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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 497586 7 pageshttpdxdoiorg1011552013497586

Research ArticleProperties of Expected Residual Minimization Model fora Class of Stochastic Complementarity Problems

Mei-Ju Luo1 and Yuan Lu2

1 School of Mathematics Liaoning University Liaoning 110036 China2 School of Sciences Shenyang University Liaoning 110044 China

Correspondence should be addressed to Mei-Ju Luo meiju luoyahoocn

Received 31 January 2013 Revised 10 May 2013 Accepted 14 May 2013

Academic Editor Farhad Hosseinzadeh Lotfi

Copyright copy 2013 M-J Luo and Y LuThis 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

Expected residualminimization (ERM)modelwhichminimizes an expected residual function defined by anNCP function has beenstudied in the literature for solving stochastic complementarity problems In this paper we first give the definitions of stochastic119875-function stochastic 119875

0-function and stochastic uniformly 119875-function Furthermore the conditions such that the function is a

stochastic 119875(1198750)-function are considered We then study the boundedness of solution set and global error bounds of the expected

residual functions defined by the ldquoFischer-Burmeisterrdquo (FB) function and ldquominrdquo function The conclusion indicates that solutionsof the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variationsin stochastic complementarity problems On the other hand we employ quasi-Monte Carlo methods and derivative-free methodsto solve ERMmodel

1 Introduction

Given a vector-valued function119865 R119899 times Ω rarr R119899 the stochas-tic complementarity problems denoted by SCP(119865(119909 120596)) areto find a vector 119909

lowast such that

119909lowast

ge 0 119865 (119909lowast 120596) ge 0

(119909lowast)119879119865 (119909lowast 120596) = 0 120596 isin Ω as

(1)

where 120596 isin Ω sube R119898 is a random vector with given probabilitydistribution P and ldquoasrdquo means ldquoalmost surelyrdquo under thegiven probability measure Particularly when 119865 is an affinefunction of 119909 for any 120596 that is

119865 (119909 120596) = 119872 (120596) 119909 + 119902 (120596) 120596 isin Ω (2)

where 119872(120596) isin R119899times119899 and 119902(120596) isin R119899 the SCP(119865(119909 120596)) iscalled stochastic linear complementarity problems denotedby SLCP(119872(120596) 119902(120596)) Correspondingly problem (1) is calledstochastic nonlinear complementarity problem denoted bySNCP(119865(119909 120596)) if 119865 can not be denoted by an affine functionof 119909 for any 120596 The deterministic problems which are calledcomplementarity problems (denoted by CP(119865(119909))) have

been intensively studied More information about theoreticalanalysis numerical algorithms and applications especially ineconomics and engineering can be found in comprehensivebooks [1 2]

In practical applications some elements may involvestochastic factors In fact due to stochastic factors thefunction value of 119865 depends not only on 119909 but also onrandom vectors Hence problem (1) does not have solutionin general for almost all 120596 isin Ω To solve these problemsresearchers focus on giving reasonable deterministic refor-mulations for SCP(119865(119909 120596)) Certainly different deterministicformulations may yield different solutions that are optimalin different senses In the study of SCP(119865(119909 120596)) three typesof formulations have been proposed the expected value(EV) formulation the expected residualminimization (ERM)formulation and the SMPEC formulation [3]

The EV formulation is studied by Gurkan et al [4] Theproblem considered in [4] is actually a stochastic variationalinequality problem When applied to the SCP(119865(119909 120596)) theEV model can be stated as follows

119909lowast

ge 0 E [119865 (119909lowast 120596)] ge 0 (119909

lowast)119879E [119865 (119909

lowast 120596)] = 0 (3)

where Emeans expectation with respect to 120596

2 Journal of Applied Mathematics

TheERMmodel is first proposed byChen and Fukushima[5] for solving the SLCP(119872(120596) 119902(120596)) By employing an NCPfunction 120601 the SCP(119865(119909 120596)) (1) is transformed equivalentlyto the stochastic equations

Φ (119909 120596) = 0 120596 isin Ω as (4)

where Φ R119899 times Ω rarr R119899 is defined by

Φ (119909 120596) = (

120601 (1199091 1198651

(119909 120596))

120601 (119909119899 119865119899

(119909 120596))

) (5)

and 119909119894denotes the 119894th component of the vector 119909 Here 120601

R119899 rarr R is an NCP function which has the property

120601 (119886 119887) = 0 lArrrArr 119886 ge 0 119887 ge 0 119886119887 = 0 (6)

Then the ERM formulation for (1) is given by

min119909isinR119899+

120579 (119909) = E [Φ (119909 120596)2] (7)

The NCP functions employed in [5] include the Fischer-Burmeister function which is defined by

120601FB (119886 119887) = radic1198862 + 1198872 minus (119886 + 119887) (8)

and the min function

120601min (119886 119887) = min 119886 119887 (9)

In particular it is known [6 7] that there exist the followingrelations between these two functions

2

radic2 + 2

1003816100381610038161003816120601min1003816100381610038161003816 le

1003816100381610038161003816120601FB1003816100381610038161003816 le (radic2 + 2)

1003816100381610038161003816120601min1003816100381610038161003816 (10)

As observed in [5] the ERM formulations with differentNCP functions may have different properties Subsequentlythe ERM formulation for SCP(119865(119909 120596)) has been studied in[6 8ndash13] Note that Fang et al [8] propose a new concept ofstochastic matrice 119872(sdot) is called a stochastic 119877

0matrix if

P 120596 119909 ge 0 119872 (120596) 119909 ge 0 119909119879119872 (120596) 119909 = 0 = 1 997904rArr 119909 = 0

(11)

Moreover Zhang and Chen [11] introduce a new conceptof stochastic 119877

0function which can be regarded as a

generalization of the stochastic 1198770matrix given in [8]

Throughout this paper we suppose that the sample spaceΩ is nonempty and compact set and that the function 119865(119909 120596)

is continuous with respect to 119909 and 120596 On the other hand wewill use the following notations 119868(119909) = 119894 119909

119894= 0 and

119869(119909) = 119894 119909119894

= 0 for a given vector 119909 isin R119899 ⟨119897 119899⟩ representsthe set 119897 119897 + 1 119897 + 119899 for natural numbers 119897 and 119906 with119897 lt 119906 119909

+= max119909 0 for any given vector 119909 sdot refers to the

Euclidean normThe remainder of the paper is organized as follows

in Section 2 we introduce the concepts of a stochastic 119875-function a stochastic119875

0-function and a stochastic uniformly

119875-function which can be regarded as a generalization ofthe deterministic 119875 119875

0-function and uniformly 119875-function

or an extension of stochastic 119875 matrix and stochastic 1198750

matrix [14] In addition some properties of a stochastic119875(1198750)-function are given In Section 3 we show the suffi-

cient conditions for the solution set of ERM problem tobe nonempty and bounded In Section 4 we discuss errorbounds of SCP(119865(119909 120596)) In Section 5 an algorithm will begiven to solve ERM model We then give conclusions inSection 6

2 Stochastic 119875(1198750)-Function

It is well known that the 119875-function 1198750-function and

uniformly119875-function play an important role in the nonlinearcomplementarity problems theory [1] We will introducea new concept of stochastic 119875-function 119875

0-function and

uniformly 119875-function which can be regarded as a general-ization of their deterministic form or stochastic 119875 matrix andstochastic 119875

0matrix

Definition 1 (see [14]) 119872(sdot) is called a stochastic119875(1198750)-matrix

if there exists 119894 isin 119869(119909) such that for every 119909 = 0 in R119899

P 120596 119909119894(119872 (120596) 119909)

119894gt 0 (ge 0) gt 0 (12)

Definition 2 A function 119865 R119899 times Ω rarr R119899 is a stochastic119875(1198750)-function if there exist 119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that

for every 119909 = 119910 in R119899

119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0

(13)

Definition 3 A function 119865 R119899 times Ω rarr R119899 is a stochasticuniformly 119875-function if there exists a positive constant 120572 and119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that for every 119909 = 119910 in R119899

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) ge 120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2 gt 0

(14)

Clearly every stochastic uniformly 119875-function must be astochastic 119875-function which in turn must be a stochastic 119875

0-

function We further cite the definition of ldquoequicoerciverdquo in[11] More information about this definition can be found in[11]

Definition 4 (see [11]) We say that 119865 R119899 times Ω rarr R119899 isequicoercive on D sube R119899 if for any 119909

119896 sube D satisfying

119909119896 rarr infin the existence of 120596

119896 sube suppΩ with

lim119896rarrinfin

119865119894(119909119896 120596119896) = infin(lim

119896rarrinfin(minus119865119894(119909119896 120596119896))+

= infin) forsome 119894 isin ⟨1 119899⟩ implies that

P 120596 lim119896rarrinfin

119865119894(119909119896 120596) = infin

gt 0 (P 120596 lim119896rarrinfin

(minus119865119894(119909119896 120596))+

= infin gt 0)

(15)

Journal of Applied Mathematics 3

where

suppΩ

= 120596 isin Ω int119861120596(120596])capΩ

119889119865 (120596) gt 0 for any ] gt 0

(16)

and 119861120596(120596 ]) = 120596 120596 minus 120596 lt ] and 119865(120596) is the distribution

function of 120596

More details about suppΩ were included in [8]

Proposition 5 If 119865 is a stochastic 1198750-function then 119865 + 120576119909 is

a stochastic 119875-function for every 120576 gt 0

Proof From the definition of stochastic 1198750-function there

exist 119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that for every 119909 = 119910

119909119894

= 119910119894 P 120596 (119909

119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) ge 0 gt 0

(17)

Hence we have

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) + 120576119909

119894minus (119865119894(119910 120596) + 120576119910

119894)) gt 0

= P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) + 120576(119909

119894minus 119910119894)2gt 0

ge P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) ge 0 gt 0

(18)

This proposition gives the relationship between stochastic1198750-function and stochastic 119875-function

Proposition 6 Let 119865 be an affine function of 119909 for any 120596 isin Ω

defined by (2)Then 119865 is a stochastic 119875(1198750)-function if and only

if 119872(sdot) is a stochastic 119875(1198750) matrix

Proof By the definition of stochastic 119875(1198750)-function we

have that there exist 119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that for every119909 = 119910

119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0

(19)

which is equivalent to

119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119872 (120596) (119909 minus 119910))

119894gt 0 (ge 0) gt 0

(20)

when119865 is defined by (2) Set 119911 = 119909minus119910 then 119911 = 0 andwe havethat formulation (20) holds if and only if there exists 119894 isin 119869(119911)

such that for every 119911 = 0

P 120596 119911119894(119872 (120596) 119911)

119894gt 0 (ge 0) gt 0 (21)

Hence 119872(sdot) is a stochastic 119875(1198750) matrix

Proposition 7 119865 is a stochastic 119875(1198750)-function if and only if

there exists a 120596 isin suppΩ such that 119865(sdot 120596) is a 119875(1198750)-function

Proof For the ldquoif rdquo part suppose on the contrary that119865 is nota stochastic 119875(119875

0)-function and then there exist 119909 119910 119909 = 119910 in

R119899 for any 119894 isin ⟨1 119899⟩ satisfying

119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(22)

On the other hand since 119865(sdot 120596) is a 119875(1198750)-function then

for 119909 119910 there exist 119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) (23)

Notice that 120596 isin suppΩ by the definition of suppΩ in (16)we have

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0 (24)

This contradicts formulation (22)Therefore 119865 is a stochastic119875(1198750)-functionNow for the ldquoonly if rdquo part suppose on the contrary that

there does not exist a 120596 isin suppΩ such that 119865(sdot 120596) is a 119875(1198750)-

function Then for any 119894 isin ⟨1 119899⟩ 120596 isin suppΩ there exists119909 119910 119909 = 119910 in R119899 such that

119909119894

= 119910119894

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) le 0 (lt 0)

(25)

which means that

119909119894

= 119910119894

P 120596 isin suppΩ (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596))

gt 0 (ge 0) = 0

(26)

By the definition of suppΩ in (16) we have P120596 isin Ω

suppΩ = 0 Hence formulation (26) is equivalent to

119909119894

= 119910119894

P 120596 isin Ω (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(27)

which contradicts definition (20) Therefore there exists a120596 isin suppΩ such that 119865(sdot 120596) is a 119875(119875

0)-function

Theorem 8 Suppose that 119891(119909) = E[119865(119909 120596)] is a 119875(1198750)-

function Then 119865 is a stochastic 119875(1198750)-function

Proof Suppose on the contrary that 119865 is not a stochastic119875(1198750)-function then there exist 119909 119910 119909 = 119910 in R119899 for any 119894 isin

⟨1 119899⟩ satisfying

119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(28)

This means that

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) le 0 (lt 0) (29)


always holds for any 119894 isin ⟨1 119899⟩ and 120596 isin Ω Furthermorefollowing from (29) we have

E [(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596))] le 0 (lt 0) (30)

that is

(119909119894minus 119910119894) (119891119894(119909) minus 119891

119894(119910)) le 0 (lt 0) (31)

which contradicts the definition of119875(1198750)-functionTherefore

119865 is a stochastic 119875(1198750)-function

Note that there is at most one solution (may not bea solution) for the EV model stochastic complementarityproblems if 119891(119909) = E[119865(119909 120596)] is a 119875(119875

0)-function

3 Boundedness of Solution Set

Theorem 9 Suppose that 119865 is a stochastic uniformly 119875-function and 119865 is equicoercive on R119899 Then the solution setof ERM model (7) defined by 120601min and 120601FB is nonempty andbounded

Proof Suppose on the contrary that the ERMmodel definedby 120601min is not boundedThus there exist a sequence 119909

119896 sub R119899+

with 119909119896 rarr infin (119896 rarr infin) and a constant 119888 isin R

+ such that

120579 (119909119896) le 119888 for forall119896 (32)

Define the index set 119868 sube 1 119899 by

119868 = 119894 | 119909119896

119894 is unbounded (33)

By assumption we have 119868 = 0 We now define a sequence119910119896 sube R119899 as follows

119910119896

119894=

0 if 119894 isin 119868

119909119896

119894if 119894 notin 119868

(34)

From the definition of 119910119896 and the fact that 119865 is a stochastic

uniformly 119875-function we obtain that for any 119909119896 119910119896 there

exists 119894 such that

P 120596 (119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596) minus 119865

119894(119910119896 120596))

ge 12057210038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

gt 0

(35)

and hence there are 120596119896

isin suppΩ satisfying

(119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596119896) minus 119865119894(119910119896 120596119896)) ge 120572

10038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

(36)

Take subsequence 119909119896119894 119910119896119894 such that the corresponding sub-

script of (36) is 119895 Noting that 119895 isin 119868 and taking (36) intoaccount we have

120572sum

119895isin119868

(119909119896119894

119895)2

le 119909119896119894

119895(119865119895(119909119896119894 120596119896119894) minus 119865

119895(119910119896119894 120596119896119894))

le radicsum

119895isin119868

(119909119896119894

119895)2

sdotradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816

(37)

from which we get

120572radicsum

119895isin119868

(119909119896119894

119895)2

leradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816 (38)

By definition the sequence 119910119896119894 remains bounded From the

continuity of 119865 it follows that the sequence 119865119895(119910119896119894 120596119896119894) is

also bounded for every 119895 isin 119868 Hence taking a limit in (38)we obtain that there is at least one index 119895 isin 119868 such that

119909119896119894

119895997888rarr infin 119865

119895(119909119896119894 120596119896119894) 997888rarr infin (39)

Since 119865 is equicoercive on R119899 we have

P120596 lim119896119894rarrinfin

119865119895(119909119896119894 120596) = infin gt 0 (40)

Let

Ω1

= 120596 lim119896119894rarrinfin

min (119909119896119894

119895 119865119895(119909119896119894 120596)) = infin (41)

ThenPΩ1 gt 0 By Fatoursquos Lemma [15] we have

EΩ1

[lim inf119896119894rarrinfin

(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

le lim inf119896119894rarrinfin

EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

(42)

Since

lim inf119896119894rarrinfin

(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

= infin (43)

on Ω1andPΩ

1 gt 0 then the left-hand side of formulation

(42) is infinite Therefore


EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

] = infin (44)

Moreover it is easy to find

120579 (119909119896119894) = E [

10038171003817100381710038171003817Φ(119909119896119894 119865(119909119896119894 120596))

10038171003817100381710038171003817

2

]

ge EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

997888rarr infin

(45)

as 119896119894

rarr infin This contradicts formulation (32) Hence thesolution set of ERM model (7) defined by 120601min is nonemptyand bounded Similar results about 120601FB can be obtained byrelation formulation (10)

4 Robust Solution

As we show both EV model and ERM model give decisionsby a deterministic formulation However the decisions maynot be the best or may be even infeasible for each individualevent In fact we should take risk into account to make


a priori decision in many cases Naturally it is necessary toknow how good or how bad the decision which we have givencan be In this section we study the robustness of solutions ofthe ERM model Let SOL(119865(119909 120596)) denote the solution set ofSCP(119865(119909 120596)) and define the distance from a point 119909 to theset SOL(119865(119909 120596)) by

dist (119909 SOL (119865 (119909 120596))) = inf1199091015840isinSOL(119865(119909120596))

10038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817

(46)

Theorem 10 Assume that Ω = 1205961 1205962 120596

119873 sub R119898

and 120596 takes values 1205961 120596

119873 with respective probabilities1199011 119901

119873 Furthermore suppose that for every120596 isin Ω119865(119909 120596)

is uniformly P-function and Lipschitz continuous with respectto 119909 Then there is a constant 119862 gt 0 such that

E [dist (119909 SOL (119865 (119909 120596)))] le 119862 sdot radic120579 (119909) (47)

where 120579(119909) is defined by 120601min or 120601FB

Proof For any fixed 120596119894 since 119865(119909 120596

119894) is uniformly 119875-

function and Lipschitz continuous from Corollary 319 of[16] we have unique solution 119909(120596

119894) of CP(119865(119909 120596

119894)) and there

exists a constant 119862119894such that

10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

le 119862119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817 (48)

Letting 119862 = ((radic2 + 2)2)max1198621 119862

119873 we have

E2 [dist (119909 SOL (119865 (119909 120596)))]

= E2 [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817]

le E [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

2

]

le

119873

sum

119894=1

119901119894sdot 1198622

119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817

2

le

119873

sum

119894=1

119901119894sdot 1198622

119894sdot (

radic2 + 2

2)

2

times

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

le 1198622

119873

sum

119894=1

119901119894

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

= 1198622120579 (119909)

(49)

where the first inequality follows from Cauchy-Schwarzinequality the second inequality follows from formulation(48) and the third inequality follows from formulation (10)This completes the proof of the theorem

Theorem 10 particularly shows that for the solution 119909lowast of

(7)

E [dist (119909lowast SOL (119865 (119909 120596)))] le 119862 sdot radic120579 (119909

lowast) (50)

This inequality indicates that the expected distance to thesolution set SOL(119865(119909 120596)) for 120596 isin Ω is also likely to be smallat the solution 119909

lowast of (7) In other words we may expectthat a solution of the ERM formulation (7) has a minimumsensitivity with respect to random parameter variations inSCP(119865(119909 120596)) In this sense solutions of (7) can be regardedas robust solutions for SCP(119865(119909 120596))

5 Quasi-Monte Carlo and Derivative-FreeMethods for Solving ERM Model

Note that the ERM model (7) included an expectation func-tionwhich is generally difficult to be evaluated exactlyHencein this section we first employ a quasi-Monte Carlo methodto obtain approximation problems of (7) for numericalintegration Then we consider derivative-free methods tosolve these approximation problems

By the quasi-Monte Carlo method we obtain the follow-ing approximation problem of (7)

min119909isinR119899+

120579119873

(119909) =1

119873sum

120596119894isinΩ119873

10038171003817100381710038171003817Φ (119909 120596

119894)10038171003817100381710038171003817

2

120588 (120596119894) (51)

where Ω119873

= 120596119894

| 119894 = 1 2 119873 is a set of observationsgenerated by a quasi-Monte Carlo method such that Ω

119873sube

Ω and 120588(120596) stands for the probability density functionIn the rest of this paper we assume that the probabilitydensity function 120588 is continuous on Ω For each 119873 120579

119873(119909)

is continuously differentiable function We denote by 119909119873 the

optimal solutions of approximation problems (51) We areinterested in the situation where the first-order derivatives of120579119873

(119909) cannot be explicitly calculated or approximated

Condition 1 Given a point 1199090

ge 0 the level set

119871 = 119909 ge 0 | 119891 (119909) le 119891 (1199090) (52)

is compact

Condition 2 If 119909119873

119896 and 119910

119873

119896 are sequences of points such

that 119909119873

119896ge 0 119910

119873

119896ge 0 converging to some 119909

119873 and 119868119873

119896sube

119868(119909119873

) = 119894 | 119909119873

119894= 0 for all 119896 then

dist (119879119868119873

119896

(119909119873

119896) 119879119868119873

119896

(119910119873

119896)) 997888rarr 0 (53)

where dist(1198791 1198792) = max

1198891isin11987911198891=1

min1198892isin1198792

1198891

minus 1198892 and

119879119868119873

119896

(119909) = 119889119873

119896isin R119899 | 119889

119873

119896119894ge 0 forall119894 isin 119868

119873

119896

Condition 3 For every 119909119873

ge 0 there exist scalars 120575 gt 0 and120578 gt 0 such that

min119911ge0

119911 minus 119909 le 120578

119899

sum

119894=1

max (minus119909119894 0)

forall119909 isin 119909 isin R119899 | 119909 minus 119909 le 120575

(54)

Condition 4 Given 119909119873

119896and 120598

119873

119896gt 0 the set of search

directions

119863119873

119896= 119889119873119895

119896 119895 = 1 119903

119873

119896 with 10038171003817100381710038171003817

119889119873119895

119896

10038171003817100381710038171003817= 1 (55)


satisfing 119903119873

119896is uniformly bounded and cone119863119873

119896 = 119879(119909

119873

119896

120598119873

119896) Here

cone 119863119873

119896

= 1198891198731

1198961205731

+ sdot sdot sdot + 119889119873119903119873

119896

119896120573119903119873

119896 1205731

ge 0 120573119903119873

119896 ge 0

119879 (119909119873

119896 120598119873

119896) = 119889

119873

119896isin R119899 | 119889

119873

119896119894ge 0 119909

119873

119896119894le 120598119873

119896

(56)

Under Conditions 1 2 and 3 and by choosing 119863119873

119896

satisfying Condition 4 with 120598119873

119896rarr 0 then the following

generated iterates have at least one cluster point that is astationary point of (51) for each 119873

Algorithm 11 Parameters 119909119873

0ge 0

119873

0gt 0 120574

119873gt 0 120579

119873

1isin

(0 1) 120579119873

2isin (0 1) 120598

119873

0gt 0

Step 1 Set 119896119873

= 0

Step 2 Choose a set of directions 119863119873

119896= 119889119873119895

119896 119895 = 1 119903

119873

119896

satisfying Condition 4

Step 3

(a) Set 119895 = 1 119910119873119895

119896= 119909119873

119896 119873119895

119896= 119873

119896

(b) Compute the maximum stepsize 120572119873119895

119896such that 119910

119873119895

119894119896+

120572119873119895

119896119889119873119895

119894119896ge 0 for all 119894 Set

119873119895

119896= min120572

119873119895

119896 119873119895

119896

(c) If 119873119895

119896gt 0 and 120579

119873(119910119873119895

119896) le 120579

119873(119910119873119895

119896) minus 120574(

119873119895

119896)2

set 119873119895+1

119896= 120572119873119895

119896 otherwise set 120572

119873119895

119896= 0 119910

119873119895+1

119896=

119910119873119895

119896 119873119895+1

119896= 120579119873

1119873119895

119896

(d) If 120572119873119895

119896= 120572119873119895

119896 set 120598119873

119896+1= 120598119873

119896 and go to Step 4

(e) If 119895 lt 119903119873

119896 set 119895 = 119895 + 1 and go to Step 3(b) Otherwise

set 120598119873

119896+1= 120579119873

2120598119873

119896and go to Step 4

Step 4 Find 119909119873

119896+1ge 0 such that 120579

119873(119909119873

119896+1) le 120579

119873(119910119873119895+1

119896) Set

119873

119896+1= 119873119895+1

119896 119903119896

= 119895 119896 = 119896 + 1 and go to Step 2

For this algorithm it is easy to proof that if 119909119873

119896is the

sequence produced by algorithm under Conditions 1ndash4 then119909119873

119896is bounded and there exists at least one cluster pointwhich

is a stationary point of problem (51) for each 119873

6 Conclusions

The SCP(119865(119909 120596)) has a wide range of applications in engi-neering and economics Therefore it is meaningful andinteresting to study this problem In this paper we give thedefinitions of stochastic 119875-function stochastic 119875

0-function

and stochastic uniformly 119875-function which can be regardedas a generalization of the deterministic formulation or anextension of a stochastic 119877

0function given in [11] Moreover

we consider the conditions when the function is a stochastic119875(1198750)-function Furthermore we show that the involved

function being a stochastic uniformly 119875-function and equi-coercive [11] are sufficient conditions for the solution set of theexpected residualminimization problem to be nonempty andbounded Finally we illustrate that the ERM formulation pro-duces robust solutions with minimum sensitivity in violationof feasibility with respect to random parameter variationsin SCP(119865(119909 120596)) On the other hand we employ a quasi-Monte Carlo method to obtain approximation problems of(7) for dealing numerical integration and further considerderivative-free methods to solve these approximation prob-lems

Acknowledgments

This work was supported by NSFC Grants no 11226238 andno 11226230 and predeclaration fund of state project ofLiaoning university 2012 2012LDGY01 and University Sci-entific Research Projects of School of Education Departmentof Liaoning Province 2012 2012427

References

[1] F Facchinei and J S Pang Finite-Dimensional VariationalInequalities and Complementarity Problems Springer NewYork NY USA 2003

[2] RW Cottle J-S Pang and R E StoneTheLinear Complemen-tarity Problem Computer Science and Scientific ComputingAcademic Press Boston Mass USA 1992

[3] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[4] G Gurkan A Y Ozge and S M Robinson ldquoSample-pathsolution of stochastic variational inequalitiesrdquo MathematicalProgramming vol 84 no 2 pp 313ndash333 1999

[5] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[6] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[7] P Tseng ldquoGrowth behavior of a class of merit functions for thenonlinear complementarity problemrdquo Journal of OptimizationTheory and Applications vol 89 no 1 pp 17ndash37 1996

[8] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[9] G-H Lin X Chen and M Fukushima ldquoNew restrictedNCP functions and their applications to stochastic NCP andstochastic MPECrdquo Optimization vol 56 no 5-6 pp 641ndash9532007

[10] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property of

an expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[11] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008


[12] C Zhang and X Chen ldquoSmoothing projected gradient methodand its application to stochastic linear complementarity prob-lemsrdquo SIAM Journal onOptimization vol 20 no 2 pp 627ndash6492009

[13] G L Zhou and L Caccetta ldquoFeasible semismooth Newtonmethod for a class of stochastic linear complementarity prob-lemsrdquo Journal of OptimizationTheory and Applications vol 139no 2 pp 379ndash392 2008

[14] X L Li H W Liu and Y K Huang ldquoStochastic 119875 matrix and1198750matrix linear complementarity problemrdquo Journal of Systems

Science and Mathematical Sciences vol 31 no 1 pp 123ndash1282011

[15] K L Chung A Course in Probability Theory Academic PressNew York NY USA 2nd edition 1974

[16] B Chen and P T Harker ldquoSmooth approximations to nonlinearcomplementarity problemsrdquo SIAM Journal on Optimizationvol 7 no 2 pp 403ndash420 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


TheERMmodel is first proposed byChen and Fukushima[5] for solving the SLCP(119872(120596) 119902(120596)) By employing an NCPfunction 120601 the SCP(119865(119909 120596)) (1) is transformed equivalentlyto the stochastic equations

Φ (119909 120596) = 0 120596 isin Ω as (4)

where Φ R119899 times Ω rarr R119899 is defined by

Φ (119909 120596) = (

120601 (1199091 1198651

(119909 120596))

120601 (119909119899 119865119899

(119909 120596))

) (5)

and 119909119894denotes the 119894th component of the vector 119909 Here 120601

R119899 rarr R is an NCP function which has the property

120601 (119886 119887) = 0 lArrrArr 119886 ge 0 119887 ge 0 119886119887 = 0 (6)

Then the ERM formulation for (1) is given by

min119909isinR119899+

120579 (119909) = E [Φ (119909 120596)2] (7)

The NCP functions employed in [5] include the Fischer-Burmeister function which is defined by

120601FB (119886 119887) = radic1198862 + 1198872 minus (119886 + 119887) (8)

and the min function

120601min (119886 119887) = min 119886 119887 (9)

In particular it is known [6 7] that there exist the followingrelations between these two functions

2

radic2 + 2

1003816100381610038161003816120601min1003816100381610038161003816 le

1003816100381610038161003816120601FB1003816100381610038161003816 le (radic2 + 2)

1003816100381610038161003816120601min1003816100381610038161003816 (10)

As observed in [5] the ERM formulations with differentNCP functions may have different properties Subsequentlythe ERM formulation for SCP(119865(119909 120596)) has been studied in[6 8ndash13] Note that Fang et al [8] propose a new concept ofstochastic matrice 119872(sdot) is called a stochastic 119877

0matrix if

P 120596 119909 ge 0 119872 (120596) 119909 ge 0 119909119879119872 (120596) 119909 = 0 = 1 997904rArr 119909 = 0

(11)

Moreover Zhang and Chen [11] introduce a new conceptof stochastic 119877

0function which can be regarded as a

generalization of the stochastic 1198770matrix given in [8]

Throughout this paper we suppose that the sample spaceΩ is nonempty and compact set and that the function 119865(119909 120596)

is continuous with respect to 119909 and 120596 On the other hand wewill use the following notations 119868(119909) = 119894 119909

119894= 0 and

119869(119909) = 119894 119909119894

= 0 for a given vector 119909 isin R119899 ⟨119897 119899⟩ representsthe set 119897 119897 + 1 119897 + 119899 for natural numbers 119897 and 119906 with119897 lt 119906 119909

+= max119909 0 for any given vector 119909 sdot refers to the

Euclidean normThe remainder of the paper is organized as follows

in Section 2 we introduce the concepts of a stochastic 119875-function a stochastic119875

0-function and a stochastic uniformly

119875-function which can be regarded as a generalization ofthe deterministic 119875 119875

0-function and uniformly 119875-function

or an extension of stochastic 119875 matrix and stochastic 1198750

matrix [14] In addition some properties of a stochastic119875(1198750)-function are given In Section 3 we show the suffi-

cient conditions for the solution set of ERM problem tobe nonempty and bounded In Section 4 we discuss errorbounds of SCP(119865(119909 120596)) In Section 5 an algorithm will begiven to solve ERM model We then give conclusions inSection 6

2 Stochastic 119875(1198750)-Function

It is well known that the 119875-function 1198750-function and

uniformly119875-function play an important role in the nonlinearcomplementarity problems theory [1] We will introducea new concept of stochastic 119875-function 119875

0-function and

uniformly 119875-function which can be regarded as a general-ization of their deterministic form or stochastic 119875 matrix andstochastic 119875

0matrix

Definition 1 (see [14]) 119872(sdot) is called a stochastic119875(1198750)-matrix

if there exists 119894 isin 119869(119909) such that for every 119909 = 0 in R119899

P 120596 119909119894(119872 (120596) 119909)

119894gt 0 (ge 0) gt 0 (12)

Definition 2 A function 119865 R119899 times Ω rarr R119899 is a stochastic119875(1198750)-function if there exist 119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that

for every 119909 = 119910 in R119899

119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0

(13)

Definition 3 A function 119865 R119899 times Ω rarr R119899 is a stochasticuniformly 119875-function if there exists a positive constant 120572 and119894 isin 119869(119909 119910) 119894 isin ⟨1 119899⟩ such that for every 119909 = 119910 in R119899

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) ge 120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2 gt 0

(14)

Clearly every stochastic uniformly 119875-function must be astochastic 119875-function which in turn must be a stochastic 119875

0-

function We further cite the definition of ldquoequicoerciverdquo in[11] More information about this definition can be found in[11]

Definition 4 (see [11]) We say that 119865 R119899 times Ω rarr R119899 isequicoercive on D sube R119899 if for any 119909

119896 sube D satisfying

119909119896 rarr infin the existence of 120596

119896 sube suppΩ with

lim119896rarrinfin

119865119894(119909119896 120596119896) = infin(lim

119896rarrinfin(minus119865119894(119909119896 120596119896))+

= infin) forsome 119894 isin ⟨1 119899⟩ implies that

P 120596 lim119896rarrinfin

119865119894(119909119896 120596) = infin

gt 0 (P 120596 lim119896rarrinfin

(minus119865119894(119909119896 120596))+

= infin gt 0)

(15)


where

suppΩ

= 120596 isin Ω int119861120596(120596])capΩ

119889119865 (120596) gt 0 for any ] gt 0

(16)


function of 120596






119909119894

= 119910119894 P 120596 (119909

119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) ge 0 gt 0

(17)

Hence we have

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) + 120576119909

119894minus (119865119894(119910 120596) + 120576119910

119894)) gt 0

= P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) + 120576(119909

119894minus 119910119894)2gt 0

ge P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) ge 0 gt 0

(18)







119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0

(19)


119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119872 (120596) (119909 minus 119910))

119894gt 0 (ge 0) gt 0

(20)



P 120596 119911119894(119872 (120596) 119911)

119894gt 0 (ge 0) gt 0 (21)







119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(22)



(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) (23)


P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0 (24)




119909119894

= 119910119894

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) le 0 (lt 0)

(25)

which means that

119909119894

= 119910119894


119894(119910 120596))

gt 0 (ge 0) = 0

(26)



119909119894

= 119910119894


119894(119910 120596)) gt 0 (ge 0) = 0

(27)


0)-function





119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(28)

This means that

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) le 0 (lt 0) (29)



E [(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596))] le 0 (lt 0) (30)

that is

(119909119894minus 119910119894) (119891119894(119909) minus 119891

119894(119910)) le 0 (lt 0) (31)




0)-function




119896 sub R119899+


+ such that

120579 (119909119896) le 119888 for forall119896 (32)


119868 = 119894 | 119909119896



119910119896

119894=

0 if 119894 isin 119868

119909119896

119894if 119894 notin 119868

(34)




P 120596 (119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596) minus 119865

119894(119910119896 120596))

ge 12057210038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

gt 0

(35)



(119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596119896) minus 119865119894(119910119896 120596119896)) ge 120572

10038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

(36)



120572sum

119895isin119868

(119909119896119894

119895)2

le 119909119896119894

119895(119865119895(119909119896119894 120596119896119894) minus 119865

119895(119910119896119894 120596119896119894))

le radicsum

119895isin119868

(119909119896119894

119895)2

sdotradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816

(37)

from which we get

120572radicsum

119895isin119868

(119909119896119894

119895)2

leradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816 (38)




119909119896119894

119895997888rarr infin 119865

119895(119909119896119894 120596119896119894) 997888rarr infin (39)



119865119895(119909119896119894 120596) = infin gt 0 (40)

Let

Ω1

= 120596 lim119896119894rarrinfin

min (119909119896119894

119895 119865119895(119909119896119894 120596)) = infin (41)


EΩ1


(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]


EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

(42)

Since


(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

= infin (43)

on Ω1andPΩ




EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

] = infin (44)


120579 (119909119896119894) = E [

10038171003817100381710038171003817Φ(119909119896119894 119865(119909119896119894 120596))

10038171003817100381710038171003817

2

]

ge EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

997888rarr infin

(45)

as 119896119894


4 Robust Solution





10038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817

(46)


119873 sub R119898

and 120596 takes values 1205961 120596







119894) is uniformly 119875-


119894) of CP(119865(119909 120596

119894)) and there


10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

le 119862119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817 (48)


119873 we have

E2 [dist (119909 SOL (119865 (119909 120596)))]

= E2 [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817]

le E [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

2

]

le

119873

sum

119894=1

119901119894sdot 1198622

119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817

2

le

119873

sum

119894=1

119901119894sdot 1198622

119894sdot (

radic2 + 2

2)

2

times

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

le 1198622

119873

sum

119894=1

119901119894

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

= 1198622120579 (119909)

(49)



(7)


lowast) (50)






min119909isinR119899+

120579119873

(119909) =1

119873sum

120596119894isinΩ119873

10038171003817100381710038171003817Φ (119909 120596

119894)10038171003817100381710038171003817

2

120588 (120596119894) (51)

where Ω119873

= 120596119894


119873sube


119873(119909)





ge 0 the level set

119871 = 119909 ge 0 | 119891 (119909) le 119891 (1199090) (52)

is compact


119896 and 119910

119873


that 119909119873

119896ge 0 119910

119873


119873 and 119868119873

119896sube

119868(119909119873

) = 119894 | 119909119873

119894= 0 for all 119896 then

dist (119879119868119873

119896

(119909119873

119896) 119879119868119873

119896

(119910119873

119896)) 997888rarr 0 (53)

where dist(1198791 1198792) = max

1198891isin11987911198891=1

min1198892isin1198792

1198891

minus 1198892 and

119879119868119873

119896

(119909) = 119889119873

119896isin R119899 | 119889

119873

119896119894ge 0 forall119894 isin 119868

119873

119896



min119911ge0

119911 minus 119909 le 120578

119899

sum

119894=1

max (minus119909119894 0)


(54)


119896and 120598

119873


directions

119863119873

119896= 119889119873119895

119896 119895 = 1 119903

119873

119896 with 10038171003817100381710038171003817

119889119873119895

119896

10038171003817100381710038171003817= 1 (55)




119896 = 119879(119909

119873

119896

120598119873

119896) Here

cone 119863119873

119896

= 1198891198731

1198961205731

+ sdot sdot sdot + 119889119873119903119873

119896

119896120573119903119873

119896 1205731

ge 0 120573119903119873

119896 ge 0

119879 (119909119873

119896 120598119873

119896) = 119889

119873

119896isin R119899 | 119889

119873

119896119894ge 0 119909

119873

119896119894le 120598119873

119896

(56)


119896





0ge 0

119873

0gt 0 120574

119873gt 0 120579

119873

1isin

(0 1) 120579119873

2isin (0 1) 120598

119873

0gt 0

Step 1 Set 119896119873

= 0


119896= 119889119873119895

119896 119895 = 1 119903

119873

119896


Step 3

(a) Set 119895 = 1 119910119873119895

119896= 119909119873

119896 119873119895

119896= 119873

119896


119896such that 119910

119873119895

119894119896+

120572119873119895

119896119889119873119895

119894119896ge 0 for all 119894 Set

119873119895

119896= min120572

119873119895

119896 119873119895

119896

(c) If 119873119895

119896gt 0 and 120579

119873(119910119873119895

119896) le 120579

119873(119910119873119895

119896) minus 120574(

119873119895

119896)2

set 119873119895+1

119896= 120572119873119895


119873119895

119896= 0 119910

119873119895+1

119896=

119910119873119895

119896 119873119895+1

119896= 120579119873

1119873119895

119896

(d) If 120572119873119895

119896= 120572119873119895

119896 set 120598119873

119896+1= 120598119873


(e) If 119895 lt 119903119873


set 120598119873

119896+1= 120579119873

2120598119873


Step 4 Find 119909119873

119896+1ge 0 such that 120579

119873(119909119873

119896+1) le 120579

119873(119910119873119895+1

119896) Set

119873

119896+1= 119873119895+1

119896 119903119896

= 119895 119896 = 119896 + 1 and go to Step 2


119896is the




6 Conclusions


0-function





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

suppΩ

= 120596 isin Ω int119861120596(120596])capΩ

119889119865 (120596) gt 0 for any ] gt 0

(16)


function of 120596






119909119894

= 119910119894 P 120596 (119909

119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) ge 0 gt 0

(17)

Hence we have

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) + 120576119909

119894minus (119865119894(119910 120596) + 120576119910

119894)) gt 0

= P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) + 120576(119909

119894minus 119910119894)2gt 0

ge P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) ge 0 gt 0

(18)







119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0

(19)


119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119872 (120596) (119909 minus 119910))

119894gt 0 (ge 0) gt 0

(20)



P 120596 119911119894(119872 (120596) 119911)

119894gt 0 (ge 0) gt 0 (21)







119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(22)



(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) (23)


P 120596 (119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) gt 0 (24)




119909119894

= 119910119894

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) le 0 (lt 0)

(25)

which means that

119909119894

= 119910119894


119894(119910 120596))

gt 0 (ge 0) = 0

(26)



119909119894

= 119910119894


119894(119910 120596)) gt 0 (ge 0) = 0

(27)


0)-function





119909119894

= 119910119894

P 120596 (119909119894minus 119910119894) (119865119894 (119909 120596) minus 119865

119894(119910 120596)) gt 0 (ge 0) = 0

(28)

This means that

(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596)) le 0 (lt 0) (29)



E [(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596))] le 0 (lt 0) (30)

that is

(119909119894minus 119910119894) (119891119894(119909) minus 119891

119894(119910)) le 0 (lt 0) (31)




0)-function




119896 sub R119899+


+ such that

120579 (119909119896) le 119888 for forall119896 (32)


119868 = 119894 | 119909119896



119910119896

119894=

0 if 119894 isin 119868

119909119896

119894if 119894 notin 119868

(34)




P 120596 (119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596) minus 119865

119894(119910119896 120596))

ge 12057210038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

gt 0

(35)



(119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596119896) minus 119865119894(119910119896 120596119896)) ge 120572

10038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

(36)



120572sum

119895isin119868

(119909119896119894

119895)2

le 119909119896119894

119895(119865119895(119909119896119894 120596119896119894) minus 119865

119895(119910119896119894 120596119896119894))

le radicsum

119895isin119868

(119909119896119894

119895)2

sdotradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816

(37)

from which we get

120572radicsum

119895isin119868

(119909119896119894

119895)2

leradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816 (38)




119909119896119894

119895997888rarr infin 119865

119895(119909119896119894 120596119896119894) 997888rarr infin (39)



119865119895(119909119896119894 120596) = infin gt 0 (40)

Let

Ω1

= 120596 lim119896119894rarrinfin

min (119909119896119894

119895 119865119895(119909119896119894 120596)) = infin (41)


EΩ1


(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]


EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

(42)

Since


(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

= infin (43)

on Ω1andPΩ




EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

] = infin (44)


120579 (119909119896119894) = E [

10038171003817100381710038171003817Φ(119909119896119894 119865(119909119896119894 120596))

10038171003817100381710038171003817

2

]

ge EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

997888rarr infin

(45)

as 119896119894


4 Robust Solution





10038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817

(46)


119873 sub R119898

and 120596 takes values 1205961 120596







119894) is uniformly 119875-


119894) of CP(119865(119909 120596

119894)) and there


10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

le 119862119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817 (48)


119873 we have

E2 [dist (119909 SOL (119865 (119909 120596)))]

= E2 [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817]

le E [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

2

]

le

119873

sum

119894=1

119901119894sdot 1198622

119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817

2

le

119873

sum

119894=1

119901119894sdot 1198622

119894sdot (

radic2 + 2

2)

2

times

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

le 1198622

119873

sum

119894=1

119901119894

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

= 1198622120579 (119909)

(49)



(7)


lowast) (50)






min119909isinR119899+

120579119873

(119909) =1

119873sum

120596119894isinΩ119873

10038171003817100381710038171003817Φ (119909 120596

119894)10038171003817100381710038171003817

2

120588 (120596119894) (51)

where Ω119873

= 120596119894


119873sube


119873(119909)





ge 0 the level set

119871 = 119909 ge 0 | 119891 (119909) le 119891 (1199090) (52)

is compact


119896 and 119910

119873


that 119909119873

119896ge 0 119910

119873


119873 and 119868119873

119896sube

119868(119909119873

) = 119894 | 119909119873

119894= 0 for all 119896 then

dist (119879119868119873

119896

(119909119873

119896) 119879119868119873

119896

(119910119873

119896)) 997888rarr 0 (53)

where dist(1198791 1198792) = max

1198891isin11987911198891=1

min1198892isin1198792

1198891

minus 1198892 and

119879119868119873

119896

(119909) = 119889119873

119896isin R119899 | 119889

119873

119896119894ge 0 forall119894 isin 119868

119873

119896



min119911ge0

119911 minus 119909 le 120578

119899

sum

119894=1

max (minus119909119894 0)


(54)


119896and 120598

119873


directions

119863119873

119896= 119889119873119895

119896 119895 = 1 119903

119873

119896 with 10038171003817100381710038171003817

119889119873119895

119896

10038171003817100381710038171003817= 1 (55)




119896 = 119879(119909

119873

119896

120598119873

119896) Here

cone 119863119873

119896

= 1198891198731

1198961205731

+ sdot sdot sdot + 119889119873119903119873

119896

119896120573119903119873

119896 1205731

ge 0 120573119903119873

119896 ge 0

119879 (119909119873

119896 120598119873

119896) = 119889

119873

119896isin R119899 | 119889

119873

119896119894ge 0 119909

119873

119896119894le 120598119873

119896

(56)


119896





0ge 0

119873

0gt 0 120574

119873gt 0 120579

119873

1isin

(0 1) 120579119873

2isin (0 1) 120598

119873

0gt 0

Step 1 Set 119896119873

= 0


119896= 119889119873119895

119896 119895 = 1 119903

119873

119896


Step 3

(a) Set 119895 = 1 119910119873119895

119896= 119909119873

119896 119873119895

119896= 119873

119896


119896such that 119910

119873119895

119894119896+

120572119873119895

119896119889119873119895

119894119896ge 0 for all 119894 Set

119873119895

119896= min120572

119873119895

119896 119873119895

119896

(c) If 119873119895

119896gt 0 and 120579

119873(119910119873119895

119896) le 120579

119873(119910119873119895

119896) minus 120574(

119873119895

119896)2

set 119873119895+1

119896= 120572119873119895


119873119895

119896= 0 119910

119873119895+1

119896=

119910119873119895

119896 119873119895+1

119896= 120579119873

1119873119895

119896

(d) If 120572119873119895

119896= 120572119873119895

119896 set 120598119873

119896+1= 120598119873


(e) If 119895 lt 119903119873


set 120598119873

119896+1= 120579119873

2120598119873


Step 4 Find 119909119873

119896+1ge 0 such that 120579

119873(119909119873

119896+1) le 120579

119873(119910119873119895+1

119896) Set

119873

119896+1= 119873119895+1

119896 119903119896

= 119895 119896 = 119896 + 1 and go to Step 2


119896is the




6 Conclusions


0-function





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E [(119909119894minus 119910119894) (119865119894(119909 120596) minus 119865

119894(119910 120596))] le 0 (lt 0) (30)

that is

(119909119894minus 119910119894) (119891119894(119909) minus 119891

119894(119910)) le 0 (lt 0) (31)




0)-function




119896 sub R119899+


+ such that

120579 (119909119896) le 119888 for forall119896 (32)


119868 = 119894 | 119909119896



119910119896

119894=

0 if 119894 isin 119868

119909119896

119894if 119894 notin 119868

(34)




P 120596 (119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596) minus 119865

119894(119910119896 120596))

ge 12057210038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

gt 0

(35)



(119909119896

119894minus 119910119896

119894) (119865119894(119909119896 120596119896) minus 119865119894(119910119896 120596119896)) ge 120572

10038171003817100381710038171003817119909119896

minus 11991011989610038171003817100381710038171003817

2

(36)



120572sum

119895isin119868

(119909119896119894

119895)2

le 119909119896119894

119895(119865119895(119909119896119894 120596119896119894) minus 119865

119895(119910119896119894 120596119896119894))

le radicsum

119895isin119868

(119909119896119894

119895)2

sdotradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816

(37)

from which we get

120572radicsum

119895isin119868

(119909119896119894

119895)2

leradic

sum

119895isin119868

10038161003816100381610038161003816119865119895(119909119896119894 120596

119896119894) minus 119865

119895(119910119896119894 120596

119896119894)

10038161003816100381610038161003816 (38)




119909119896119894

119895997888rarr infin 119865

119895(119909119896119894 120596119896119894) 997888rarr infin (39)



119865119895(119909119896119894 120596) = infin gt 0 (40)

Let

Ω1

= 120596 lim119896119894rarrinfin

min (119909119896119894

119895 119865119895(119909119896119894 120596)) = infin (41)


EΩ1


(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]


EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

(42)

Since


(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

= infin (43)

on Ω1andPΩ




EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

] = infin (44)


120579 (119909119896119894) = E [

10038171003817100381710038171003817Φ(119909119896119894 119865(119909119896119894 120596))

10038171003817100381710038171003817

2

]

ge EΩ1

[(min (119909119896119894

119895 119865119895(119909119896119894 120596)))

2

]

997888rarr infin

(45)

as 119896119894


4 Robust Solution





10038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817

(46)


119873 sub R119898

and 120596 takes values 1205961 120596







119894) is uniformly 119875-


119894) of CP(119865(119909 120596

119894)) and there


10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

le 119862119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817 (48)


119873 we have

E2 [dist (119909 SOL (119865 (119909 120596)))]

= E2 [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817]

le E [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

2

]

le

119873

sum

119894=1

119901119894sdot 1198622

119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817

2

le

119873

sum

119894=1

119901119894sdot 1198622

119894sdot (

radic2 + 2

2)

2

times

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

le 1198622

119873

sum

119894=1

119901119894

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

= 1198622120579 (119909)

(49)



(7)


lowast) (50)






min119909isinR119899+

120579119873

(119909) =1

119873sum

120596119894isinΩ119873

10038171003817100381710038171003817Φ (119909 120596

119894)10038171003817100381710038171003817

2

120588 (120596119894) (51)

where Ω119873

= 120596119894


119873sube


119873(119909)





ge 0 the level set

119871 = 119909 ge 0 | 119891 (119909) le 119891 (1199090) (52)

is compact


119896 and 119910

119873


that 119909119873

119896ge 0 119910

119873


119873 and 119868119873

119896sube

119868(119909119873

) = 119894 | 119909119873

119894= 0 for all 119896 then

dist (119879119868119873

119896

(119909119873

119896) 119879119868119873

119896

(119910119873

119896)) 997888rarr 0 (53)

where dist(1198791 1198792) = max

1198891isin11987911198891=1

min1198892isin1198792

1198891

minus 1198892 and

119879119868119873

119896

(119909) = 119889119873

119896isin R119899 | 119889

119873

119896119894ge 0 forall119894 isin 119868

119873

119896



min119911ge0

119911 minus 119909 le 120578

119899

sum

119894=1

max (minus119909119894 0)


(54)


119896and 120598

119873


directions

119863119873

119896= 119889119873119895

119896 119895 = 1 119903

119873

119896 with 10038171003817100381710038171003817

119889119873119895

119896

10038171003817100381710038171003817= 1 (55)




119896 = 119879(119909

119873

119896

120598119873

119896) Here

cone 119863119873

119896

= 1198891198731

1198961205731

+ sdot sdot sdot + 119889119873119903119873

119896

119896120573119903119873

119896 1205731

ge 0 120573119903119873

119896 ge 0

119879 (119909119873

119896 120598119873

119896) = 119889

119873

119896isin R119899 | 119889

119873

119896119894ge 0 119909

119873

119896119894le 120598119873

119896

(56)


119896





0ge 0

119873

0gt 0 120574

119873gt 0 120579

119873

1isin

(0 1) 120579119873

2isin (0 1) 120598

119873

0gt 0

Step 1 Set 119896119873

= 0


119896= 119889119873119895

119896 119895 = 1 119903

119873

119896


Step 3

(a) Set 119895 = 1 119910119873119895

119896= 119909119873

119896 119873119895

119896= 119873

119896


119896such that 119910

119873119895

119894119896+

120572119873119895

119896119889119873119895

119894119896ge 0 for all 119894 Set

119873119895

119896= min120572

119873119895

119896 119873119895

119896

(c) If 119873119895

119896gt 0 and 120579

119873(119910119873119895

119896) le 120579

119873(119910119873119895

119896) minus 120574(

119873119895

119896)2

set 119873119895+1

119896= 120572119873119895


119873119895

119896= 0 119910

119873119895+1

119896=

119910119873119895

119896 119873119895+1

119896= 120579119873

1119873119895

119896

(d) If 120572119873119895

119896= 120572119873119895

119896 set 120598119873

119896+1= 120598119873


(e) If 119895 lt 119903119873


set 120598119873

119896+1= 120579119873

2120598119873


Step 4 Find 119909119873

119896+1ge 0 such that 120579

119873(119909119873

119896+1) le 120579

119873(119910119873119895+1

119896) Set

119873

119896+1= 119873119895+1

119896 119903119896

= 119895 119896 = 119896 + 1 and go to Step 2


119896is the




6 Conclusions


0-function





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












10038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817

(46)


119873 sub R119898

and 120596 takes values 1205961 120596







119894) is uniformly 119875-


119894) of CP(119865(119909 120596

119894)) and there


10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

le 119862119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817 (48)


119873 we have

E2 [dist (119909 SOL (119865 (119909 120596)))]

= E2 [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817]

le E [10038171003817100381710038171003817119909 minus 119909 (120596

119894)10038171003817100381710038171003817

2

]

le

119873

sum

119894=1

119901119894sdot 1198622

119894

10038171003817100381710038171003817min 119909 119865 (119909 120596

119894)

10038171003817100381710038171003817

2

le

119873

sum

119894=1

119901119894sdot 1198622

119894sdot (

radic2 + 2

2)

2

times

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

le 1198622

119873

sum

119894=1

119901119894

119899

sum

119895=1

(radic1198652

119895(119909 120596119894) + 119909

2

119895minus (119865119895(119909 120596119894) + 119909119895))

2

= 1198622120579 (119909)

(49)



(7)


lowast) (50)






min119909isinR119899+

120579119873

(119909) =1

119873sum

120596119894isinΩ119873

10038171003817100381710038171003817Φ (119909 120596

119894)10038171003817100381710038171003817

2

120588 (120596119894) (51)

where Ω119873

= 120596119894


119873sube


119873(119909)





ge 0 the level set

119871 = 119909 ge 0 | 119891 (119909) le 119891 (1199090) (52)

is compact


119896 and 119910

119873


that 119909119873

119896ge 0 119910

119873


119873 and 119868119873

119896sube

119868(119909119873

) = 119894 | 119909119873

119894= 0 for all 119896 then

dist (119879119868119873

119896

(119909119873

119896) 119879119868119873

119896

(119910119873

119896)) 997888rarr 0 (53)

where dist(1198791 1198792) = max

1198891isin11987911198891=1

min1198892isin1198792

1198891

minus 1198892 and

119879119868119873

119896

(119909) = 119889119873

119896isin R119899 | 119889

119873

119896119894ge 0 forall119894 isin 119868

119873

119896



min119911ge0

119911 minus 119909 le 120578

119899

sum

119894=1

max (minus119909119894 0)


(54)


119896and 120598

119873


directions

119863119873

119896= 119889119873119895

119896 119895 = 1 119903

119873

119896 with 10038171003817100381710038171003817

119889119873119895

119896

10038171003817100381710038171003817= 1 (55)




119896 = 119879(119909

119873

119896

120598119873

119896) Here

cone 119863119873

119896

= 1198891198731

1198961205731

+ sdot sdot sdot + 119889119873119903119873

119896

119896120573119903119873

119896 1205731

ge 0 120573119903119873

119896 ge 0

119879 (119909119873

119896 120598119873

119896) = 119889

119873

119896isin R119899 | 119889

119873

119896119894ge 0 119909

119873

119896119894le 120598119873

119896

(56)


119896





0ge 0

119873

0gt 0 120574

119873gt 0 120579

119873

1isin

(0 1) 120579119873

2isin (0 1) 120598

119873

0gt 0

Step 1 Set 119896119873

= 0


119896= 119889119873119895

119896 119895 = 1 119903

119873

119896


Step 3

(a) Set 119895 = 1 119910119873119895

119896= 119909119873

119896 119873119895

119896= 119873

119896


119896such that 119910

119873119895

119894119896+

120572119873119895

119896119889119873119895

119894119896ge 0 for all 119894 Set

119873119895

119896= min120572

119873119895

119896 119873119895

119896

(c) If 119873119895

119896gt 0 and 120579

119873(119910119873119895

119896) le 120579

119873(119910119873119895

119896) minus 120574(

119873119895

119896)2

set 119873119895+1

119896= 120572119873119895


119873119895

119896= 0 119910

119873119895+1

119896=

119910119873119895

119896 119873119895+1

119896= 120579119873

1119873119895

119896

(d) If 120572119873119895

119896= 120572119873119895

119896 set 120598119873

119896+1= 120598119873


(e) If 119895 lt 119903119873


set 120598119873

119896+1= 120579119873

2120598119873


Step 4 Find 119909119873

119896+1ge 0 such that 120579

119873(119909119873

119896+1) le 120579

119873(119910119873119895+1

119896) Set

119873

119896+1= 119873119895+1

119896 119903119896

= 119895 119896 = 119896 + 1 and go to Step 2


119896is the




6 Conclusions


0-function





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119896 = 119879(119909

119873

119896

120598119873

119896) Here

cone 119863119873

119896

= 1198891198731

1198961205731

+ sdot sdot sdot + 119889119873119903119873

119896

119896120573119903119873

119896 1205731

ge 0 120573119903119873

119896 ge 0

119879 (119909119873

119896 120598119873

119896) = 119889

119873

119896isin R119899 | 119889

119873

119896119894ge 0 119909

119873

119896119894le 120598119873

119896

(56)


119896





0ge 0

119873

0gt 0 120574

119873gt 0 120579

119873

1isin

(0 1) 120579119873

2isin (0 1) 120598

119873

0gt 0

Step 1 Set 119896119873

= 0


119896= 119889119873119895

119896 119895 = 1 119903

119873

119896


Step 3

(a) Set 119895 = 1 119910119873119895

119896= 119909119873

119896 119873119895

119896= 119873

119896


119896such that 119910

119873119895

119894119896+

120572119873119895

119896119889119873119895

119894119896ge 0 for all 119894 Set

119873119895

119896= min120572

119873119895

119896 119873119895

119896

(c) If 119873119895

119896gt 0 and 120579

119873(119910119873119895

119896) le 120579

119873(119910119873119895

119896) minus 120574(

119873119895

119896)2

set 119873119895+1

119896= 120572119873119895


119873119895

119896= 0 119910

119873119895+1

119896=

119910119873119895

119896 119873119895+1

119896= 120579119873

1119873119895

119896

(d) If 120572119873119895

119896= 120572119873119895

119896 set 120598119873

119896+1= 120598119873


(e) If 119895 lt 119903119873


set 120598119873

119896+1= 120579119873

2120598119873


Step 4 Find 119909119873

119896+1ge 0 such that 120579

119873(119909119873

119896+1) le 120579

119873(119910119873119895+1

119896) Set

119873

119896+1= 119873119895+1

119896 119903119896

= 119895 119896 = 119896 + 1 and go to Step 2


119896is the




6 Conclusions


0-function





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







