R E S E A R C H
Open Access
An algorithm for solving a multi-valued
variational inequality
Changjie Fang
*, Shenglan Chen and Chunde Yang
*Correspondence:
Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China
Abstract
We propose a new extragradient method for solving a multi-valued variational inequality. It is showed that the method converges globally to a solution of the multi-valued variational inequality, provided the multi-valued mapping is continuous with nonempty compact convex values. Preliminary computational experience is also reported.
MSC: 47H04; 47H10; 47J20; 47J25
Keywords: variational inequality; multi-valued mapping; extragradient method
1 Introduction
We consider the following multi-valued variational inequality, denoted byMVI(F,C): to findx*∈Candξ∈F(x*) such that
ξ,y–x*≥, ∀y∈C, (.)
whereCis a nonempty closed convex set inRn,Fis a multi-valued mapping fromCinto
Rnwith nonempty values, and·,·and · denote the inner product and the norm inRn,
respectively.
Extragradient-type algorithms have been extensively studied in the literature; see [– ]. Various algorithms for solving the multi-valued variational inequality have been ex-tensively studied in the literature [–]. The well-known proximal point algorithm [] requires the multi-valued mappingFto be monotone. [] proposes a projection algo-rithm for solving the multi-valued variational inequality with a pseudomonotone map-ping. In [], choosingui∈F(xi) needs solving a single-valued variational inequality; see
the expression (.) in []. [] presents a double projection algorithm, which is an im-provement of [], so thatui∈F(xi) can be taken arbitrarily. In [], however, choosing the
hyperplane needs computing the supremum and hence is computationally expensive. To overcome this difficulty, [] introduces an extragradient algorithm for solving the multi-valued variational inequality in which computing the supremum is avoided. In this paper, we present a new extragradient method for solving the multi-valued variational inequal-ity. In our method,ui∈F(xi) can be taken arbitrarily. Moreover, the main difference of
our method from those of [, , ] is the procedure of Armijo-type linesearch. We also present numerical tests to compare our Algorithm . with those in [, ].
This paper is organized as follows. In Section , we present the algorithm details. We prove the preliminary results for convergence analysis in Section . Numerical results are reported in the last section.
2 Algorithms
Let us recall the definition of a continuous multi-valued mapping.F is said to be upper semicontinuous atx∈Cif for every open setVcontainingF(x), there is an open setU con-tainingxsuch thatF(y)⊂Vfor ally∈C∩U. F is said to be lower semicontinuous atx∈C
if given any sequencexkconverging toxand anyy∈F(x), there exists a sequenceyk∈F(xk)
that converges toy.Fis said to be continuous atx∈Cif it is both upper semicontinuous and lower semicontinuous atx. IfFis single-valued, then both upper semicontinuity and lower semicontinuity reduce to the continuity ofF.
Fis called pseudomonotone onCin the sense of Karamardian [] if for anyx,y∈C,
v,x–y ≥ for somev∈F(y)⇒ u,x–y ≥ for allu∈F(x). (.)
LetSbe the solution set of (.), that is, those pointsx*∈Csatisfying (.). Throughout this paper, we assume that the solution setSof problem (.) is nonempty andFis contin-uous onCwith nonempty compact convex values satisfying the following property:
ζ,y–x ≥, ∀y∈C,∀ζ∈F(y),∀x∈S. (.)
The property (.) holds ifFis pseudomonotone onC.
LetPCdenote the projector ontoC, and letμ> be a parameter.
Proposition . x∈C andξ∈F(x)solve problem(.)if and only if
rμ(x,ξ) :=x–PC(x–μξ) = .
Algorithm . Choosex∈Cand two parametersγ,σ∈(, ). Seti= .
Step . Chooseui∈F(xi) and letkibe the smallest nonnegative integer satisfying
vi∈F
PC
xi–γkiui
, (.)
γkiui–vi,rγki(xi,ui)
≤σrγki(xi,ui)
. (.)
Setηi=γki. Ifrηi(xi,ui) = , stop.
Step . Computexi+:=PC(xi–αidi), where
di=rηi(xi,ui) +ηivi, (.)
αi=
( –σ)rηi(xi,ui)
di
. (.)
Remark . Let us compare the above algorithm with those in [, , ]. First, Aimijo-type linesearch procedures in the four algorithms are different. [, , ] use different proce-dures which replace (.) by the following ones:
vi,rμ(xi,ui)
≥σrμ(xi,ui) or
ui–vi,rμ(xi,ui)
≤σrμ(xi,ui),
whereμis required to be strictly less than or /σ, andvi∈F(xi–γkirμ(xi,ui)). In our
algo-rithm,μcan change according to the value ofηiin each iteration andvi∈F(PC(xi–γkiui)).
Secondly, the way to generate the next iterate is different. In [, ], the next iterate is a pro-jection of the current iterate onto the intersection of the feasible setCand a hyperplane, while in our algorithm as well as in [] the next iterate is a projection onto the feasible setC. In addition, the searching directions in [] and our algorithm are also different.
Lemma . Let C be a closed convex subset ofRn.For any x,y∈Rnand z∈C,the following
statements hold.
(i) x–PC(x),z–PC(x) ≤.
(ii) PC(x) –PC(y)≤ x–y–PC(x) –x+y–PC(y).
Proof See [].
The proof of the following lemma is easy and we omit it (see Lemma . in [] for example).
Lemma . For any x∈Rn,ξ∈F(x)andμ> ,
min{,μ}r(x,ξ)≤rμ(x,ξ)≤max{,μ}r(x,ξ).
We first show that Algorithm . is well defined.
Proposition . If xi is not a solution of problem(.),then there exists a nonnegative
integer kisatisfying(.)and(.).
Proof Suppose that for allkand allv∈F(PC(xi–γkui)), we have
γkui–v,rγk(xi,ui)
>σrγk(xi,ui)
,
and hence,
γkui–v>σrγk(xi,ui).
Therefore,
ui–v>
σ
γkrγk(xi,ui)
≥ σ
γkmin
,γkr(xi,ui)
where the second inequality follows from Lemma . and the equality follows fromγ ∈
(, ) andk≥. SincePC(·) is continuous andxi∈C,PC(xi–γkui)→xi(k→ ∞). SinceF
is lower semicontinuous,ui∈F(xi) andPC(xi–γkui)→xi(k→ ∞), there isvk∈F(PC(xi–
γkui)) such thatvk→ui(k→ ∞). Therefore,
ui–vk>σr(xi,ui), ∀k. (.)
Letk→ ∞in (.), we have
=ui–ui ≥σr(xi,ui)> .
This contradiction completes the proof.
3 Main results
Now we obtain the following auxiliary result that will be used for proving the convergence of Algorithm ..
Theorem . If the assumption(.)holds and xi∈/S,then for any x*∈S,
di,xi–x*
≥( –σ)rηi(xi,ξi)
> . (.)
Proof Letx*∈S. Sinceu
i∈F(xi) andηi> , it follows from (.) that
ηiui,xi–x*
≥. (.)
Similarly, we have
ηivi,PC(xi–ηiui) –x*
≥, (.)
becausevi∈F(PC(xi–ηiui)). Sincex*∈C, from Lemma .(i) we have
xi–ηiui–PC(xi–ηiui),PC(xi–ηiui) –x*
≥. (.)
It follows from (.), (.) and (.) that
di,xi–x*
=rηi(xi,ui) +ηivi,xi–x
*
=rηi(xi,ui) +ηi(vi–ui),xi–x
*+η
iui,xi–x*
≥rηi(xi,ui) +ηi(vi–ui),xi–x
*
=rηi(xi,ui) +ηi(vi–ui),rηi(xi,ui)
+xi–ηiui–PC(xi–ηiui),PC(xi–ηiui) –x*
+ηivi,PC(xi–ηiui) –x*
≥rηi(xi,ui) +ηi(vi–ui),rηi(xi,ui)
Therefore,
di,xi–x*
≥rηi(xi,ui) +ηi(vi–ui),rηi(xi,ui)
=rηi(xi,ui)
–ηi
ui–vi,rηi(xi,ui)
≥rηi(xi,ui)
–σrηi(xi,ui)
= ( –σ)rηi(xi,ui)
, (.)
where the second inequality follows from (.). This completes the proof.
Theorem . If F:C→Rnis continuous with nonempty compact convex values on C and
the assumption(.)holds,then the sequence{xi}generated by Algorithm.converges to
a solution x of(.).
Proof Letx*∈S. It follows from Lemma .(ii), Lemma ., (.), (.) and (.) that
xi+–x*≤xi–x*–αidi
=xi–x*
– αi
di,xi–x*
+αidi
≤xi–x*–
( –σ)rηi(xi,ui)
di
≤xi–x*
–( –σ) (min{η
i, }r(xi,ui))
di
=xi–x*
–( –σ) η
ir(xi,ui)
rηi(xi,ui) +ηivi
. (.)
It follows that the sequence{xi+–x*}is nonincreasing, and hence is a convergent se-quence. Therefore,{xi}is bounded. SinceFis continuous with compact values,
Proposi-tion . in [] implies that{F(xi) :i∈N}is a bounded set, and so are{ui},{rηi(xi,ui)}and
{vi}. Thus,{rηi(xi,ui) +ηivi}is bounded. Then there exists a positive numberMsuch that rηi(xi,ui) +ηivi≤M.
It follows from (.) that
xi+–x*≤xi–x*– ( –σ)M–ηir(xi,ui). (.)
Therefore,
lim i→∞ηi
r(xi,ui)= . (.)
By the boundedness of{xi}, there exists a convergent subsequence{xij}converging tox.
Ifxis a solution of problem (.), we show next that the whole sequence{xi}converges
to x. Replacing x*by xin the preceding argument, we obtain that the sequence{x
subsequence of{xi–x}converges to zero. This shows that the whole sequence{xi–x}
converges to zero, hencelimi→∞xi=x.
Suppose now thatxis not a solution of problem (.). We show first thatki in
Algo-rithm . cannot tend to∞. SinceFis continuous with compact values, Proposition . in [] implies that{F(xi) :i∈N}is a bounded set, and so the sequence{ui}is bounded.
Therefore, there exists a subsequence{uij}converging tou. SinceFis upper
semicontin-uous with compact values, Proposition . in [] implies thatFis closed, and sou∈F(x). By the definition ofki, we have
γki–u
i–v,rγki–(xi,ui)
>σrγki–(xi,ui), ∀v∈F
PC
xi–γki–ui
, (.)
and hence
γki–ui–v>σrγki–(xi,ui), ∀v∈F
PC
xi–γki–ui
. (.)
Therefore,
ui–v>
σ γki–
rγki–(xi,ui)
≥ σ
γki–min
,γki–r(xi,ui)
=σr(xi,ui), ∀v∈F
PC
xi–γki–ui
,∀ki≥, (.)
where the second inequality follows from Lemma . and the equality follows fromγ ∈
(, ).
Ifkij→ ∞, thenPC(xij–γ kij–
uij)→x. The lower continuity ofF, in turn, implies the
existence ofuij∈F(PC(xij–γkij–uij)) such thatuij converges tou. Therefore,
uij–uij>σr(xij,uij). (.)
Lettingj→ ∞, we obtain the contradiction
≥σr(x,u)> , (.)
beingr(·,·) continuous. Therefore,{ki}is bounded and so is{ηi}.
By the boundedness of{ηi}, it follows from (.) thatlimi→∞r(xi,ui)= . Sincer(·,·)
is continuous and the sequences{xi}and{ui}are bounded, there exists an accumulation
point (x,u) of{(xi,ui)}such thatr(x,u) = . This implies thatxsolves the variational
in-equality (.). Similar to the preceding proof, we obtain thatlimi→∞xi=x.
For anyδ> , define
P(δ) :=(x,ξ)∈C×Rn:ξ∈F(x),r(x,ξ)≤δ.
We say thatFis Lipschitz continuous onCif there exists a constantL> such that, for allx,y∈C,H(F(x),F(y))≤Lx–y, whereHdenotes the Hausdorff metric.
Theorem . In addition to the assumptions in Theorem.,if F is Lipschitz continuous
with modulus L> and if there exist positive constants c andδsuch that
dist(x,S)≤cr(x,ξ), ∀(x,ξ)∈P(δ), (.)
then there is a constantα> such that for sufficiently large i,
dist(xi,S)≤
αi+dist–(x,S).
Proof Putη:=min{/,L–γ σ}. We first prove thatη
i>ηfor alli. By the construction of
ηi, we haveηi∈(, ]. Ifηi= , then clearlyηi> ≥η. Now we assume thatηi< . Since
ηi=γki, it follows that the nonnegative integerki≥. Thus the construction ofkiimplies
that
γki–u
i–v,rγki–(xi,ui)
>σrγki–(xi,ui), ∀v∈F
PC
xi–γki–ui
, (.)
and hence, aski≥,
ui–v>
σ
γki–rγki–(xi,ui), ∀v∈F
PC
xi–γki–ui
.
Sinceui∈F(xi) andF is compact-valued, the definition of the Hausdorff metric implies
the existence ofvi∈F(PC(xi–γki–ui)) such that
σ
γki–rγki–(xi,ui)<ui–vi ≤H
F(xi),F
PC
xi–γki–ui
≤Lrγki–(xi,ui).
Thereforeηi>L–γ σ≥η.
Letx*∈P
S(xi). By (.) and (.), we obtain that for sufficiently largei,
dist(xi+,S)≤xi+–x*
≤xi–x*
– ( –σ)M–ηir(xi,ui)
≤xi–x*– ( –σ)M–ηr(xi,ui)
≤dist(xi,S) – ( –σ)M–ηc–dist(xi,S),
where the second inequality follows fromηi>η.
Writeαfor ( –σ)M–ηc–. Applying Lemma in Chapter of [], we have
dist(xi,S)≤dist(x,S)/
αidist(x,S) + = /
αi+dist–(x,S).
Table 1 Example 4.1
ε Algorithm 2.2 [6, Algorithm 1] It. (Num.) CPU (Sec.) It. (Num.) CPU (Sec.) 10–3 38 0.640625 67 0.546875 10–5 66 0.8125 120 0.828125 10–7 96 1.10938 173 1.15625
Table 2 Example 4.1
ε Algorithm 2.2 [11, Algorithm 1] It. (Num.) CPU (Sec.) It. (Num.) CPU (Sec.) 10–3 38 0.640625 71 0.96875
10–5 66 0.8125 126 1.53125
10–7 96 1.10938 181 2.14063
4 Numerical experiments
In this section, we present some numerical experiments for the proposed algorithm. The MATLAB codes are run on a PC (with CPU Intel P-T) under MATLAB Version ...(R) Service Pack . We compare the performance of our Algorithm ., [, Algorithm ] and [, Algorithm ]. In Tables and , ‘It.’ denotes the number of iteration and ‘CPU’ denotes the CPU time in seconds. The toleranceεmeans whenrμ(x,ξ) ≤ε, the procedure stops.
Example . Let n= ,
C:=
x∈Rn+:
n
i=
xi=
and F:C→Rnbe defined by
F(x) :=(t,t–x,t–x) :t∈[, ]
.
Then the set C and the mapping F satisfy the assumptions of Theorem.and(, , )is
a solution of the multi-valued variational inequality.Example.is tested in[, ].We
chooseσ = .,γ = .for our algorithm;σ= .,γ = .,μ= for Algorithmin[];
σ= .,γ = .,μ= for Algorithmin[].We use x= (, ., .)as the initial point
(Tableand Table).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.
Acknowledgements
This work is partially supported by Natural Science Foundation Project of CQ CSTC (No. 2010BB9401), Science and Technology Project of Chongqing Municipal Education Committee of China (Nos. KJ110509 and KJ100513) and Foundation of Chongqing University of Posts and Telecommunications for the Scholars with Doctorate (No. A2012-04).
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