R E S E A R C H
Open Access
The subgradient double projection method
for variational inequalities in a Hilbert space
Lian Zheng
**Correspondence: [email protected] Department of Mathematics and Computer Science, Yangtze Normal University, Fuling, Chongqing 408100, China
Abstract
We present a modification of the double projection algorithm proposed by Solodov and Svaiter for solving variational inequalities (VI) in a Hilbert space. The main modification is to use the subgradient of a convex function to obtain a hyperplane, and the second projection onto the intersection of the feasible set and a halfspace is replaced by projection onto the intersection of two halfspaces. In addition, we propose a modified version of our algorithm that is to find a solution of VI which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for our algorithms.
MSC: 90C25; 90C30
Keywords: variational inequalities; nonexpansive mapping; subgradient; double projection algorithm; weak convergence
1 Introduction
LetHbe a real Hilbert space,C⊂Hbe a nonempty, closed and convex set, and letf :C→ Hbe a mapping. The inner product and norm are denoted by·,·and · , respectively. We writexkxto indicate that the sequence{xk}converges weakly toxandxk→xto
indicate that the sequence{xk}converges strongly tox. For each pointx∈H, there exists
a unique nearest point inC, which is called the projection ofxontoCand denoted by PC(x). That is
PC(x) =arg min
y–x|y∈C.
It is well known that the projection operator is nonexpansive (i.e., Lipschitz continuous with a Lipschitz constant ), and hence is also a bounded mapping.
We consider the following variational inequality problem denoted byVI(C,f): find a vectorx∗∈Csuch that
fx∗,y–x∗≥ for ally∈C. (.)
The variational inequality problem was first introduced by Hartman and Stampacchia [] in . In recent years, many iterative projection-type algorithms have been proposed and analyzed for solving the variational inequality problem; see [] and the references therein. To implement these algorithms, one has to compute the projection onto the fea-sible setC, or onto some related set.
In , Solodov and Svaiter [] proposed an algorithm for solving theVI(C,f) in an Euclidean space, known as the double projection algorithm due to the fact that one needs to implement two projections in each iteration. One is onto the feasible setC, and the other is onto the intersection of the feasible setCand the halfspace. More precisely, they presented the following algorithm.
Algorithm . Choose an initial pointx, parametersμ> ,σ,γ ∈(, ) and setk= .
Step . Havingxk, compute
yk=PC
xk–μfxk .
Stop ifxk=yk; otherwise, go to Step .
Step . Computezk=xk–ηk(xk–yk), whereηk=γmk withmkbeing the smallest
non-negative integermsuch that
fxk–γmxk–yk,xk–yk≥σxk–yk. (.) Step . Compute
xk+=PC∩Hk
xk, where
Hk=
x∈Rn|fzk,x–zk≤. (.) Letk:=k+ and return to Step .
Although [] shows that Algorithm . can get a longer stepsize, and hence is a better algorithm than the extragradient method proposed by Korpelevich [] in theory, there is still the need to calculate two projections onto the feasible setCand onto a related set C∩Hk at each iteration. If the set Cis simple enough (e.g., Cis a halfspace or a ball)
so that projections onto it and the related set are easily executed, then Algorithm . is particularly useful. But ifCis a general closed and convex set, one has to solve the two problemsminx∈Cx– (xk–μf(xk))andmin
x∈C∩Hkx–x
kat each iteration to get the
next iteratexk+. This might seriously affect the efficiency of Algorithm ..
Recently, Censoret al.[, ] presented a subgradient extragradient projection method for solvingVI(C,f). Inspired by the above works, in this paper we present a modification of Algorithm . in a Hilbert space. Our algorithm replaces an iteratek PC∩Hk byPCk∩Hk, whereCk is a halfspace constructed by the subgradient and contains the feasible setC,
andCk∩Hk is the intersection of two halfspaces. Observe that the projection onto the
intersection of two halfspaces is easily computable. In addition, we propose a modified version of our algorithm that is to find a solution of VI which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for our algorithms.
2 Preliminaries
In this section, we recall some useful definitions and results which will be used in this paper.
Lemma . Let⊂H be a closed and convex set.Then for any x∈H and z∈,
() P(x) –z
≤ x–z–P(x) –x
;
() P(x) –x,z–P(x)
≥.
The next property is known as the Opial condition []. Any Hilbert space has this prop-erty.
Condition .(Opial) For any sequence{xk}inHthat converges weakly tox(xkx),
lim inf k→∞
xk–x<lim inf
k→∞
xk–y, y=x.
Lemma . Let H be a real Hilbert space and D be a nonempty,closed and convex subset of H.Let the sequence{xk} ⊂H be Fejér-monotone with respect to D,i.e.,for every u∈D,
xk+–u≤xk–u, ∀k≥.
Then{PD(xk)}converges strongly to some z∈D.
Proof See [, Lemma .].
Lemma . Let H be a real Hilbert space,{αk}be a real sequence satisfying <a≤αk≤
b< for all k≥,and let{νk}and{ωk}be two sequences in H such that for someσ≥,
lim sup k→∞
νk≤σ,
lim sup k→∞
ωk≤σ
and
lim k→∞
αkνk+ ( –αk)ωk=σ.
Then
lim k→∞ν
k–ωk= .
Proof See [, Lemma .].
The next fact is known as the demiclosedness principle [].
Lemma . Let H be a real Hilbert space,D be a closed and convex subset of H and S:D→ H be a nonexpansive mapping.Then I–S(I is the identity operator on H)is demiclosed at y∈H,i.e.,for any sequence{xk}in D such that xkx¯∈D and(I–S)xk→y,we have
Lemma . Let H be a real Hilbert space,h be a real-valued function on H and K be the set{x∈H:h(x)≤}.If K is nonempty and h is Lipschitz continuous with modulusθ> , then
dist(x,K)≥θ–h(x), ∀x∈H, (.)
wheredist(x,K)denotes the distance from x to K.
Proof It is clear that (.) holds for allx∈K. Hence, it suffices to show that (.) holds for allx∈H\K. Letx∈Hbutx∈/K. SinceKis closed, there existsy∈Ksuch thatx–y=
dist(x,K). So, for an arbitrary positive numberε, there existsz∈Ksuch that
x–z ≤dist(x,k) +ε.
Sincehis Lipschitz continuous with modulusθ onH, we have
h(x) –h(z)≤θx–z ≤θdist(x,k) +εθ.
It follows fromz∈Kthath(z)≤. Thus we have
h(x)≤h(x) –h(z)≤h(x) –h(z)≤θdist(x,K) +θ ε.
Letε→+, we obtain the desired result.
Remark . The idea of Lemma . is from Lemma . of []. In Lemma ., if we take K:=K∩C, whereCis a closed subset ofHandK∩C=∅, then (.) still holds. In fact, for eachx∈H, sinceCandC∩Kare closed, we haveminy∈K∩Cx–yandminy∈Kx–y
exist, and
min
y∈K∩Cx–y ≥miny∈Kx–y,
that is,dist(x,C∩K)≥dist(x,K).
In this paper, we assume that the convex setCsatisfies the following assumptions: () The setCis given by
C=x∈H|c(x)≤, (.)
wherec:H→Ris a convex (not necessarily differentiable) function andCis nonempty. Note that the differentiability ofc(x) is not assumed, therefore the setCis quite general. For example, any system of inequalitiescj(x)≤,j∈J, wherecj(x) is convex andJis an
arbitrary index set, is the same as the single inequalityc(x)≤ withc(x) =sup{cj(x)|j∈J}.
In fact, every closed convex set can be represented in this way,e.g., takec(x) =dist(x,C), where dist is the distance function; see [, Section .(c)].
() For anyx∈H, at least one subgradientξ∈∂c(x) can be calculated, where∂c(x) is the subdifferential ofc(x) atxand is defined as follows:
Denote
Ck=
x∈H|cxk+ξk,x–xk≤, (.) whereξk∈∂c(xk).
Proposition . For every nonnegative integer k,let xk∈H and C
kbe defined as in(.).
Then for any x∈H,we have
PCk(x) =
⎧ ⎨ ⎩
x–c(xk)+ξξkk,x–x kξk if c(xk) +ξk,x–xk> ,
x otherwise. (.)
Proof See [].
Remark . () From the definition of subdifferential, we haveC⊆Ck for allk. In fact,
for anyx∈Candξk∈∂c(xk), we have
cxk+ξk,x–xk≤c(x)≤, i.e.,x∈Ckand henceC⊆Ck.
() From Proposition ., we can observe thatPCk can be explicitly represented with-out resorting to projection operator, thus its computation is easy. Recently, Ckis often
regarded as the projection region in the algorithm of the split feasibility problem, see [– ].
3 The subgradient double projection algorithm To this end, the following assumptions are needed.
Assumption
(A) The solution set of problem (.), denoted bySOL(C,f), is nonempty.
(A) For allx∈H, lety=PC[x–μf(x)]and[x,y]be a closed line segment joiningxand
y,f satisfies
f(z),z–x∗≥, ∀z∈[x,y],x∗∈SOL(C,f).
(A) The mappingf is continuous and bounded on a bounded set ofH.
Remark . () Iff is Lipschitz continuous, thenf is bounded on a bounded set of H, but the continuity and boundedness cannot ensure the Lipschitz continuity. For example, let H=R= (–∞, +∞) andf(x) =x,x∈H. It is easy to see thatf is continuous and bounded
on a bounded set ofH, but we can see thatf is not Lipschitz continuous. So, our assump-tion (A) is weaker than Lipschitz continuous. Recently, the literature [] proposed two subgradient extragradient algorithms forVI(C,f) in a Hilbert space and established weak convergence theorems for them under the assumptions of Lipschitz continuity and mono-tonicity off.
In this paper, we establish weak convergence theorem of subgradient double projection methods forVI(C,f) in a Hilbert space under assumptions (A)-(A).
Algorithm . Selectx∈C,α> ,β≥,σ> ,μ∈(,σ–),γ ∈(, ). Setk= .
Step . Forxk∈C, define
yk=PC
xk–μfxk.
Ifxk=yk, stop; else go to Step .
Step . Computezk= ( –ηk)xk+ηkyk, whereηk=γmkwithmkbeing the smallest
non-negative integermsatisfying
fxk–fxk–γmxk–yk,xk–yk≤σxk–yk. (.)
Step . Computexk+=P Ck∩Hk(x
k), where
Ck=
x∈H|cxk+ξk,x–xk≤,
Hk={v∈H:hk(v)≤}is a halfspace defined by the function
hk(v) =
αηk
xk–yk+βfxk+αμfzk,v–xk+αηk( –μσ)xk–yk
, (.)
andξk∈∂c(xk).
Letk=k+ and return to Step .
Remark . () SinceCkandHkare halfspaces, and by Proposition ., the projection onto
Ck∩Hkcan be directly calculated, so our Algorithm . can be more easily implemented
than Algorithm ..
() Ifyk=xkfor some positive integerk, thenxkis a solution of problem (.). In fact,
supposeyk=xk. Sinceyk=P
C(xk–μf(xk)), it follows from Lemma .() that
xk–μfxk–xk,x–xk≤, ∀x∈C,
this means thatxkis a solution of problem (.).
4 Convergence of the subgradient double projection algorithm
Now, we turn to consider the convergence of Algorithm .. Certainly, if an algorithm terminates within finite steps,e.g., stepk, thenxkis a solution of problem (.). So, in the
following analysis, we assume that our algorithm always generates an infinite sequence.
Lemma . Let x∗∈SOL(f,C)and the function hkbe defined by(.).Then
hk
xk≥αηk( –μσ)xk–yk
and hk
x∗≤. (.)
In particular,if xk=yk,then h
Proof
hk
xk=αηk
xk–yk+βfxk+αμfzk,xk–xk+αηk( –μσ)xk–yk
=αηk( –μσ)xk–yk
.
Ifxk=yk, thenh
k(xk) > becauseμσ< . In the following, we prove thathk(x∗)≤. Since
yk=PC
xk–μfxk, by () of Lemma ., we have
xk–yk–μfxk,yk–x∗≥. (.) By assumption (A),
μfxk,xk–x∗≥. (.)
Adding inequalities (.) and (.), we obtain
xk–yk–μfxk,yk–xk+xk–yk,xk–x∗≥. (.) We have
xk–x∗,αηk
xk–yk+βfxk+αμfzk =αηk
xk–x∗,xk–yk+βfxk,xk–x∗+αμfzk,xk–x∗
≥αηk
xk–yk–μfxk,xk–yk+αμηk
fzk,xk–yk =αηkxk–yk–αμηk
fxk–fzk,xk–yk
≥αηkxk–yk
–αμσ ηkxk–yk
=ηkα( –μσ)xk–yk
,
where the first inequality follows from (A) and (.) and the last one follows from (.). It follows that
hk
x∗=αηk
xk–yk+βfxk+αμfzk,x∗–xk+ηkα( –μσ)xk–yk
= –αηk
xk–yk+βfxk+αμfzk,xk–x∗+ηkα( –μσ)xk–yk
≤–ηkα( –μσ)xk–yk
+ηkα( –μσ)xk–yk
= .
The proof is completed.
Lemma . Suppose assumptions(A)-(A)hold and the sequences{xk}and{yk}are
gen-eralized by Algorithm.,then there exists a positive number M such that
xk+–x∗≤xk–x∗–M–( –μσ)αηkyk–xk, ∀k, (.)
Proof From Lemma . and Remark .(), we getSOL(C,f)⊆Hk∩Ck, and hencex∗∈
Hk∩Ck. Sincexk+=PCk∩Hk(xk), it follows from Lemma .() that
xk+–x∗≤xk–x∗–xk+–xk
=xk–x∗–distxk,Ck∩Hk
. (.)
It follows that the sequence{xk–x∗}is nonincreasing, and hence is a convergent se-quence. Thus,{xk}is bounded and
lim k→∞dist
xk,Ck∩Hk
= . (.)
Sincef and the projection operatorPC are continuous and bounded, we obtain that the
sequence {yk} and hence the sequences{f(xk)}and{f(zk)} are bounded, and for some
M> ,
αηk
xk–yk+βfxk+αμfzk≤M, ∀k. (.) Clearly, each function hk is Lipschitz continuous on H with modulus M. Applying
Lemma . and Remark ., we obtain that
distxk,Ck∩Hk
≥M–hk
xk≥M–αηk( –μσ)xk–yk,
which, together with (.), yields that
xk+–x∗≤xk–x∗–M–( –μσ)αηkyk–xk, ∀k.
Theorem . Suppose assumptions(A)-(A)hold,then any sequence{xk}generalized by
Algorithm.weakly converges to some solution u∗∈SOL(C,f).
Proof By Lemma ., the sequence{xk}is bounded and
lim k→∞ηkx
k–yk
= . (.)
Iflim supk→∞ηk> , then we havelim infk→∞xk–yk= . Therefore there exists a weak
accumulation pointx¯of{xk}, and two subsequences{xkj}and{ykj}of{xk}and{yk}, respec-tively, such that
lim j→∞x
kj–ykj= (.)
and
xkjx,¯ asj→ ∞. (.)
Sincexkj– (xkj–ykj) =P
C(xkj–μf(xkj)), it follows from Lemma .() that
Lettingj→ ∞, taking into account (.), (.) and the continuity off, we obtain
f(x),¯ x–x¯≥, x∈C,
i.e.,x¯is a solution of problem (.). In order to show that the entire sequence{xk}weakly
converges to x, assume that there exists another subsequence¯ {xk¯j}of {xk} that weakly converges to somex¯=x¯andx¯∈SOL(C,f). It follows from Lemma . that the sequences
{xk–x¯}and{xk–x¯}are decreasing. By the Opial condition, we have
lim k→∞x
k–x¯=lim inf
j→∞
xkj–x¯<lim inf
j→∞
xkj–x¯
= lim k→∞
xk–x¯=lim inf
j→∞ x ¯ kj–x¯
<lim inf
j→∞ x ¯
kj–x¯= lim
k→∞
xk–x¯,
which is a contradiction. Thusx¯=x. This implies that the sequence¯ {xk}converges weakly
tox¯∈SOL(C,f).
Suppose now thatlimj→∞ηkj= . By the choice ofηk, (.) implies that
σxkj–ykj<fxkj–fxkj–γmkj–xkj–ykj,xkj–ykj =fxkj–fxkj–γ–η
kj
xkj–ykj,xkj–ykj.
Since{xk}and{yk}are bounded andf is continuous, we obtain by lettingj→ ∞ that limj→∞xkj–ykj= . Applying the similar proof to the previous case, we get the desired
result.
Remark . If the mappingf is pseudomonotone onC,i.e.,
f(y),x–y≥ ⇒ f(x),x–y≥, ∀x,y∈C,
thenSOL(f,C) is a closed and convex set. In fact, iff is pseudomonotone onC, then for anyx∗∈SOL(C,f), we have
f(x),x–x∗≥, ∀x∈C.
Hence, it suffices to show that the solution setSOL(C,f) can be characterized as the in-tersection of a family of halfspaces. That is,
SOL(C,f) =
x∈C
y∈C|f(x),x–y≥
since
SOL(C,f) =
x∈C
From the pseudomonotonicity off, we obtain
x∈C
y∈C|f(y),x–y≥⊆
x∈C
y∈C|f(x),y–x≥.
So, we need only to prove
x∈C
y∈C|f(y),x–y≥⊇
x∈C
y∈C|f(x),x–y≥. (.)
We suppose that the conclusion (.) does not hold, then there existyandxinCsuch
that
f(x),x–y
≥ for allx∈C, (.)
and
f(y),x–y
< . (.)
In (.), takingx= ( –t)y+tx,t∈(, ), we obtain
f( –t)y+tx
,x–y
≥.
Lettingt→+, it follows from the continuity off that
f(y),x–y
≥,
which contradicts (.). We obtain the desired conclusion. Therefore the solution set
SOL(C,f) is closed and convex.
In Theorem ., ifSOL(f,C) is a closed set, then we can furthermore obtain
u∗= lim
k→∞PSOL(C,f)
xk.
In fact, we take
uk=PSOL(C,f)
xk.
By Lemma .() and notingx¯∈SOL(C,f), we have
¯
x–uk,uk–xk≥.
By Lemma .,{uk}converges strongly to someu∗∈SOL(C,f). Therefore
¯
x–u∗,u∗–x¯≥,
5 The modified subgradient double projection algorithm
In this section, we present a modified subgradient double projection algorithm which finds a solution of theVI(C,f) which is also a fixed point of a given nonexpansive mapping. Let S:H→Hbe a nonexpansive mapping and denote byFix(S) its fixed point set,i.e.,
Fix(S) =x∈H|S(x) =x.
Let{αk}∞k=⊂[c,d] for somec,d∈(, ).
Algorithm . Selectx∈C,α> ,β≥,σ> ,μ∈(,σ–),γ ∈(, ). Setk= .
Step . Forxk∈C, define
yk=PC
xk–μfxk.
Ifxk=yk, stop; else go to Step .
Step . Computezk= ( –ηk)xk+ηkyk, whereηk=γmkwithm
kbeing the smallest
non-negative integermsatisfying
fxk–fxk–γmxk–yk,xk–yk≤σxk–yk. (.)
Step . Compute
xk+=αkxk+ ( –αk)SPCk∩Hk
xk,
where
Ck=
x∈H|cxk+ξk,x–xk≤,
Hk={v∈H:hk(v)≤}being a halfspace defined by the function
hk(v) =
αηk
xk–yk+βfxk+αμfzk,v–xk+αηk( –μσ)xk–yk
, (.)
andξk∈∂c(xk). Letk=k+ and return to Step .
6 Convergence of the modified subgradient double projection algorithm In this section, we establish a weak convergence theorem for Algorithm .. We assume that the following condition holds:
Fix(S)∩SOL(C,f)=∅.
We also recall that in a Hilbert spaceH,
λx+ ( –λ)y=λx+ ( –λ)y–λ( –λ)x–y (.)
Theorem . Suppose that assumptions(A)-(A)hold,then any sequence{xk}
general-ized by Algorithm.weakly converges to some solution u∗∈Fix(S)∩SOL(C,f).
Proof Denotetk=P Ck∩Hk(x
k) for allk. Letu∈Fix(S)∩SOL(C,f). By the definition ofxk+,
we obtain
xk+–u=αkxk+ ( –αk)S
tk–u =αk
xk–u+ ( –αk)
Stk–u (.)
=αkxk–u
+ ( –αk)S
tk–u–αk( –αk)xk–S
tk
≤αkxk–u
+ ( –αk)S
tk–S(u)
≤αkxk–u
+ ( –αk)tk–u
≤αkxk–u
+ ( –αk)xk–u
–tk–xk
=xk–u– ( –αk)tk–xk (.) =xk–u– ( –αk)dist
xk,Ck∩Hk
, (.)
where the third equality follows from (.), the second inequality follows from the nonex-pansion ofSand the third inequality follows from Lemma .(). In (.), using the proof similar to those of Lemma . and Theorem ., we obtain that there exists subsequences
{xkj}and{ykj}of{xk}and{yk}, respectively, such that
lim j→∞
xkj–ykj= . (.)
By (.), we obtain that{xk–u}is a convergent sequence,i.e., there exists someσ> such that
lim k→∞x
k–u=σ (.)
and
lim k→∞x
k–tk= (.)
and{xk}and{tk}are bounded. Hence we may assume, without loss of generality, that{xkj} weakly converges to somex¯∈H. We now show that
¯
x∈Fix(S)∩SOL(C,f).
By the triangle inequality,
tk–yk≤tk–xk+xk–yk, (.)
so by (.) and (.), we obtain
lim j→∞t
Sincetkj– (tkj–ykj) =P
C[xkj–μf(xkj)], it follows from Lemma .() that
x–tkj+tkj–ykj,xkj–μfxkj–tkj+tkj–ykj≤ for allx∈C.
Passing to the limitj→ ∞in the above inequality, taking into account (.), (.),C⊂Ckj and the continuity off, we obtain
f(x),¯ x–x¯≥ for allx∈C,
i.e.,x¯∈SOL(C,f). It is now left to show thatx¯∈Fix(S). SinceSis nonexpansive, we obtain
Stk–u=Stk–S(u)≤tk–u≤xk–u. (.)
By (.),
lim sup k→∞
Stk–u≤σ. (.)
Furthermore,
lim k→∞
αk
xk–u+ ( –αk)
Stk–u = lim
k→∞αkx
k+ ( –α k)S
tk–u = lim
k→∞x
k+–u=σ. (.)
So, applying Lemma ., we obtain
lim k→∞S
tk–xk= (.)
since
Sxk–xk=Sxk–Stk+Stk–xk
≤Sxk–Stk+Stk–xk
≤xk–tk+Stk–xk.
It follows from (.) and (.) that
lim k→∞
Sxk–xk= . (.)
SinceSis nonexpansive onH,xkj weakly converges tox¯and
lim j→∞
(I–S)
xkj= lim
j→∞
xkj–Sxkj= , (.)
we obtain by Lemma . that (I–S)(x) = , which means that¯ x¯∈Fix(S). Now, again by using similar arguments to those used in the proof of Theorem ., we obtain that the entire sequence{xk}weakly converges tox. Therefore the sequence¯ {xk}weakly converges
7 Conclusion
In this paper, a new double projection algorithm for the variational inequality problem has been presented. The main advantage of the proposed method is that the second projection at each iteration is onto the intersection set of two halfspaces, which is implemented very easily. When the feasible setCof VI is a quite general set, our algorithm is more effective than the double projection method proposed by Solodov and Svaiter. It is natural to ask whether it is possible to replace the first projection onto the halfspace or the intersection set of halfspaces. This would be an interesting topic in further research.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This work was supported by the Educational Science Foundation of Chongqing, Chongqing of China (grant KJ111309).
Received: 25 December 2012 Accepted: 10 May 2013 Published: 27 May 2013 References
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doi:10.1186/1687-1812-2013-136
Cite this article as:Zheng:The subgradient double projection method for variational inequalities in a Hilbert space.
