research article properties of expected residual
TRANSCRIPT
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)
4 Journal of Applied Mathematics
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
lim inf119896119894rarrinfin
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
Journal of Applied Mathematics 5
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)
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Journal of Applied Mathematics
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
lim inf119896119894rarrinfin
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
Journal of Applied Mathematics 5
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)
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Journal of Applied Mathematics
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
lim inf119896119894rarrinfin
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
Journal of Applied Mathematics 5
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)
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
[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
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
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
lim inf119896119894rarrinfin
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
Journal of Applied Mathematics 5
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)
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
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)
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of