119894(119909) = 0 be the set of active

constraints of 119892 at 119909 Then we can conclude from (70)ndash(72)that

120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)

Together with (68) and (69) there exist 120573119894ge 0 with 119894 isin J(119909)

120574 isin R119904 and (119909lowast minus119910lowast) isin 119873(gph119876 (minus119909 119902(119909))) that is 119909lowast isin119863lowast

119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0

119894 isin 1 2 119903 J(119909) 120574 isin R119904 with 120574 = 120583 120574 and 120591 = 120583Then we have

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)


Combining Proposition 12 and Theorem 15 we immedi-ately have the following corollary

Corollary 16 Let 119909 isin 119878 be a local efficient solution for(MOPEC) Suppose that the constraint system of (MOPEC) hasa local error bound with order 1 at 119909 or equivalently the set-valued map 119878 R119903+119904+119899+119898 999490999473 R119899 defined by (26) is calmwith order 1 at (0R119903+119904+119899+119898 119909) Then 119909 is a M-stationary pointfor (MOPEC)

Remark 17 Recently Kanzow and Schwartz [26] discussedthe enhanced Fritz-John conditions for a smooth scalar opti-mization problemwith equilibrium constraints and proposedsome new constraint qualifications for the enhanced M-stationary condition In particular they obtained some suf-ficient conditions for the existence of a local error bound forthe constraint system and the exactness of penalty functionswith order 1 by using an appropriate condition SubsequentlyYe and Zhang [27] extendedKanzow and Schwartzrsquos results tothe nonsmooth case It is worth noting that the exactness ofthe penalty function with order 1 in [26 27] was establishedby using various qualification conditions whichwere actuallysufficient for the local error bound property of the constraintsystem see [28 29] for more details However just asshown in Theorem 13 the exactness for the two types ofmultiobjective penalty functions with order 120590 is obtainedby means of the equivalent (MOPEC-) calmness conditionwhich is associated with not only the objective function butalso the constraint system Simultaneously it follows fromTheorem 10 and Proposition 12 that the (MOPEC-) calmnesscondition is weaker than the local error bound property ofthe constraint system

4 Applications

The main purpose of this section is to apply the obtainedresults for (MOPEC) to a multiobjective optimization prob-lem with complementarity constraints (in short (MOPCC))and a multiobjective optimization problem with weak vectorvariational inequality constraints (in short (MOPWVVI))and establish corresponding calmness conditions and M-stationary conditions

41 Applications to (MOPCC) In this subsection we con-sider the followingmultiobjective optimization problemwithcomplementarity constraints

(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)

where 119891 R119899 rarr R119901 is locally Lipschitz 119892 R119899 rarr R119903ℎ R119899 rarr R119904 119866 119867 R119899 rarr R119897 are continuously Frechet


differentiable and Θ is a nonempty and closed subset of R119899As usual we denote

1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)

Obviously the feasible set 119878 = 119909 isin R119899 | 119892(119909) isin minusR119903

+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879

119867(119909) = 0 119909 isin Θ is aclosed subset of R119899 It is easy to verify that (MOPCC) can bereformulated as a special case of (MOPEC) if we let119898 = 2119897

119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)

where 119862 = (119886 119887) isin R2 | 0 le minus119886 perp minus119887 ge 0 Note that119876 is constant and equals to 119862119897 Then the parametric form119878(119906 V 119910 119911) of 119878 with parameter (119906 V 119910 119911) isin R119903+119904+2119897 is

119878 (119906 V 119910 119911) = 119909 isin Θ | 119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)

Clearly for the set-valued map 119878 R119903+119904+2119897 999490999473 R119899 one has119878(0R119903+119904+2119897) = 119878

Inspired by Definitions 7 and 9 we give the followingconcepts called (MOPCC-) calm and local error bound for(MOPCC)

Definition 18 Given 120590 gt 0 and 119909 isin 119878 being a localefficient (resp local weak efficient) solution for (MOPCC)then (MOPCC) is said to be (MOPCC-) calm with order 120590at 119909 if and only if there exist 120575 gt 0 and119872 gt 0 such that forall (119906 V 119910 119911) isin B(0R119903+119904+2119897 120575) and all 119909 isin 119878(119906 V 119910 119911) capB(119909 120575)one has

119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)

Definition 19 Given 120590 gt 0 and 119909 isin 119878 then the constraintsystem of (MOPCC) is said to have a local error bound withorder 120590 at 119909 if and only if there exist 120575 gt 0 and119872 gt 0 suchthat for all (119906 V 119910 119911) isin B(0R119903+119904+2119897 120575) 0R119903+119904+2119897 and all 119909 isin119878(119906 V 119910 119911) cap B(119909 120575) one has

119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)

Similarly it follows fromTheorem 10 that if the constraintsystem of (MOPCC) has a local error bound with order120590 at 119909 then (MOPCC) is (MOPCC-) calm with order 120590at 119909 Moreover by Proposition 12 the constraint system of(MOPCC) has a local error bound with order 120590 at 119909 if and

only if the set-valued map 119878 R119903+119904+2119897 999490999473 R119899 is calm withorder 120590 at (0R119903+119904+2119897 119909) Specially if we take 119901 = 1 and 120590 = 1thenDefinitions 18 and 19 reduce toDefinitions 33 and 36 in[4] respectively Simultaneously the corresponding results toPropositions 34 and 37 in [4] also hold

As mentioned in the introduction there have beenvarious stationary concepts proposed for (MOPCC) Here weonly recall the notion of the M-stationary point

Definition 20 (see [4]) A point 119909 isin 119878 is called aM-stationarypoint of (MOPCC) if and only if there exists a Lagrangemultiplier 120582lowast = (120582119891 120582119892 120582ℎ 120582119866 120582119867) isin R119901+119903+119904+2119897 with 120582119891 isin R

119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866

119894isin R forall119894 isin 119868

0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)

The following formula for the Mordukhovich normalcone of the set 119862 is useful in the sequel

Lemma 21 (see [4]) For every (119886 119887) isin 119862 we have

119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)

We now apply Theorem 15 to establish a M-stationarycondition for (MOPCC) by virtue of the (MOPCC-) calmnesscondition

Theorem 22 Suppose that 119909 isin 119878 is a local weak efficientsolution for (MOPCC) and (MOPCC) is (MOPCC-) calm withorder 1 at 119909 Then 119909 is a M-stationary point of (MOPCC)

Proof As stated above (MOPCC) is equivalent to (MOPCC)with119898 = 2119897 and 119902119876 given by (78) Note that the (MOPCC-)calmness of (MOPCC) implies the (MOPEC-) calmness of(MOPCC) Then it follows from Theorem 15 that there exist


120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and

(119909lowast 119910lowast) isin R119899+2119897 with 119909lowast isin 119863lowast119876(119909 minus119902(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)

Note that 119876(119909) = 119862119897 for all 119909 isin R119899 Then it follows thatgph119876 = R119899 times 119862119897 and

119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))

= 0R119899 times 119873 (119862 (minus1198661 (119909) minus1198671(119909)))

times sdot sdot sdot times 119873 (119862 (minus119866119897(119909) minus119867

119897(119909)))

(85)

Together with Lemma 21 we have for every (119909lowast 119910lowast) isin R119899+2119897

with 119909lowast isin 119863lowast119876(119909 minus119902(119909))(119910lowast)

119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)

Taking 120582lowast = (120582119891 120582119892 120582ℎ 120582119866 120582119867) isin R119901+119903+119904+2119897 with 120582119891 = 120582120582119892 = 120573 120582ℎ = 120574 120582119866

119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897

and substituting (86) and (87) into (84) thenwe can concludethat 119909 isin 119878 is a M-stationary point of (MOPCC) with respectto the Lagrange multiplier 120582lowast

42 Applications to (MOPWVVI) Consider the followingmultiobjective optimization problem with weak vector varia-tional inequality constraints

(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)

where 119891 R119899 rarr R119901 is locally Lipschitz 119892 R119899 rarr R119903 andℎ R119899 rarr R119904 are continuously Frechet differentiable Θ isthe solution set of the weak vector variational inequality (inshort (WVVI)) find a vector 119909 isin Θ such that

(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

notin minus intR119898

+ forall119908 isin Θ

(89)

where119865119894 R119899

rarr R119899 119894 = 1 2 119898 are continuously Frechetdifferentiable andΘ is a nonempty closed and convex subsetof R119899 In the sequel we denote 119878 = 119909 isin R119899 | 119892(119909) isin

minusR119903

+ ℎ(119909) = 0R119904 119909 isin Θ by the feasible set of (MOPWVVI)

Then it is clear that 119878 is closedTake 119890

0= (1 1 1) isin R119898 Then it follows from

Lemma 6 that 119909 isin R119899 is a solution of (WVVI) if and onlyif

119909 isin Θ inf119908isinΘ

1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)

Moreover sinceΘ is nonempty closed and convex and 1205851198900

isa convex function we can conclude fromTheorem 815 in [7]and Proposition 5 that 119909 is a solution of (WVVI) if and onlyif

0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)

This shows that there exists some 120577 = (1205771 1205772 120577

119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)

Given 119909 isin 119878 being a local efficient (resp local weakefficient) solution for (MOPWVVI) then 119909 is a solution of(MOPWVVI) Next we define the concept of (MOPWVVI-)calmness with order 120590 gt 0 at 119909 for (MOPWVVI) with respectto the corresponding 120577 = (120577

1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)

Definition 23 Given 120590 gt 0 119909 isin 119878 being a local efficient(resp local weak efficient) solution for (MOPWVVI) and120577 = (120577

1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then

(MOPWVVI) is said to be (MOPWVVI-) calm with order 120590at 119909 if and only if there exists 119872 gt 0 such that for everysequence (119906

119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr

0R119903+119904+119899+119898 and every sequence 119909119896 sub Θ satisfying 119892(119909119896) + 119906

119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)


Obviously we can define a similar local error boundcondition at a local weak efficient solution 119909 isin 119878 for(MOPWVVI) with respect to 120577 = (120577

1 1205772 120577

119898) isin R119898

+

with sum119898

119894=1120577119894= 1 satisfying (92) Moreover we can obtain

a corresponding relationship between the (MOPWVVI-)calmness condition and the local error bound conditionHowever we omit the details here for simplicity

Next we establish a M-stationary condition for (MOP-WVVI) under the (MOPWVVI-) calmness with order 1assumption

Theorem 24 Suppose that 119909 isin 119878 is a local weak efficientsolution for (MOPWVVI) and 120577 = (120577

1 1205772 120577

119898) isin R119898

+

with sum119898

119894=1120577119894= 1 satisfy (92) If in addition (MOPWVVI) is

(MOPWVVI-) calm with order 1 at 119909 then there exist 120582 isin R119901

+

with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and (119909lowast 119910lowast) isin R2119899

with 119909lowast isin 119863lowast119873Θ(119909 minussum

119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)

where the set-valued map 119873Θ R119899 999490999473 R119899 is defined by

119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899

Proof Consider the problem (MOPEC) with 119902(119909) =

sum119898

119894=1120577119894119865119894(119909) and 119876(119909) = 119873(Θ 119909) for all 119909 isin R119899 Obviously

119909 is a feasible point of (MOPEC) and the feasible set of(MOPEC) is contained in 119878 By assumption 119909 is a local weakefficient solution for (MOPEC) Moreover it is easy to verifythat the (MOPWVVI-) calmness of (MOPWVVI) with order1 at 119909 implies the (MOPEC-) calmness of (MOPEC) withorder 1 at 119909 Thus together with 119865

119894 119894 = 1 2 119898 being con-

tinuously Frechet differentiable and nabla119902(119909) = sum119898

119894=1120577119894nabla119865

119894(119909)

we immediately complete the proof by Theorem 15

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referee forhisher valuable comments and suggestions which helpedto improve the paper This research was supported by theNational Natural Science Foundation of China (Grant no11171362) and the Fundamental Research Funds for theCentral Universities (Grant no CDJXS12100021)

References

[1] Z Luo J Pang D Ralph and S Wu ldquoExact penalizationand stationarity conditions of mathematical programs with

equilibrium constraintsrdquoMathematical Programming B vol 75no 1 pp 19ndash76 1996

[2] J J Ye and X Y Ye ldquoNecessary optimality conditions foroptimization problems with variational inequality constraintsrdquoMathematics of Operations Research vol 22 no 4 pp 977ndash9971997

[3] J J Ye andQ J Zhu ldquoMultiobjective optimization problemwithvariational inequality constraintsrdquo Mathematical ProgrammingB vol 96 no 1 pp 139ndash160 2003

[4] M L Flegel and C Kanzow ldquoOn M-stationary points formathematical programs with equilibrium constraintsrdquo Journalof Mathematical Analysis and Applications vol 310 no 1 pp286ndash302 2005

[5] Y Chen and M Florian ldquoThe nonlinear bilevel programmingproblem formulations regularity and optimality conditionsrdquoOptimization vol 32 pp 120ndash145 1995

[6] F H ClarkeOptimization andNonsmooth AnalysisWiley NewYork NY USA 1983

[7] R T Rockafellar and R J B Wets Variational AnalysisSpringer Berlin Germany 1998

[8] B S Mordukhovich Variational Analysis and GeneralizedDifferentiation Volume I Basic Theory Volume II ApplicationsSpringer Berlin Germany 2006

[9] J V Burke ldquoCalmness and exact penalizationrdquo SIAM Journal onControl and Optimization vol 29 no 2 pp 493ndash497 1991

[10] A Fiacco and G McCormic Nonlinear Programming Sequen-tial Unconstrained Minimization Techniques John Wiley ampSons New York NY USA 1987

[11] A M Rubinov B M Glover and X Q Yang ldquoDecreasingfunctions with applications to penalizationrdquo SIAM Journal onOptimization vol 10 no 1 pp 289ndash313 1999

[12] A M Rubinov and X Q Yang Lagrange-Type Functionsin Constrained Non-Convex Optimzation Kluwer AcademicPublishers New York NY USA 2003

[13] X X Huang and X Q Yang ldquoNonlinear Lagrangian formultiobjective optimization and applications to duality andexact penalizationrdquo SIAM Journal on Optimization vol 13 no3 pp 675ndash692 2002

[14] T Q Bao P Gupta and B S Mordukhovich ldquoNecessaryconditions in multiobjective optimization with equilibriumconstraintsrdquo Journal of Optimization Theory and Applicationsvol 135 no 2 pp 179ndash203 2007

[15] S M Robinson ldquoStability theory for systems of inequalitiespart II differentiable nonlinear systemsrdquo SIAM Journal onNumerical Analysis vol 13 no 4 pp 497ndash513 1976

[16] J S Treiman ldquoThe linear nonconvex generalized gradient andLagrangemultipliersrdquo SIAM Journal on Optimization vol 5 pp670ndash680 1995

[17] A Gopfert H Riahi C Tammer and C Zalinescu VariationalMethods in Partially Ordered Spaces Springer Berlin Germany2003

[18] G Y Chen and X Q Yang ldquoCharacterizations of variabledomination structures via nonlinear scalarizationrdquo Journal ofOptimizationTheory and Applications vol 112 no 1 pp 97ndash1102002

[19] M Durea and C Tammer ldquoFuzzy necessary optimality condi-tions for vector optimization problemsrdquo Optimization vol 58no 4 pp 449ndash467 2009

[20] C R Chen S J Li and Z M Fang ldquoOn the solution semi-continuity to a parametric generalized vector quasivariational


inequalityrdquo Computers and Mathematics with Applications vol60 no 8 pp 2417ndash2425 2010

[21] A L Dontchev and R T Rockafellar Implicit Functions andSolutionMappings Springer DordrechtTheNetherlands 2009

[22] A D Ioffe ldquoMetric regularity and subdifferential calculusrdquoRussian Mathematical Surveys vol 55 no 3 pp 501ndash558 2000

[23] R Henrion A Jourani and J Outrata ldquoOn the calmness of aclass of multifunctionrdquo SIAM Journal on Optimization vol 13no 2 pp 603ndash618 2002

[24] X Y Zheng and K F Ng ldquoMetric subregularity and constraintqualifications for convex generalized equations in banachspacesrdquo SIAM Journal on Optimization vol 18 no 2 pp 437ndash460 2007

[25] J Jahn Vecor Optimization Theory Applications and ExtensionSpringer Berlin Germany 2004

[26] C Kanzow and A Schwartz ldquoMathematical programs withequilibrium constraints enhanced Fritz John-conditions newconstraint qualifications and improved exact penalty resultsrdquoSIAM Journal on Optimization vol 20 no 5 pp 2730ndash27532010

[27] J J Ye and J Zhang ldquoEnhanced Karush-Kuhn-Tucker condi-tions for mathematical programs with equilibrium constraintsrdquoJournal of Optimization Theory and Applications 2013

[28] A SchwartzMathematical programs with complementarity con-straints theory methods and applications [PhD dissertation]Institute of Applied Mathematics and Statistics University ofWurzburg 2011

[29] L Guo J J Ye and J Zhang ldquoMathematical programs withgeometric constraints in Banach spaces enhanced optimalityexact penalty and sensitivityrdquo SIAM Journal on Optimizationvol 23 pp 2295ndash2319 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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



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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


models arisen in practical applications such as mathematicalprograms with geometric constraints mathematical pro-grams with complementarity constraints and mathematicalprograms with variational inequality constraints For moredetails we refer to [1ndash4] It is worth noting that when 119876 isa general closed set-valued map even if119876(119909) is a fixed closedsubset of R119898 for all 119909 isin R119899 the general constraint system(1) fails to satisfy the standard linear independence constraintqualification and Mangasarian-Fromovitz constraint qualifi-cation at any feasible point [5]Thus it is a hardwork to estab-lish Karush-Kuhn-Tucker (in short KKT) necessary optimal-ity conditions for (MOPEC) Recently by virtue of advancedtools of variational analysis and various coderivatives forset-valued maps developed in [6ndash8] and references thereinsome necessary optimality conditions including the strongMordukhovich Clarke and Bouligand stationary conditionsare obtained by using different reformulations under somegeneralized constraint qualifications Simultaneously Ye andZhu [3] claimed that the Mordukhovich stationary (in shortM-stationary) condition is the strongest stationary conditionexcept the strong stationary condition which is equivalentto the classical KKT condition and proposed some newconstraint qualifications for M-stationary conditions to hold

It is well known that the penalization method is a veryimportant and effective tool for dealing with optimizationtheories and numerical algorithms of constrained extremumproblems In scalar optimization with equality and inequalityconstraints the classical exact penalty function with order 1was extensively used to investigate optimality conditions andconvergence analysis see [6 9 10] and references thereinClarke [6] derived some Fritz-John necessary optimalityconditions for a constrained mathematical programmingproblem on a Banach space by virtue of exact penaltyfunctions with order 1 Moreover Burke [9] showed that theexistence of an exact penalization function is equivalent to acalmness condition involving with the objective function andthe equality and inequality constrained system SubsequentlyFlegel and Kanzow [4] demonstrated that the correspondingrelationships still held in a generalized bilevel programmingproblem and a mathematical programming problem withcomplementarity constraints respectively Simultaneouslythey obtained some KKT necessary optimality conditionsby using exact penalty formulations and nonsmooth anal-ysis Recently the classical penalization theory has beenwidely generalized by various kinds of Lagrangian functionsespecially the augmented Lagrangian function introducedby Rockafellar and Wets [7] and the nonlinear Lagrangianfunction proposed by Rubinov et al [11] It has also beenproved that the exactness of these types of penalty functionsis equivalent to some generalized calmness conditions seemore details in [11 12]

However to the best of our knowledge there are onlya few papers devoted to study the penalty method for con-strainedmultiobjective optimization problems especially for(MOPEC) Huang and Yang [13] first introduced a vector-valued nonlinear Lagrangian and penalty functions formulti-objective optimization problems with equality and inequalityconstraints and obtained some relationships between theexact penalization property and a generalized calmness-type

condition Moreover Mordukhovich [8] and Bao et al [14]investigated some more general optimization problems withequilibrium constraints by methods of modern variationalanalysis It is worth noting that the standard Mangasarian-Fromovitz constraint qualification and error bound conditionfor a nonlinear programming problem with equality andinequality constraints implies the calmness condition see[6 15] for details Taking into account this fact it is neces-sary to further investigate the calmness condition and thepenalty method for constrained multiobjective optimizationproblems

The main motivation of this work is that there hasbeen no study on the penalization method and M-stationarycondition for (MOPEC) by using an appropriate calmnesscondition associated with the objective function and theconstraint system Although there have been many papersdealing with constrained multiobjective optimization prob-lems for example [3 8] and references therein the KKTnecessary optimality conditions are obtained under somegeneralized qualification conditions only involved with theconstraint system Inspired by the ideas reported in [3 4 6 813] we introduce a so-called (MOPEC-) calmness conditionwith order 120590 gt 0 at a local efficient (weak efficient) solutionassociated with the objective function and the constraintsystem for (MOPEC) and show that the (MOPEC-) calmnesscondition can be implied by an error bound condition of theconstraint system Moreover we establish some equivalentrelationships between the exact penalization property withorder 120590 and the (MOPEC-) calmness condition Simultane-ously we apply a nonlinear scalar technical to obtain a KKTnecessary optimality condition for (MOPEC) by using Mor-dukhovich generalized differentiation and the (MOPEC-)calmness condition with order 1

The organization of this paper is as follows In Section 2we recall some basic concepts and tools generally used invariational analysis and set-valued analysis In Section 3 weintroduce a (MOPEC-) calmness condition for (MOPEC)and establish some relationships between the exact penal-ization property and the (MOPEC-) calmness conditionMoreover we obtain a KKT necessary optimality conditionunder the (MOPEC-) calmness condition with order 1 InSection 4 we apply the obtained results to a multiobjectiveoptimization problem with complementarity constraints anda multiobjective optimization problem with weak vectorvariational inequality constraints respectively

2 Notations and Preliminaries

Throughout this paper all vectors are viewed as columnvectors Since all the norms on finite dimensional spaces areequivalent we take specially the sum norm on R119899 and theproduct space R119899 times R119898 for simplicity that is for all 119909 =

(1199091 119909

2 119909

119899)119879

isin R119899 119909 = |1199091| + |119909

2| + sdot sdot sdot + |119909

119899| and

for all (119909 119910) isin R119899 times R119898 (119909 119910) = 119909 + 119910 As usual wedenote by 119909119879 the transposition of 119909 and by ⟨119909 119910⟩ = 119909119879119910 theinner product of vectors 119909 and 119910 respectively For a givenmap 119891 R119899 rarr R119901 and a vector 120582 isin R119901 the function⟨120582 119891⟩ R119899 rarr R is defined by ⟨120582 119891⟩(119909) = ⟨120582 119891(119909)⟩ for





lim119909rarr119909


119909 minus 119909= 0 (4)


lowast


lowast


lim119909rarr119909119906rarr119909119909 = 119906


119909 minus 119906= 0 (5)



119910isin119878119909 minus 119910


(119878 119909) =

119909lowast

isin R119899

| lim sup119909

119878

997888rarr119909

⟨119909lowast

119909 minus 119909⟩

119909 minus 119909le 0

(6)

where 119909 119878


119873(119878 119909) = 119909lowast

isin R119899

| exist119909119899

119904


119899997888rarr 119909

lowast

with 119909lowast119899isin (119878 119909


(7)


(119878 119909) = 119873 (119878 119909) = 119909lowast

isin R119899

| ⟨119909lowast


(8)





ℎ (119909)

= 119909lowast

isin R119899

| lim inf119909rarr119909 119909 = 119909


119909 minus 119909ge 0

(9)




120597ℎ (119909) = 119909lowast

isin R119899

| exist119909119899

ℎ


119899997888rarr 119909

lowast


119899)

(10)

where 119909119899

ℎ




ℎ (119909) = 120597ℎ (119909)

= 119909lowast

isin R119899

| ⟨119909lowast


120597infin

ℎ (119909) sub 119909lowast

isin R119899

| ⟨119909lowast


= 119873 (dom ℎ 119909) (11)


119863lowast

119865 (119909 119910) (119910lowast

)

= 119909lowast

isin R119899

| (119909lowast

minus119910lowast

) isin (gph119865 (119909 119910))

forall119910lowast

isin R119901

119863lowast

119865 (119909 119910) (119910lowast

)

= 119909lowast

isin R119899

| (119909lowast

minus119910lowast

) isin 119873 (gph119865 (119909 119910))

forall119910lowast

isin R119901

(12)




(i) 119889(∙ Ω)(119909) = BR119899 cap (Ω 119909) and (Ω 119909) =

⋃120582gt0120582119889(∙ Ω)(119909)




⋃120582gt0120582120597119889(∙ Ω)(119909)








120597infin

1205931(119909) cap (minus120597

infin

1205932(119909)) = 0R119899 (13)


120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (14)



one always has

120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (15)



120597 ⟨119910lowast

120593⟩ (119909) sub 119863lowast

120593 (119909) (119910lowast


isin R119901

(16)


119863lowast

120593 (119909) (119910lowast

) = 120597 ⟨119910lowast

120593⟩ (119909) forall119910lowast

isin R119901

(17)



119910lowast

isin 120597infin

ℎ (119867 (119909)) 0R119899 isin 119863lowast

119867(119909) (119910lowast

)

119894119898119901119897119894119890119904 119910lowast

= 0Rℓ (18)

then

120597ℎ ∘ 119867 (119909) sub 119863lowast

119867(119909) (119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (19)



(119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (20)



+ the nonlinear



119901


119901

(21)


+-monotone R

119901

+-monotone


119901


120585119890(120572119890) = 120572

119910 isin R119901

| 120585119890(119910) le 120572 = 120572119890 minusR

119901

+

119910 isin R119901

| 120585119890(119910) lt 120572 = 120572119890 minus intR119901

+

(22)


120597120585119890(119910) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 ⟨120582 119910⟩ = 120585

119890(119910) (23)

Specially one has

120597120585119890(0R119901) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 (24)




119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ(25)


119878 (119906 V 119910 119911) = 119909 isin R119899

| 119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ

(26)






119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (27)



119896 119911119896) sub R119903+119904+119899+119898

with (119906119896 V119896 119910


satisfying 119892(119909119896) + 119906

119896isin R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin 119892(119909119896) +

119876(119909119896+ 119910

119896) and 119909


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (28)



119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (29)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+(30)


119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) 0R119903+119904+119899+119898 and

119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) cap B(119909 1119896) such that

119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (31)


119896 119878) of 119909

119896onto 119878 such that 119889(119909

119896 119878) = 119909

119896minus 119875(119909

119896 119878) for


1003817100381710038171003817119875 (119909119896 119878) minus 1199091003817100381710038171003817 le

1003817100381710038171003817119875 (119909119896 119878) minus 1199091198961003817100381710038171003817 +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817

= 119889 (119909119896 119878) +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817 997888rarr 0

(32)


1isin N

such that


119901


+)

forall119896 ge 1198731

(33)


1003817100381710038171003817 le 1198711003817100381710038171003817119909119896 minus 119875 (119909119896 119878)

1003817100381710038171003817 forall119896 ge 1198732

(34)


119891 (119875 (119909119896 119878)) minus 119891 (119909

119896)

= 119891 (119875 (119909119896 119878)) minus 119891 (119909)

+ (119891 (119909) minus 119891 (119909119896)) notin 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

(35)

Together with 119889(119909119896 119878) = 119909

119896minus 119875(119909

119896 119878) and (34) we can

conclude that

119889 (119909119896 119878)BR119901 sub

119896

119871

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

forall119896 ge max 1198731 119873

2

(36)


119896 119911119896) = 0R119903+119904+119899+119898 (119906119896 V119896 119910119896 119911119896) rarr 0R119903+119904+119899+119898 119909119896 isin

119878(119906119896 V119896 119910

119896 119911119896) and 119909

119896rarr 119909


119889 (119909 119878) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(37)


119889 (119909 119878)1

119901119890 isin 119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

119890 minus intR119901

+ (38)







119898 = max119908isinBR119901

120585119890(119908) (39)





120585119890(119889 (119909 119878) 119908) = 119889 (119909 119878) 120585

119890(119908) lt 119898119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

(40)

By Lemma 6 we have

119889 (119909 119878) 119908 isin 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (41)

which implies

119889 (119909 119878)BR119901 sub 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (42)


Ψ (119909) cap 119881 sub Ψ (119909) + ℓ119909 minus 119909120590

BR119899 forall119909 isin 119880 (43)





119878(119906 V 119910 119911) cap B(119909 120575)

119889 (119909 119878 (0R119903+119904+119899+119898)) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(44)








(119872119875119875)119868

min 119891 (119909) + 120588 (1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) +1003817100381710038171003817(119910 119911)

1003817100381710038171003817)120590

119890

119904119905 119911 isin 119902 (119909) + 119876 (119909 + 119910)

119909 isin Θ (119910 119911) isin R119899+119898

(45)

where 119892+(119909) = (max119892

1(119909) 0 max119892

2(119909) 0

max119892119903(119909) 0)


(119872119875119875)119868119868

min 119891 (119909)

+ 120583[1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)]120590

119890

119904119905 119909 isin Θ

(46)



119896 119910


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) such that

119891 (119909119896) + 119896 (

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

119890 isin 119891 (119909) minus intR119901

+

(47)

Take 119906119896= minus119892

+(119909119896) and V

119896= minusℎ(119909


119892(119909119896) + 119906


+and ℎ(119909

119896) + V


119902(119909119896) + 119876(119909

119896+ 119910

119896) and 119909

119896isin Θ we get 119909

119896isin 119878(119906

119896 V119896 119910

119896 119911119896)


119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (48)


+ ℎ(119909) = 0R119904 and 119892 and ℎ


+(119909119896) rarr 0R119903 and V

119896= ℎ(119909







119896 V119896 119910

119896 119911119896) it


119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910


119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(50)


119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)


Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890


+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)


119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573


119896) + 119906


+ ℎ(119909

119896) +


119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)



119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890


+minus intR119901

+= minus intR119901

+

(55)


119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)


that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910



119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909



119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817


1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)



+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













lim119909rarr119909


119909 minus 119909= 0 (4)


lowast


lowast


lim119909rarr119909119906rarr119909119909 = 119906


119909 minus 119906= 0 (5)



119910isin119878119909 minus 119910


(119878 119909) =

119909lowast

isin R119899

| lim sup119909

119878

997888rarr119909

⟨119909lowast

119909 minus 119909⟩

119909 minus 119909le 0

(6)

where 119909 119878


119873(119878 119909) = 119909lowast

isin R119899

| exist119909119899

119904


119899997888rarr 119909

lowast

with 119909lowast119899isin (119878 119909


(7)


(119878 119909) = 119873 (119878 119909) = 119909lowast

isin R119899

| ⟨119909lowast


(8)





ℎ (119909)

= 119909lowast

isin R119899

| lim inf119909rarr119909 119909 = 119909


119909 minus 119909ge 0

(9)




120597ℎ (119909) = 119909lowast

isin R119899

| exist119909119899

ℎ


119899997888rarr 119909

lowast


119899)

(10)

where 119909119899

ℎ




ℎ (119909) = 120597ℎ (119909)

= 119909lowast

isin R119899

| ⟨119909lowast


120597infin

ℎ (119909) sub 119909lowast

isin R119899

| ⟨119909lowast


= 119873 (dom ℎ 119909) (11)


119863lowast

119865 (119909 119910) (119910lowast

)

= 119909lowast

isin R119899

| (119909lowast

minus119910lowast

) isin (gph119865 (119909 119910))

forall119910lowast

isin R119901

119863lowast

119865 (119909 119910) (119910lowast

)

= 119909lowast

isin R119899

| (119909lowast

minus119910lowast

) isin 119873 (gph119865 (119909 119910))

forall119910lowast

isin R119901

(12)




(i) 119889(∙ Ω)(119909) = BR119899 cap (Ω 119909) and (Ω 119909) =

⋃120582gt0120582119889(∙ Ω)(119909)




⋃120582gt0120582120597119889(∙ Ω)(119909)








120597infin

1205931(119909) cap (minus120597

infin

1205932(119909)) = 0R119899 (13)


120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (14)



one always has

120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (15)



120597 ⟨119910lowast

120593⟩ (119909) sub 119863lowast

120593 (119909) (119910lowast


isin R119901

(16)


119863lowast

120593 (119909) (119910lowast

) = 120597 ⟨119910lowast

120593⟩ (119909) forall119910lowast

isin R119901

(17)



119910lowast

isin 120597infin

ℎ (119867 (119909)) 0R119899 isin 119863lowast

119867(119909) (119910lowast

)

119894119898119901119897119894119890119904 119910lowast

= 0Rℓ (18)

then

120597ℎ ∘ 119867 (119909) sub 119863lowast

119867(119909) (119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (19)



(119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (20)



+ the nonlinear



119901


119901

(21)


+-monotone R

119901

+-monotone


119901


120585119890(120572119890) = 120572

119910 isin R119901

| 120585119890(119910) le 120572 = 120572119890 minusR

119901

+

119910 isin R119901

| 120585119890(119910) lt 120572 = 120572119890 minus intR119901

+

(22)


120597120585119890(119910) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 ⟨120582 119910⟩ = 120585

119890(119910) (23)

Specially one has

120597120585119890(0R119901) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 (24)




119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ(25)


119878 (119906 V 119910 119911) = 119909 isin R119899

| 119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ

(26)






119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (27)



119896 119911119896) sub R119903+119904+119899+119898

with (119906119896 V119896 119910


satisfying 119892(119909119896) + 119906

119896isin R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin 119892(119909119896) +

119876(119909119896+ 119910

119896) and 119909


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (28)



119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (29)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+(30)


119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) 0R119903+119904+119899+119898 and

119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) cap B(119909 1119896) such that

119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (31)


119896 119878) of 119909

119896onto 119878 such that 119889(119909

119896 119878) = 119909

119896minus 119875(119909

119896 119878) for


1003817100381710038171003817119875 (119909119896 119878) minus 1199091003817100381710038171003817 le

1003817100381710038171003817119875 (119909119896 119878) minus 1199091198961003817100381710038171003817 +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817

= 119889 (119909119896 119878) +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817 997888rarr 0

(32)


1isin N

such that


119901


+)

forall119896 ge 1198731

(33)


1003817100381710038171003817 le 1198711003817100381710038171003817119909119896 minus 119875 (119909119896 119878)

1003817100381710038171003817 forall119896 ge 1198732

(34)


119891 (119875 (119909119896 119878)) minus 119891 (119909

119896)

= 119891 (119875 (119909119896 119878)) minus 119891 (119909)

+ (119891 (119909) minus 119891 (119909119896)) notin 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

(35)

Together with 119889(119909119896 119878) = 119909

119896minus 119875(119909

119896 119878) and (34) we can

conclude that

119889 (119909119896 119878)BR119901 sub

119896

119871

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

forall119896 ge max 1198731 119873

2

(36)


119896 119911119896) = 0R119903+119904+119899+119898 (119906119896 V119896 119910119896 119911119896) rarr 0R119903+119904+119899+119898 119909119896 isin

119878(119906119896 V119896 119910

119896 119911119896) and 119909

119896rarr 119909


119889 (119909 119878) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(37)


119889 (119909 119878)1

119901119890 isin 119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

119890 minus intR119901

+ (38)







119898 = max119908isinBR119901

120585119890(119908) (39)





120585119890(119889 (119909 119878) 119908) = 119889 (119909 119878) 120585

119890(119908) lt 119898119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

(40)

By Lemma 6 we have

119889 (119909 119878) 119908 isin 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (41)

which implies

119889 (119909 119878)BR119901 sub 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (42)


Ψ (119909) cap 119881 sub Ψ (119909) + ℓ119909 minus 119909120590

BR119899 forall119909 isin 119880 (43)





119878(119906 V 119910 119911) cap B(119909 120575)

119889 (119909 119878 (0R119903+119904+119899+119898)) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(44)








(119872119875119875)119868

min 119891 (119909) + 120588 (1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) +1003817100381710038171003817(119910 119911)

1003817100381710038171003817)120590

119890

119904119905 119911 isin 119902 (119909) + 119876 (119909 + 119910)

119909 isin Θ (119910 119911) isin R119899+119898

(45)

where 119892+(119909) = (max119892

1(119909) 0 max119892

2(119909) 0

max119892119903(119909) 0)


(119872119875119875)119868119868

min 119891 (119909)

+ 120583[1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)]120590

119890

119904119905 119909 isin Θ

(46)



119896 119910


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) such that

119891 (119909119896) + 119896 (

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

119890 isin 119891 (119909) minus intR119901

+

(47)

Take 119906119896= minus119892

+(119909119896) and V

119896= minusℎ(119909


119892(119909119896) + 119906


+and ℎ(119909

119896) + V


119902(119909119896) + 119876(119909

119896+ 119910

119896) and 119909

119896isin Θ we get 119909

119896isin 119878(119906

119896 V119896 119910

119896 119911119896)


119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (48)


+ ℎ(119909) = 0R119904 and 119892 and ℎ


+(119909119896) rarr 0R119903 and V

119896= ℎ(119909







119896 V119896 119910

119896 119911119896) it


119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910


119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(50)


119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)


Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890


+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)


119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573


119896) + 119906


+ ℎ(119909

119896) +


119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)



119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890


+minus intR119901

+= minus intR119901

+

(55)


119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)


that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910



119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909



119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817


1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)



+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












⋃120582gt0120582120597119889(∙ Ω)(119909)








120597infin

1205931(119909) cap (minus120597

infin

1205932(119909)) = 0R119899 (13)


120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (14)



one always has

120597 (1205931+ 120593

2) (119909) sub 120597120593

1(119909) + 120597120593

2(119909) (15)



120597 ⟨119910lowast

120593⟩ (119909) sub 119863lowast

120593 (119909) (119910lowast


isin R119901

(16)


119863lowast

120593 (119909) (119910lowast

) = 120597 ⟨119910lowast

120593⟩ (119909) forall119910lowast

isin R119901

(17)



119910lowast

isin 120597infin

ℎ (119867 (119909)) 0R119899 isin 119863lowast

119867(119909) (119910lowast

)

119894119898119901119897119894119890119904 119910lowast

= 0Rℓ (18)

then

120597ℎ ∘ 119867 (119909) sub 119863lowast

119867(119909) (119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (19)



(119910lowast

) 119910lowast

isin 120597ℎ (119867 (119909)) (20)



+ the nonlinear



119901


119901

(21)


+-monotone R

119901

+-monotone


119901


120585119890(120572119890) = 120572

119910 isin R119901

| 120585119890(119910) le 120572 = 120572119890 minusR

119901

+

119910 isin R119901

| 120585119890(119910) lt 120572 = 120572119890 minus intR119901

+

(22)


120597120585119890(119910) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 ⟨120582 119910⟩ = 120585

119890(119910) (23)

Specially one has

120597120585119890(0R119901) = 120582 isin R

119901

+|

119901

sum119894=1

120582119894= 1 (24)




119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ(25)


119878 (119906 V 119910 119911) = 119909 isin R119899

| 119892 (119909) + 119906 isin minusR119903

+ ℎ (119909) + V = 0R119904

119911 isin 119902 (119909) + 119876 (119909 + 119910) 119909 isin Θ

(26)






119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (27)



119896 119911119896) sub R119903+119904+119899+119898

with (119906119896 V119896 119910


satisfying 119892(119909119896) + 119906

119896isin R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin 119892(119909119896) +

119876(119909119896+ 119910

119896) and 119909


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (28)



119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (29)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+(30)


119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) 0R119903+119904+119899+119898 and

119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) cap B(119909 1119896) such that

119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (31)


119896 119878) of 119909

119896onto 119878 such that 119889(119909

119896 119878) = 119909

119896minus 119875(119909

119896 119878) for


1003817100381710038171003817119875 (119909119896 119878) minus 1199091003817100381710038171003817 le

1003817100381710038171003817119875 (119909119896 119878) minus 1199091198961003817100381710038171003817 +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817

= 119889 (119909119896 119878) +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817 997888rarr 0

(32)


1isin N

such that


119901


+)

forall119896 ge 1198731

(33)


1003817100381710038171003817 le 1198711003817100381710038171003817119909119896 minus 119875 (119909119896 119878)

1003817100381710038171003817 forall119896 ge 1198732

(34)


119891 (119875 (119909119896 119878)) minus 119891 (119909

119896)

= 119891 (119875 (119909119896 119878)) minus 119891 (119909)

+ (119891 (119909) minus 119891 (119909119896)) notin 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

(35)

Together with 119889(119909119896 119878) = 119909

119896minus 119875(119909

119896 119878) and (34) we can

conclude that

119889 (119909119896 119878)BR119901 sub

119896

119871

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

forall119896 ge max 1198731 119873

2

(36)


119896 119911119896) = 0R119903+119904+119899+119898 (119906119896 V119896 119910119896 119911119896) rarr 0R119903+119904+119899+119898 119909119896 isin

119878(119906119896 V119896 119910

119896 119911119896) and 119909

119896rarr 119909


119889 (119909 119878) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(37)


119889 (119909 119878)1

119901119890 isin 119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

119890 minus intR119901

+ (38)







119898 = max119908isinBR119901

120585119890(119908) (39)





120585119890(119889 (119909 119878) 119908) = 119889 (119909 119878) 120585

119890(119908) lt 119898119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

(40)

By Lemma 6 we have

119889 (119909 119878) 119908 isin 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (41)

which implies

119889 (119909 119878)BR119901 sub 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (42)


Ψ (119909) cap 119881 sub Ψ (119909) + ℓ119909 minus 119909120590

BR119899 forall119909 isin 119880 (43)





119878(119906 V 119910 119911) cap B(119909 120575)

119889 (119909 119878 (0R119903+119904+119899+119898)) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(44)








(119872119875119875)119868

min 119891 (119909) + 120588 (1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) +1003817100381710038171003817(119910 119911)

1003817100381710038171003817)120590

119890

119904119905 119911 isin 119902 (119909) + 119876 (119909 + 119910)

119909 isin Θ (119910 119911) isin R119899+119898

(45)

where 119892+(119909) = (max119892

1(119909) 0 max119892

2(119909) 0

max119892119903(119909) 0)


(119872119875119875)119868119868

min 119891 (119909)

+ 120583[1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)]120590

119890

119904119905 119909 isin Θ

(46)



119896 119910


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) such that

119891 (119909119896) + 119896 (

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

119890 isin 119891 (119909) minus intR119901

+

(47)

Take 119906119896= minus119892

+(119909119896) and V

119896= minusℎ(119909


119892(119909119896) + 119906


+and ℎ(119909

119896) + V


119902(119909119896) + 119876(119909

119896+ 119910

119896) and 119909

119896isin Θ we get 119909

119896isin 119878(119906

119896 V119896 119910

119896 119911119896)


119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (48)


+ ℎ(119909) = 0R119904 and 119892 and ℎ


+(119909119896) rarr 0R119903 and V

119896= ℎ(119909







119896 V119896 119910

119896 119911119896) it


119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910


119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(50)


119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)


Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890


+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)


119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573


119896) + 119906


+ ℎ(119909

119896) +


119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)



119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890


+minus intR119901

+= minus intR119901

+

(55)


119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)


that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910



119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909



119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817


1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)



+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (27)



119896 119911119896) sub R119903+119904+119899+119898

with (119906119896 V119896 119910


satisfying 119892(119909119896) + 119906

119896isin R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin 119892(119909119896) +

119876(119909119896+ 119910

119896) and 119909


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (28)



119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (29)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+(30)


119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) 0R119903+119904+119899+119898 and

119909119896isin 119878(119906

119896 V119896 119910

119896 119911119896) cap B(119909 1119896) such that

119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (31)


119896 119878) of 119909

119896onto 119878 such that 119889(119909

119896 119878) = 119909

119896minus 119875(119909

119896 119878) for


1003817100381710038171003817119875 (119909119896 119878) minus 1199091003817100381710038171003817 le

1003817100381710038171003817119875 (119909119896 119878) minus 1199091198961003817100381710038171003817 +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817

= 119889 (119909119896 119878) +

1003817100381710038171003817119909119896 minus 1199091003817100381710038171003817 997888rarr 0

(32)


1isin N

such that


119901


+)

forall119896 ge 1198731

(33)


1003817100381710038171003817 le 1198711003817100381710038171003817119909119896 minus 119875 (119909119896 119878)

1003817100381710038171003817 forall119896 ge 1198732

(34)


119891 (119875 (119909119896 119878)) minus 119891 (119909

119896)

= 119891 (119875 (119909119896 119878)) minus 119891 (119909)

+ (119891 (119909) minus 119891 (119909119896)) notin 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

(35)

Together with 119889(119909119896 119878) = 119909

119896minus 119875(119909

119896 119878) and (34) we can

conclude that

119889 (119909119896 119878)BR119901 sub

119896

119871

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus intR119901

+

forall119896 ge max 1198731 119873

2

(36)


119896 119911119896) = 0R119903+119904+119899+119898 (119906119896 V119896 119910119896 119911119896) rarr 0R119903+119904+119899+119898 119909119896 isin

119878(119906119896 V119896 119910

119896 119911119896) and 119909

119896rarr 119909


119889 (119909 119878) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(37)


119889 (119909 119878)1

119901119890 isin 119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

119890 minus intR119901

+ (38)







119898 = max119908isinBR119901

120585119890(119908) (39)





120585119890(119889 (119909 119878) 119908) = 119889 (119909 119878) 120585

119890(119908) lt 119898119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

(40)

By Lemma 6 we have

119889 (119909 119878) 119908 isin 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (41)

which implies

119889 (119909 119878)BR119901 sub 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (42)


Ψ (119909) cap 119881 sub Ψ (119909) + ℓ119909 minus 119909120590

BR119899 forall119909 isin 119880 (43)





119878(119906 V 119910 119911) cap B(119909 120575)

119889 (119909 119878 (0R119903+119904+119899+119898)) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(44)








(119872119875119875)119868

min 119891 (119909) + 120588 (1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) +1003817100381710038171003817(119910 119911)

1003817100381710038171003817)120590

119890

119904119905 119911 isin 119902 (119909) + 119876 (119909 + 119910)

119909 isin Θ (119910 119911) isin R119899+119898

(45)

where 119892+(119909) = (max119892

1(119909) 0 max119892

2(119909) 0

max119892119903(119909) 0)


(119872119875119875)119868119868

min 119891 (119909)

+ 120583[1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)]120590

119890

119904119905 119909 isin Θ

(46)



119896 119910


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) such that

119891 (119909119896) + 119896 (

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

119890 isin 119891 (119909) minus intR119901

+

(47)

Take 119906119896= minus119892

+(119909119896) and V

119896= minusℎ(119909


119892(119909119896) + 119906


+and ℎ(119909

119896) + V


119902(119909119896) + 119876(119909

119896+ 119910

119896) and 119909

119896isin Θ we get 119909

119896isin 119878(119906

119896 V119896 119910

119896 119911119896)


119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (48)


+ ℎ(119909) = 0R119904 and 119892 and ℎ


+(119909119896) rarr 0R119903 and V

119896= ℎ(119909







119896 V119896 119910

119896 119911119896) it


119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910


119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(50)


119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)


Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890


+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)


119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573


119896) + 119906


+ ℎ(119909

119896) +


119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)



119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890


+minus intR119901

+= minus intR119901

+

(55)


119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)


that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910



119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909



119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817


1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)



+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120585119890(119889 (119909 119878) 119908) = 119889 (119909 119878) 120585

119890(119908) lt 119898119872

1003817100381710038171003817(119906 V 119910 119911)1003817100381710038171003817120590

(40)

By Lemma 6 we have

119889 (119909 119878) 119908 isin 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (41)

which implies

119889 (119909 119878)BR119901 sub 1198981198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (42)


Ψ (119909) cap 119881 sub Ψ (119909) + ℓ119909 minus 119909120590

BR119899 forall119909 isin 119880 (43)





119878(119906 V 119910 119911) cap B(119909 120575)

119889 (119909 119878 (0R119903+119904+119899+119898)) lt 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

(44)








(119872119875119875)119868

min 119891 (119909) + 120588 (1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) +1003817100381710038171003817(119910 119911)

1003817100381710038171003817)120590

119890

119904119905 119911 isin 119902 (119909) + 119876 (119909 + 119910)

119909 isin Θ (119910 119911) isin R119899+119898

(45)

where 119892+(119909) = (max119892

1(119909) 0 max119892

2(119909) 0

max119892119903(119909) 0)


(119872119875119875)119868119868

min 119891 (119909)

+ 120583[1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)]120590

119890

119904119905 119909 isin Θ

(46)



119896 119910


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) such that

119891 (119909119896) + 119896 (

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

119890 isin 119891 (119909) minus intR119901

+

(47)

Take 119906119896= minus119892

+(119909119896) and V

119896= minusℎ(119909


119892(119909119896) + 119906


+and ℎ(119909

119896) + V


119902(119909119896) + 119876(119909

119896+ 119910

119896) and 119909

119896isin Θ we get 119909

119896isin 119878(119906

119896 V119896 119910

119896 119911119896)


119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 isin 119891 (119909) minus intR119901

+ (48)


+ ℎ(119909) = 0R119904 and 119892 and ℎ


+(119909119896) rarr 0R119903 and V

119896= ℎ(119909







119896 V119896 119910

119896 119911119896) it


119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and


119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910


119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(50)


119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)


Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890


+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)


119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573


119896) + 119906


+ ℎ(119909

119896) +


119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)



119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890


+minus intR119901

+= minus intR119901

+

(55)


119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)


that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910



119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909



119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817


1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)



+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

le1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817

=10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 forall119896 isin N

(49)

which implies (119892+(119909119896) + ℎ(119909

119896) + (119910

119896 119911119896))

120590

le

(119906119896 V119896 119910


119891 (119909119896) + 119896(

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 [(1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817(119910119896 119911119896)

1003817100381710038171003817)120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

] 119890 isin 119891 (119909)

minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901


(50)


119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) and

(119909119896 119910

119896 119911119896) rarr (119909 0R119899+119898)


Θ cap B(119909 1119896) such that

119891 (119909119896)

+ 119896[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

119890


+

(51)

Note that

119889 ((119909119896 minus119902 (119909

119896)) gph119876) = inf

120573isin119876(120572)

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572 120573)1003817100381710038171003817

(52)


119896) isin R119899+119898 with 120573

119896isin

119876(120572119896) such that

1003817100381710038171003817(119909119896 minus119902 (119909119896)) minus (120572119896 120573119896)1003817100381710038171003817

le (1 +1

119896) 119889 ((119909

119896 minus119902 (119909

119896)) gph119876)

(53)

Take 119906119896= minus119892

+(119909119896) V

119896= minusℎ(119909

119896) 119910

119896= 120572

119896minus 119909

119896 and 119911

119896=

119902(119909119896) + 120573


119896) + 119906


+ ℎ(119909

119896) +


119896isin

119878(119906119896 V119896 119910

119896 119911119896) since 119909

119896isin Θ and

(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

le (1 +1

119896)120590

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590

(54)



119891 (119909119896) +

119896120590+1

(119896 + 1)120590

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 minus 119891 (119909)

= 119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890 minus 119891 (119909)

+119896120590+1

(119896 + 1)120590(10038171003817100381710038171199061198961003817100381710038171003817 +

1003817100381710038171003817V1198961003817100381710038171003817 +

1003817100381710038171003817(119910119896 119911119896)1003817100381710038171003817)120590

minus (1 +1

119896)120590

times [1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817

+ 119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590 119890


+minus intR119901

+= minus intR119901

+

(55)


119901

+with sum119901

119894=1120582119894= 1 we get

[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le1

119896

119901

sum119894=1

120582119894(119891

119894(119909) minus 119891

119894(119909

119896)) forall119896 isin N

(56)


that [119892+(119909119896) + ℎ(119909

119896) + 119889((119909

119896 minus119902(119909

119896)) gph119876)]120590 rarr 0

which implies (119906119896 V119896 119910



119896 V119896 119910

119896 119911119896) 119909

119896rarr 119909



119896 V119896 119910

119896 119911119896) isin B(0R119903+119904+119899+119898 1119896) and 119909119896 isin B(119909 1119896)

with 119909119896isin Θ 119892(119909

119896) + 119906


+ ℎ(119909

119896) + V

119896= 0R119904 and

119911119896isin 119902(119909

119896) + 119876(119909

119896+ 119910

119896) that is (119909

119896+ 119910

119896 119911119896minus 119902(119909

119896)) isin gph119876


[1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

le [1003817100381710038171003817119892 (119909119896) minus (119892 (119909119896) + 119906119896)

1003817100381710038171003817

+1003817100381710038171003817ℎ (119909119896) minus (ℎ (119909119896) + V

119896)1003817100381710038171003817


1003817100381710038171003817]120590

=1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

forall119896 isin N

(57)



+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+ we get

119891 (119909119896) + 119896 [

1003817100381710038171003817119892+ (119909119896)1003817100381710038171003817 +

1003817100381710038171003817ℎ (119909119896)1003817100381710038171003817

+119889 ((119909119896 minus119902 (119909

119896)) gph119876)]120590119890

= 119891 (119909119896) + 119896

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890

+ 119896 ([1003817100381710038171003817119892+ (119909119896)

1003817100381710038171003817 +1003817100381710038171003817ℎ (119909119896)

1003817100381710038171003817 + 119889 ((119909119896 minus119902 (119909119896)) gph119876)]120590

minus1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)

1003817100381710038171003817120590

) 119890 isin 119891 (119909) minus intR119901

+minus intR119901

+

= 119891 (119909) minus intR119901



119896 sub Θ and 119909

119896rarr 119909



min 120601 (119909)

119904119905 119909 isin Ω(59)


+with sum119901

119894=1120582119894= 1 such that

0R119899 isin

119901

sum119894=1

120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (60)



119899


119899 sub R119899


have 119909119899 sub Ω and Φ(119909

119899) = 120585

119890(120601(119909






119899 sub Ω and 119909



sub 119863lowast

120601 (119909) (120582) 120582 isin 120597120585119890(0R119901) + 119873 (Ω 119909)

= 119863lowast

120601 (119909) (120582) 120582 isin R119901

+

119901

sum119894=1

120582119894= 1 + 119873 (Ω 119909)

(63)


+with sum119901

119894=1120582119894= 1 such

that


120601 (119909) (120582) + 119873 (Ω 119909) (64)


0R119899 isin 120597 ⟨120582 120601⟩ (119909) + 119873 (Ω 119909) (65)




119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(66)


rarr R definedby

T (119909) =1003817100381710038171003817119892+ (119909)

1003817100381710038171003817 + ℎ (119909) + 119889 ((119909 minus119902 (119909)) gph119876)


(67)



119901

+with

sum119901

119894=1120582119894= 1 such that

0R119899 isin 120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) + 119873 (Θ 119909) (68)


119894=1120582119894= 1 and Proposition 3

we have

120597 ⟨120582 119891 (∙) + 120583T (∙) 119890⟩ (119909) sub 120597 ⟨120582 119891⟩ (119909) + 120583120597T (119909) (69)

120597T (119909) sub 1205971003817100381710038171003817119892+ (∙)

1003817100381710038171003817 (119909) + 120597 ℎ (∙) (119909)

+ 120597119889 ((∙ minus119902 (∙)) gph119876) (119909) (70)


120597max 0 119892119894(∙) (119909) =

0R119899 if 119892119894(119909) lt 0

[0 1] nabla119892119894(119909) if 119892

119894(119909) = 0

(71)


for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










for all 119894 isin 1 2 119904

1205971003816100381610038161003816ℎ119894 (∙)

1003816100381610038161003816 (119909) = [minus1 1] nablaℎ119894 (119909)

120597119889 ((∙ minus119902 (∙)) gph119876) (119909)


(119909lowast

minus119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(72)




120597T (119909) sub sum119894isinJ(119909)

[0 1] nabla119892119894(119909) +

119904

sum119894=1

[minus1 1] nablaℎ119894(119909)

+ 119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

) (119909lowast

minus119910lowast

)

isin 119873 (gph119876 (minus119909 119902 (119909)))

(73)



119876(minus119909 119902(119909))(119910lowast

) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909)

+ 120583( sum119894isinJ(119909)

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

))) + 119873 (Θ 119909)

= 120597 ⟨120582 119891⟩ (119909) + sum119894isinJ(119909)

120583120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120583 120574119894nablaℎ

119894(119909) + 120583 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

(74)

Take 120573 isin R119903

+with 120573

119894= 120583120573

119894 119894 isin J(119909) and 120573

119894= 0


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909)

+

119904

sum119894=1

120574119894nablaℎ

119894(119909) + 120591 (119909

lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 119894 = 1 2 119903

(75)





4 Applications



(MOPCC)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119866 (119909) isin R119897

+ 119867 (119909) isin R

119897

+ 119866(119909)

119879

119867(119909) = 0

119909 isin Θ

(76)




1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198680+= 119894 | 119866

119894(119909) = 0119867

119894(119909) gt 0

11986800= 119894 | 119866

119894(119909) = 0119867

119894(119909) = 0

119868+0= 119894 | 119866

119894(119909) gt 0119867

119894(119909) = 0

(77)


+ ℎ(119909) =

0R119904 119866(119909) isin R119897

+ 119867(119909) isin R119897

+ 119866(119909)

119879


119902 (119909) =(

1198661(119909)

1198671(119909)

119866119897(119909)

119867119897(119909)

) 119876 (119909) = 119862119897


(78)



+ ℎ (119909) + V = 0R119904

119866 (119909) + 119910 isin R119897

+ 119867 (119909) + 119911 isin R

119897

+

(119866 (119909) + 119910)119879

(119867 (119909) + 119911) = 0

(79)




119891 (119909) +1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (80)


119889 (119909 119878)BR119901 sub 1198721003817100381710038171003817(119906 V 119910 119911)

1003817100381710038171003817120590

119890 minus intR119901

+ (81)





119901

+

and sum119901

119894=1120582119891

119894= 1 such that

0R119899 isin 120597 ⟨120582119891

119891⟩ (119909) +

119903

sum119894=1

120582119892

119894nabla119892

119894(119909) +

119904

sum119894=1

120582ℎ

119894nablaℎ

119894(119909)

minus

119897

sum119894=1

[120582119892

119894nabla119866

119894(119909) + 120582

119867

119894nabla119867

119894(119909)] + 119873 (Θ 119909)

120582119866

119894= 0 forall119894 isin 119868

+0 120582

119866


0+

120582119867

119894= 0 forall119894 isin 119868

0+ 120582

119867


+0

either 120582119866119894gt 0 120582

119867

119894gt 0 or 120582

119866

119894120582119867

119894= 0 forall119894 isin 119868

00

120582119892

isin R119903

+ 120582

119892

119894119892119894(119909) = 0 forall119894 = 1 2 119903

(82)



119873(119862 (119886 119887)) =

(1198891 119889

2) | 119889

1isin R 119889

2= 0 119894119891 119886 = 0 gt 119887

1198891= 0 119889

2isin R 119894119891 119886 lt 0 = 119887

either 1198891gt 0

1198892gt 0

or 11988911198892= 0 119894119891 119886 = 0 = 119887

(83)





120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120582 isin R119901

+with sum119901

119894=1120582119894= 1 120573 isin R119903

+ 120574 isin R119904 120591 gt 0 and


0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 (119909lowast

+ (nabla119902 (119909))lowast

(119910lowast

)) + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(84)


119873(gph119876 (119909 minus119902 (119909)))

= 119873 (R119899

119909) times 119873 (119862119897

minus119902 (119909))



119897(119909)))

(85)



119909lowast

= 0R119899

minus119910lowast

isin

(

1205781

1205791

120578119897

120579119897

)

| 120578119894isin R 120579

119894= 0

120578119894= 0 120579

119894isin R

either 120578119894gt 0

120579119894gt 0

or 120578119894120579119894= 0

if 119894 isin 1198680+

if 119894 isin 119868+0

if 119894 isin 11986800

(86)

Moreover we get

nabla119902 (119909) =(

(

nabla1198661(119909)

119879

nabla1198671(119909)

119879

nabla119866

119897(119909)

119879

nabla119867119897(119909)

119879

)

)

(87)


119894= 120591120578

119894 and 120582119867

119894= 120591120579

119894 for all 119894 isin 1 2 119897



(MOPWVVI)

min 119891 (119909)

st 119892 (119909) isin minusR119903

+

ℎ (119909) = 0R119904

119909 isin Θ

(88)


(⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879



(89)

where119865119894 R119899


minusR119903






1205851198900

( (⟨1198651(119909) 119908 minus 119909⟩ ⟨119865

2(119909) 119908 minus 119909⟩

⟨119865119898(119909) 119908 minus 119909⟩)

119879

) = 0

(90)



0R119899 isin 1205971205851198900

((⟨1198651(119909) ∙ minus 119909⟩ ⟨119865

2(119909) ∙ minus 119909⟩

⟨119865119898(119909) ∙ minus 119909⟩)

119879

) (119909) + 119873 (Θ 119909)

sub (1198651(119909) 119865

2(119909) 119865

119898(119909)) 120597120585

1198900

(0R119898) + 119873 (Θ 119909)

(91)


119898) isin

R119898

+with sum119898

119894=1120577119894= 1 such that

0R119899 isin

119898

sum119894=1

120577119894119865119894(119909) + 119873 (Θ 119909) (92)


1 1205772 120577

119898) isin R119898

+with sum119898

119894=1120577119894=

1 satisfying (92)


1 1205772 120577

119898) isin R119898

+withsum119898

119894=1120577119894= 1 satisfying (92) then


119896 V119896 119910

119896 119911119896) sub R119903+119904+119899+119898 with (119906

119896 V119896 119910

119896 119911119896) rarr


119896isin

R119903

+ ℎ(119909

119896) + V

119896= 0R119904 119911119896 isin sum

119898

119894=1120577119894119865119894(119909119896) + 119873(Θ 119909

119896+ 119910

119896) and


119891 (119909119896) + 119872

1003817100381710038171003817(119906119896 V119896 119910119896 119911119896)1003817100381710038171003817120590

119890 notin 119891 (119909) minus intR119901

+ (93)



1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 1205772 120577

119898) isin R119898

+

with sum119898





1 1205772 120577

119898) isin R119898

+

with sum119898



+

with sum119901

119894=1120582119894= 1 120573 isin R119903



119898

119894=1120577119894119865119894(119909))(119910lowast) such that

0R119899 isin 120597 ⟨120582 119891⟩ (119909) +

119903

sum119894=1

120573119894nabla119892

119894(119909) +

119904

sum119894=1

120574119894nablaℎ

119894(119909)

+ 120591 [119909lowast

+ (

119898

sum119894=1

120577119894nabla119865

119894(119909))

lowast

(119910lowast

)] + 119873 (Θ 119909)

120573119894119892119894(119909) = 0 forall119894 = 1 2 119903

(94)


119873Θ(119909) = 119873(Θ 119909) for all 119909 isin R119899


sum119898



119894 119894 = 1 2 119898 being con-


119894=1120577119894nabla119865

119894(119909)




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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