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R E S E A R C H

Open Access

Accelerated Mann and CQ algorithms for

finding a fixed point of a nonexpansive

mapping

Qiao-Li Dong

1,2*

and Han-bo Yuan

1

*Correspondence:

dongql@lsec.cc.ac.cn

1College of Science, Civil Aviation

University of China, Tianjin, 300300, China

2Tianjin Key Lab for Advanced

Signal Processing, Civil Aviation University of China, Tianjin, 300300, China

Abstract

The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective.

Keywords: nonexpansive mapping; convex minimization problem; Picard algorithm; Mann algorithm; CQ algorithm; conjugate gradient method; steepest descent method

1 Introduction

LetHbe a real Hilbert space with inner product·,·and induced norm · . Suppose that CHis nonempty, closed and convex. A mappingT:CCis said to be nonexpansive if

TxTyxy

for allx,yC. The set of fixed points ofTis defined byFix(T) :={xC:Tx=x}. In this paper, we consider the following fixed point problem.

Problem . Suppose thatT:CCis nonexpansive withFix(T)=∅. Then

findx∗∈Csuch thatTx∗=x∗.

The fixed point problems for nonexpansive mappings [–] capture various applications in diversified areas, such as convex feasibility problems, convex optimization problems, problems of finding the zeros of monotone operators, and monotone variational inequal-ities (see [, ] and the references therein). The Picard algorithm [], the Mann algorithm

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[, ], and the CQ method [] are useful fixed point algorithms to solve the fixed point problems. Meanwhile, to guarantee practical systems and networks (see,e.g., [–]) are stable and reliable, the fixed point has to be quickly found. Recently, Sakurai and Liduka [] accelerated the Halpern algorithm and obtained a fast algorithm with strong conver-gence. Inspired by their work, we focus on the Mann and the CQ algorithms and present new algorithms to accelerate the approximation of a fixed point of a nonexpansive map-ping.

We first apply the Picard algorithm to the smooth convex minimization problem and illustrate that the Picard algorithm is the steepest descent method [] for solving the minimization problem. Since conjugate gradient methods [] have been widely seen as an efficient accelerated version of most gradient methods, we introduce an accelerated Picard algorithm by combining the conjugate gradient methods and the Picard algorithm. Finally, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms.

In this paper, we propose two accelerated algorithms for finding a fixed point of a non-expansive mapping and prove the convergence of the algorithms. Finally, the numerical examples are presented to demonstrate the effectiveness and fast convergence of the ac-celerated Mann algorithm and the ineffectiveness of the acac-celerated CQ algorithm.

2 Mathematical preliminaries

2.1 Picard algorithm and our algorithm

The Picard algorithm generates the sequence{xn}∞n=as follows: givenxH,

xn+=Txn, n≥. ()

The Picard algorithm () converges to a fixed point of the mapping T ifT :CC is contractive (see,e.g., []).

WhenFix(T) is the set of all minimizers of a convex, continuously Fréchet differentiable functionalf overH, that algorithm () is the steepest descent method [] to minimizef overH. Suppose that the gradient of f, denoted by∇f, is Lipschitz continuous with a constantL>  and defineTf :HHby

Tf:=Iλf, ()

whereλ∈(, /L) andI:HHstands for the identity mapping. Accordingly,Tf satisfies the contraction condition (see,e.g., []) and

FixTf=arg min xH

f(x) :=

x∗∈H:fx∗=min xHf(x)

.

Therefore, algorithm () withT:=Tf can be expressed as follows:

dnf+:= –∇f(xn),

xn+:=Tf(xn) =xnλf(xn) =xn+λdnf+.

()

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βndfn,CGD, wheredf,CGD:= –∇f(x) and{βn}∞n=⊂(,∞), which, together with (), implies that

dnf,+CGD= λ

Tf(xn) –xn

+βndnf,CGD. ()

By replacingdnf+:= –∇f(xn) in algorithm () withdfn,+CGDdefined by (), we get the accel-erated Picard algorithm as follows:

dnf,+CGD:=λTf(xn) –xn

+βndnf,CGD, xn+:=xn+λdfn,+CGD.

()

The convergence condition of Picard algorithm is very restrictive and it does not con-verge for general nonexpansive mappings (see,e.g., []). So, in  Mann [] introduced the Mann algorithm

xn+=αnxn+ ( –αn)Txn, n≥, ()

and showed that the sequence generated by it converges to a fixed point of a nonexpansive mapping. In this paper we combine ()-() and the CQ algorithm to present two novel algorithms.

2.2 Some lemmas

We will use the following notation:

() xnxmeans that{xn}converges weakly toxandxnxmeans that{xn}converges strongly tox.

() ωw(xn) :={x:∃xnjx}denotes the weakω-limit set of{xn}.

Lemma . Let H be a real Hilbert space.There hold the following identities: (i) xy=xy– xy,y,x,yH,

(ii) tx+ ( –t)y=tx+ ( –t)yt( –t)xy,t[, ],x,yH.

Lemma . Let K be a closed convex subset of a real Hilbert space H,and let PK be the (metric or nearest point)projection from H onto K (i.e.,for xH,PKx is the only point in K such thatxPKx=inf{xz:zK}).Given xH and zK.Then z=PKx if and only if there holds the relation

xz,yz ≤ for all yK.

Lemma .(See []) Let K be a closed convex subset of H.Let{xn}be a sequence in H and uH.Let q=PKu.Suppose that{xn}is such thatωw(xn)K and satisfies the condition

xnuuq for all n.

Then xnq.

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Lemma .(See []) Assume that{an}is a sequence of nonnegative real numbers satis-fying the property

an+≤an+un, n≥,

where {un} is a sequence of nonnegative real numbers such that

n=un<∞. Then limn→∞anexists.

3 The accelerated Mann algorithm

In this section, we present the accelerated Mann algorithm and give its convergence.

Algorithm . Chooseμ∈(, ],λ> , andxHarbitrarily, and set{αn}∞n=⊂(, ), {βn}∞n=⊂[,∞). Setd:= (T(x) –x)/λ. Computedn+andxn+as follows:

⎧ ⎪ ⎨ ⎪ ⎩

dn+:=λ

T(xn) –xn

+βndn, yn:=xn+λdn+,

xn+:=μαnxn+ ( –μαn)yn.

()

We can check that Algorithm . coincides with the Mann algorithm () whenβn:≡ andμ:= .

In this section we make the following assumptions.

Assumption . The sequences{αn}∞n=and{βn}∞n=satisfy (C) ∞n=μαn( –μαn) =∞,

(C) ∞n=βn<∞. Moreover,{xn}∞n=satisfies

(C) {T(xn) –xn}∞n=is bounded.

Before doing the convergence analysis of Algorithm ., we first show the two lemmas.

Lemma . Suppose that T:HH is nonexpansive withFix(T)=∅and that Assump-tion.holds.Then{dn}∞n=and{xn–p}∞n=are bounded for any p∈Fix(T).Furthermore, limn→∞xnpexists.

Proof We have from (C) thatlimn→∞βn= . Accordingly, there existsn∈Nsuch that βn≤/ for allnn. DefineM:=max{max≤kndk, (/λ)supn∈NT(xn) –xn}. Then (C) implies thatM<∞. Assume thatdnMfor somenn. The triangle inequality ensures that

dn+=

λT(xn) –xn

+βndn

≤ 

λT(xn) –xn+βndnM, () which means thatdnMfor alln≥,i.e.,{dn}∞n=is bounded.

The definition of{yn}∞n=implies that

yn=xn+λ

λ

T(xn) –xn

+βndn

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The nonexpansivity ofTand () imply that, for anyp∈Fix(T) and for allnn,

ynp=T(xn) +λβndnpT(xn) –T(p)+λβndn

xnp+λMβn. ()

Therefore, we find

xn+–p=μαn(xnp) + ( –μαn)(ynp) ≤μαnxnp+ ( –μαn)ynpμαnxnp+ ( –μαn)

xnp+λMβn

xnp+λMβn, ()

which implies

xnpxp+λM n–

k= βk<∞.

So, we get that{xn}∞n=is bounded. From () it follows that{yn}∞n=is bounded.

In addition, using Lemma ., (C), and (), we obtainlimn→∞xnpexists.

Lemma . Suppose that T:HH is nonexpansive withFix(T)=∅and that Assump-tion.holds.Then

lim

n→∞xnT(xn)= .

Proof By ()-() and the nonexpansivity ofT, we have

xn+–T(xn+)

=μαnxn+ ( –μαn)

T(xn) +λβndn

Tμαnxn+ ( –μαn)

T(xn) +λβndn =μαn

xnT(xn)

+ ( –μαn)λβndn

+T(xn) –T

μαnxn+ ( –μαn)

T(xn) +λβndnμαnxnT(xn)+ ( –μαn)λβndn

+xn

μαnxn+ ( –μαn)

T(xn) +λβndnμαnxnT(xn)+ ( –μαn)λβndn

+ ( –μαn)xnT(xn)+ ( –μαn)λβndnxnT(xn)+ λβndn

xnT(xn)+ λβnM,

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On the other hand, for anyp∈Fix(T) and for allnn, using the triangle inequality, the Cauchy-Schwarz inequality, and Lemma .(ii), we obtain

xn+–p=μαnxn+ ( –μαn)ynp =μαnxn+ ( –μαn)

T(xn) +λβndn

p =μαn(xnp) + ( –μαn)

T(xn) –p+ ( –μαn)λβndn

μαn(xnp) + ( –μαn)

T(xn) –p+( –μαn)λβndn

+ μαn(xnp) + ( –μαn)

T(xn) –p( –μαn)λβndnμαnxnp+ ( –μαn)T(xn) –p

μαn( –μαn)T(xn) –xn

+ ( –μαn)λβnM+ μαnxnp+ ( –μαn)T(xn) –p ×( –μαn)λβnM

xnp–μαn( –μαn)T(xn) –xn

+βn( –μαn)λMxnp+ ( –μαn)λβnM

xnp–μαn( –μαn)T(xn) –xn

+βnM.

We have from Lemma . thatM:=supk( –μαk)λ{Mxkp+ ( –μαk)λβkM}is bounded. Therefore, using (C), we obtain

n

k=

μαk( –μαk)T(xk) –xk≤ xp–xn+–p+M n

k= βk<∞,

which, with (C), implies that

lim inf

n→∞T(xn) –xn= .

Due to existence of the limit ofT(xn) –xn, we have

lim

n→∞T(xn) –xn= ,

which with Lemma . implies thatωw(xn)⊂Fix(T).

Theorem . Suppose that T:HH is nonexpansive withFix(T)=∅and that

Assump-tion.holds.Then the sequence{xn}generated by Algorithm.weakly converges to a fixed point of T.

Proof To see that{xn}is actually weakly convergent, we need to show thatωω(xn) consists

of exactly one point. Takep,qωw(xn) and let{xni}and{xmj}be two subsequences of{xn}

such thatxnipandxmjq, respectively. Using Lemma . of [] and Lemma ., we

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4 The accelerated CQ algorithm

In general, the Mann algorithm () has only weak convergence (see [] for an exam-ple). However, strong convergence is often much more desirable than weak convergence in many problems that arise in infinite dimensional spaces (see [] and the references therein). In , Nakajo and Takahashi [] introduced the following modification of the Mann algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xCchosen arbitrarily,

yn:=αnxn+ ( –αn)Txn,

Cn={zC:ynzxnz}, Qn=

zC:xnz,xxn ≥

,

xn+=PCnQnx,

()

where Cis a nonempty closed convex subset of a Hilbert spaceH andT :CC is a nonexpansive mapping, andPKdenotes the metric projection fromHonto a closed convex subsetKofH.

Here, we introduce an acceleration of CQ algorithm based on Algorithm . and show its strong convergence.

Theorem . Let C be a bounded, closed and convex subset of a Hilbert space H and T:CC be nonexpansive withFix(T)=∅.Assume that{αn}∞n=is a sequence in(,a]for some <a< and{βn}∞n=⊂[,∞)such thatlimn→∞βn= .Define a sequence{xn}∞n=in C by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xC chosen arbitrarily,

dn+:=λ

T(xn) –xn

+βndn, yn:=xn+λdn+,

zn:=αnxn+ ( –αn)yn, Cn=

zC:znz≤ xnz+θn

,

Qn=

zC:xnz,xxn ≥

,

xn+=PCnQnx,

()

where

θn=λβnM[λβnM+ M]→ as n→ ∞,

M=diamC and M:=max{max≤kndk, (/λ)M},nis chosen such thatβn≤/for

all nn.Then{xn}n∞=converges in norm to PFix(T)(x).

Proof First observe thatCnis convex. Indeed, the inequality defined inCncan be rewritten as

(xnzn),z

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which is affine (and hence convex ) inz. Next we show thatFix(T)⊂Cnfor alln. Similar to the proof of (), we getdnM. Due toxnC, we have, for allp∈Fix(T)⊂C,

xnpM, ()

and

ynpxnp+λβnM,

which comes from (). Thus we get

znpαnxnp+ ( –αn)ynpxnp+λβnM,

and consequently,

znp≤ xnp+ λβnMxnp+ (λβnM) ≤ xnp+θn,

whereθn=λβnM[λβnM+ M]. So,pCnfor alln. Next we show that

Fix(T)⊂Qn for alln≥. ()

We prove this by induction. Forn= , we haveFix(T)⊂C=Q. Assume thatFix(T)⊂Qn. Sincexn+is the projection ofxontoCnQn, we have

xn+–z,xxn+ ≥, ∀zCnQn.

AsFix(T)⊂CnQn, the last inequality holds, in particular, for allz∈Fix(T). This together with the definition ofQn+implies thatFix(T)⊂Qn+. Hence () holds for alln≥.

Notice that the definition ofQnactually impliesxn=PQn(x). This together with the fact

thatFix(T)⊂Qnfurther implies

xnxpx, p∈Fix(T).

Due toq=PFix(T)(x)∈Fix(T), we have

xnxqx. ()

The fact thatxn+∈Qnasserts thatxn+–xn,xnx ≥. This together with Lemma .(i) implies

xn+–xn=(xn+–x) – (xnx) 

=xn+–x–xnx– xn+–xn,xnx

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This implies that the sequence {xnx}∞n= is increasing. Recall (), we get that limn→∞xnxexists. It turns out from () that

lim

n→∞xn+–xn= .

By the factxn+∈Cnwe get

znxn+≤ xnxn++θn,

and thus

znxn+ ≤ xnxn++

θn. ()

On the other hand, sincezn=αnxn+ ( –αn)T(xn) + ( –αn)λβndnandαna, we have

T(xn) –xn=   –αn

znxn– ( –αn)λβndn

≤ 

 –aznxn+λβndn

≤ 

 –a

znxn++xn+–xn

+λβnM

≤ 

 –a

xn+–xn+

θn

+λβnM→, ()

where the last inequality comes from ().

Lemma . and () then guarantee that every weak limit point of{xn}∞n=is a fixed point ofT. That is,ωw(xn)⊂Fix(T). This fact, with inequality () and Lemma ., ensures the strong convergence of{xn}∞n=toq=PFix(T)x.

5 Numerical examples and conclusion

In this section, we compare the original algorithms and the accelerated algorithms. The codes were written in Matlab . and run on personal computer.

Firstly, we apply the Mann algorithm () and Algorithm . to the following convex fea-sibility problem (see [, ]).

Problem .(From []) Given a nonempty, closed convex setCi⊂RN (i= , , . . . ,m),

findx∗∈C:= m

i= Ci,

where one assumes thatC=∅. Define a mappingT:RN →RN by

T:=P

m

m

i= Pi

, ()

where Pi=PCi (i= , , . . . ,m) stands for the metric projection onto Ci. SincePi (i=

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Table 1 Computational results for Problem 5.1 with different dimensions

Initial point rand(N, 1) 200×rand(N, 1) 5e 5,000e

N= 5 m= 5

Algorithm 3.1 Iter. 2 62 5 9

Sec. 0 0 0 0

Mann algorithm Iter. 64 68 73 74

Sec. 0 0 0 0

N= 1,000 m= 50

Algorithm 3.1 Iter. 4 448 7 425

Sec. 0.0156 1.0608 0.0312 1.0608

Mann algorithm Iter. 505 515 478 429

Sec. 1.2948 1.2480 1.2012 1.1076

N= 50 m= 1,000

Algorithm 3.1 Iter. 6,814 7 4 9

Sec. 38.9846 0.1092 0.0312 0.1248

Mann algorithm Iter. 8,438 7,771 8,692 6,913

Sec. 53.6799 43.4463 49.7799 38.7818

that

Fix(T) =Fix(P) m

i=

Fix(Pi) =C m

i= Ci=C.

Set λ:= ,μ:= .,αn:= /(n+ ) (n≥), andβn:= /(n+ ) in Algorithm . and αn:=μ/(n+ ) in the Mann algorithm (). In the experiment, we setCi(i= , , . . . ,m) as a closed ball with centerci∈RNand radiusri> . Thus,Pi(i= , , . . . ,m) can be computed with

Pi(x) :=

ci+cri

ix(xci) ifcix>ri,

x ifcixri.

We setri:=  (i= , , . . . ,m),c:=  andci∈(–/ √

N, /√N)N (i= , . . . ,m) were ran-domly chosen. Sete:= (, , . . . , ). In Table , ‘Iter.’ and ‘Sec.’ denote the number of iterations and the cpu time in seconds, respectively. We tookT(xn) –xn<ε= –as the stopping criterion.

Table  illustrates that, with a few exceptions, Algorithm . significantly reduces the running time and iterations needed to find a fixed point compared with the Mann algo-rithm. The advantage is more obvious, as the parametersN andmbecome larger. It is worth further research on the reason of emergence of a few exceptions.

Next, we apply the CQ algorithm () and the accelerated CQ algorithm () to the fol-lowing problem.

Problem .(From []) LetCbe the unit closed ballS(, ) ={x∈R|x}andT : S(, )→S(, ) be defined byT : (x,x,x)T→(√sin(x+z),√sin(x+z),√(x+y))T. Then

findx∗∈S(, ) such thatTx∗=x∗.

He and Yang [] showed thatTis nonexpansive and has at least a fixed point inS(, ). Take the sequenceαn=n in () and (), andβn=×n,λ= .. in (). We tested four

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Table 2 Computational results for Problem 5.2 with different initial points

Initial point (1, 0, 0) (0, 1, 0) (0, 0.5, 0.5) (0.5, 0.5, 0.5)

CQ algorithm Iter. 652 167 1,441 38

Sec. 0.1092 0.0312 0.2652 0

Accelerated CQ algorithm Iter. 577 273 1,430 84

Sec. 0.0936 0.0468 0.2496 0.0156

Table  shows that the acceleration of the CQ algorithm is ineffective, that is, the accel-erated CQ algorithm does not in fact accelerate the CQ algorithm from running time or the number of iterations. The acceleration may be eliminated by the projection onto the setsCnandQn.

6 Concluding remarks

In this paper, we accelerate the Mann and CQ algorithms to obtain the accelerated Mann and CQ algorithms, respectively. Then we present the weak convergence of the accelerated Mann algorithm and the strong convergence of the accelerated CQ algorithm under some conditions. The numerical examples illustrate that the acceleration of the Mann algorithm is effective, however, the acceleration of the CQ algorithm is ineffective.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors express their thanks to the reviewers, whose constructive suggestions led to improvements in the presentation of the results. Supported by the National Natural Science Foundation of China (No. 61379102) and Fundamental Research Funds for the Central Universities (No. 3122013D017), in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.

Received: 26 February 2015 Accepted: 30 June 2015

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Figure

Table 1 Computational results for Problem 5.1 with different dimensions
Table 2 Computational results for Problem 5.2 with different initial points

References

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