AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/350479 350479 Research Article A Strong Convergence Algorithm for the Two-Operator Split Common Fixed Point Problem in Hilbert Spaces http://orcid.org/0000-0003-3018-1319 Hong Chung-Chien 1 http://orcid.org/0000-0003-2398-7517 Huang Young-Ye 2 Zhang Qing-bang 1 Department of Industrial Management National Pingtung University of Science and Technology Pingtung 91201 Taiwan npust.edu.tw 2 Department of Accounting Information Southern Taiwan University of Science and Technology Tainan 71005 Taiwan stust.edu.tw 2014 2072014 2014 27 02 2014 13 06 2014 6 7 2014 2014 Copyright © 2014 Chung-Chien Hong and Young-Ye Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The two-operator split common fixed point problem (two-operator SCFP) with firmly nonexpansive mappings is investigated in this paper. This problem covers the problems of split feasibility, convex feasibility, and equilibrium and can especially be used to model significant image recovery problems such as the intensity-modulated radiation therapy, computed tomography, and the sensor network. An iterative scheme is presented to approximate the minimum norm solution of the two-operator SCFP problem. The performance of the presented algorithm is compared with that of the last algorithm for the two-operator SCFP and the advantage of the presented algorithm is shown through the numerical result.

1. Introduction

Throughout this paper, H denotes a real Hilbert space with inner product ·,· and its induced norm ·, I the identity mapping on H, N the set of all natural numbers, R the set of all real numbers, and PΩ the metric projection onto set Ω. x- is the upper bound of sequence {xn}, while x_ is the lower bound. For a self-mapping T on H, Fix(T) denotes the set of all fixed points of T.

It has been an interesting topic of finding zero points of maximal monotone operators. A set-valued map M:H2H with domain D(M) is called monotone if (1)x-y,u-v0 for all x, yD(M) and for any uM(x) and vM(y), where D(M) is defined to be (2)D(M)={xH:Mx}.M is said to be maximal monotone if its graph {(x,u):xH,uM(x)} is not properly contained in the graph of any other monotone operator. For a positive real number α, we denote by JαM the resolvent of a monotone operator M; that is, JαM(x)=(I+αM)-1(x) for any xH. A point vH is called a zero point of a maximal monotone operator M if 0M(v). In the sequel, we will denote the set of all zero points of A by M-10, which is equal to Fix(JαM) for any α>0. A well-known method to solve this problem is the proximal point algorithm which starts with any initial point x1H and then generates the sequence {xn} in H by (3)xn+1=JαnAxn,nN, where {αn} is a sequence of positive real numbers. This algorithm was first introduced by Martinet  and then generally studied by Rockafellar , who devised the iterative sequence {xn} by (4)xn+1=JαnAxn+en,nN, where {en} is an error sequence in H. Rockafellar showed that the sequence {xn} generated by (4) converges weakly to an element of A-10 provided that A-10 and liminfnαn>0. Since then, many authors have conducted research on modifying the sequence in (4) so that the strong convergence is guaranteed; compare  and the references therein.

On the other hand, let C and Q be nonempty closed convex subsets of two Hilbert spaces H1 and H2, respectively, and let A:H1H2 be a bounded linear mapping. The split feasibility problem (SFP) is the problem of finding a point with the property: (5)x*C,Ax*Q. The SFP was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy. The most popular algorithm for the SFP is the CQ algorithm introduced by Byrne [14, 15]. The sequence {xn} generated by the CQ algorithm converges weakly to a solution of SFP (5); compare . Under the assumption that SFP (5) has a solution, there are many algorithms designed to approximate a solution of SFP; compare  and the references therein.

Later, Censor and Segal  extended the SFP to the split common fixed point problem (SCFP) which is to find a point x* with the property: (6)x*i=1pFix(Si),Ax*j=1rFix(Tj), where Si, i=1,,p, and Tj, j=1,,r, are directed operators in Hilbert spaces. Censor and Segal  gave an algorithm for SCFP (6) in Rn spaces. Then, Moudafi  named SCFP (6) with p=1 the two-operator SCFP and gave an algorithm which generates a sequence weakly converging to the solution of the two-operator SCFP. Till very recently, Cui et al.  provided a damped projection algorithm, shown as below, to approach the solution of SCFP (6).

Assume that the solution set Ω of the SCFP is nonempty. Start with any x1H1 and generate a sequence {xn} through the iteration: (7)xn+1=(1-bn)xn+bnSn[(1-an)(xn-γnA*(I-Tn)Axn)], where {an}(0,1), {bn}[0,1], and γn(0,) satisfying that

limnan=0 and n=1an=;

liminfnbn>0;

0<γ_γnγ-<2/A2.

Then, the sequence {xn} converges strongly to p=PΩ0.

Inspired by the work of [25, 26], this paper presents another algorithm to find the minimum norm solution of two-operator SCFP. We note that the two-operator SCFP contains the SFP and the zero point problem of maximal monotone operators. Let PC and PQ be metric projections onto C and Q, respectively. Putting S1=PC and T1=PQ, the two-operator SCFP (6) is reduced to SFP (5). Let M and N be two maximal monotone operators on H1 and H2, respectively. Replacing C and Q with M-10=Fix(JαM) and N-10=Fix(JβN), respectively, in (6), the SFP becomes a two-operator SCFP: (8)Findx*H1sothatx*Fix(JαM),Ax*Fix(JβN). Putting A=I, the above two-operator SCFP is reduced to the common zero point problem of two maximal monotone operators M and N: (9)Findx*Hsothatx*M-10N-10.

Let S be S1 in the SCFP (6), and let T be T1. The target of the two-operator SCFP (6) is to find a fixed point of directed operator S. Since the definition of a directed operator is based on its fixed point set, it may be difficult to show that S is a directed operator before the two-operator SCFP is solved. Therefore, S and T are only considered as firmly nonexpansive mappings in our presented algorithm. The main result in this paper is as follows.

Let S and T be two firmly nonexpansive self-mappings on H1 and H2, respectively. Assume that the solution set Ω of the two-operator SCFP is nonempty. For any uH1, start with any x1H1 and define the sequence {xn} by (10)yn=xn-γA*(I-T)Axn,xn+1=anu+(1-an)[bnxn+(1-bn)Syn], where γ(0,1/A2) and {an} and {bn} are sequences in (0,1] satisfying that

limnan=0 and n=1an=;

liminfnbn(1-bn)>0.

Then the sequence {xn} converges strongly to p=PΩu.

The two-operator SCFP covers problems of split feasibility, convex feasibility, and equilibrium as special cases. The presented algorithm can be considered as a unified methodology for solving the aforementioned problems. In Section 4, we use the numerical result to prove that the performance of the presented algorithm is more efficient and more consistent than that of the recent damped projection algorithm .

2. Preliminaries

In order to facilitate our investigation in this paper, we recall some basic facts. A mapping S:HH is said to be

nonexpansive if (11)Sx-Syx-y,x,yH;

firmly nonexpansive if (12)Sx-Sy2x-y,Sx-Sy,x,yH;

directed if (13)Tx-x,Tx-q0,forxH,qFix(T).

It is well-known that the fixed point set Fix(S) of a nonexpansive mapping S is closed and convex; compare .

Let C be a nonempty closed convex subset of H. The metric projection PC from H onto C is the mapping that assigns each xH the unique point PCx in C with the property (14)x-PCx=minyCy-x. It is known that PC is firmly nonexpansive and characterized by the inequality, for any xH, (15)x-PCx,y-PCx0,yC.

There is a strongly convergent algorithm for a nonexpansive mapping S with Fix(S), which is related to the iteration scheme in our main result; for any uH, choose arbitrarily a point x1H and define a sequence {xn} recursively by (16)xn+1=anu+(1-an)Sxn,nN, where {an} is sequence in [0,1] satisfying (17)limnan=0,n=1an=,n=1|an+1-an|<. Then, the sequence {xn} converges strongly to PFix(S)u; compare [28, 29].

We need some lemmas that will be quoted in the sequel.

Lemma 1.

For any x,yH1 and λR, the following hold:

λx+(1-λ)y2=λx2+(1-λ)y2-λ(1-λ)x-y2;

x+y2x2+2y,x+y.

Lemma 2 (see [<xref ref-type="bibr" rid="B27">27</xref>], demiclosedness principle).

Suppose that G is a nonexpansive self-mapping on H and suppose that {xn} is a sequence in H such that {xn} converges weakly to some zH and limnxn-Gxn=0. Then, Gz=z.

Lemma 3.

Let M be a maximal monotone operator on H. Then

JαM is single-valued and firmly nonexpansive;

D(JαA)=H and Fix(JαA)=A-10.

Lemma 4 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Suppose that {zn} is a sequence of nonnegative real numbers satisfying (18)zn+1(1-an)zn+anvn,nN, where {an} and {vn} verify the following conditions:

{an}[0,1], n=1an=;

limsupnvn0.

Then limnzn=0.

Lemma 5 (see [<xref ref-type="bibr" rid="B30">30</xref>]).

Let {zn} be a sequence in R that does not decrease at infinity in the sense that there exists a subsequence {zni} such that (19)zni<zni+1,iN. For any kN, define mk=max{jk:zj<zj+1}. Then mk as k and max{zmk,zk}zmk+1,kN.

3. Main Theorems

Throughout this section, S and T denote two firmly nonexpansive self-mappings on H1 and H2, respectively, and A denotes a bounded linear operator from H1 to H2.

Under the assumption that the solution set of two-operator SCFP is nonempty, the following lemma says that the two-operator SCFP is equivalent to the fixed point problem for the operator S[I-γA*(I-T)A].

Lemma 6 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Let Ω be the solution set of two-operator SCFP (6); that is, Ω=Fix(S)A-1(Fix(T)). For any γ(0,2/A2), let U=I-γA*(I-T)A. Suppose that Ω. Then Fix(SU)=Fix(S)Fix(U)=Ω.

Theorem 7.

Let S and T be two firmly nonexpansive self-mappings on H1 and H2, respectively. Assume that the solution set Ω of the two-operator SCFP is nonempty. For any uH1, start with any x1H1 and define the sequence {xn} by (20)yn=xn-γA*(I-T)Axnxn+1=anu+(1-an)[bnxn+(1-bn)Syn], where γ(0,1/A2) and {an} and {bn} are sequences in (0,1] satisfying that

limnan=0 and n=1an=;

liminfnbn(1-bn)>0.

Then the sequence {xn} converges strongly to p=PΩu.

Proof.

Putting G=S[I-γA*(I-T)A], we see that Gxn=Syn,nN. By Lemmas 1 and 6, we have (21)xn+1-p2=an(u-p)+(1-an)kkk×[bn(xn-p)+(1-bn)(Gxn-p)]2anu-p2+(1-an)×[bnxn-p2+(1-bn)Syn-p2kkkkk-bn(1-bn)xn-Gxn2]anu-p2+(1-an)×[bnxn-p2+(1-bn)yn-p2kkkkk-bn(1-bn)xn-Gxn2]. In addition, (22)yn-p2=xn-p-γA*(I-T)Axn2=xn-p2-2γxn-p,A*(I-T)Axn+γ2A*(I-T)Axn2xn-p2-2γxn-p,A*(I-T)Axn+γ2A2(I-T)Axn2. Furthermore, since T is nonexpansive and ApFix(T), one has (23)TAxn-Ap2=(Axn-Ap)-(I-T)Axn2=Axn-Ap2-2Axn-Ap,(I-T)Axn+(I-T)Axn2=Axn-Ap2-2xn-p,A*(I-T)Axn+(I-T)Axn2Axn-Ap2, from which it follows that (24)-2xn-p,A*(I-T)Axn-(I-T)Axn2. Therefore, it follows from (21), (22), and (24) that (25)xn+1-p2anu-p2+(1-an)×[xn-p2-(1-bn)γ(1-γA2)kkkk×(I-T)Axn2-bn(1-bn)Gxn-xn2]anu-p2+(1-an)xn-p2. Hence, by induction, we see that (26)xn+1-p2max{u-p2,x1-p2}. This shows that {xn} is bounded. Now, by Lemma 1 and (22), we have (27)xn+1-p2=an(u-p)+(1-an)kkk×[bn(xn-p)+(1-bn)(Gxn-p)]2(1-an)×bn(xn-p)+(1-bn)(Gxn-p)2+2anu-p,xn+1-p=(1-an)[bnxn-p2+(1-bn)Syn-p2kkkkkkkkk-bn(1-bn)Gxn-xn2]+2anu-p,xn+1-p(1-an)[bnxn-p2+(1-bn)yn-p2kkkkkkkkk-bn(1-bn)Gxn-xn2]+2anu-p,xn+1-p(1-an)[xn-p2-(1-bn)γ(1-γA2)kkkkkkkkk×(I-T)Axn2-bn(1-bn)kkkkkkkkk×Gxn-xn2]+2anu-p,xn+1-p(1-an)[xn-p2-bn(1-bn)Gxn-xn2]+2anu-p,xn+1-p. We now carry on with the proof by considering the following two cases: (I) {xn-p} is eventually decreasing and (II) {xn-p} is not eventually decreasing.

Case I. Suppose that {xn-p} is eventually decreasing; that is, there is n0N such that {xn-p}nn0 is decreasing. In this case, limnxn-p exists in R. From inequality (27), we have (28)(1-an)bn(1-bn)Gxn-xn2(1-an)xn-p2+2anu-p,xn+1-p-xn+1-p2, which together with the boundedness of {xn} and conditions (i) and (ii) implies (29)limnGxn-xn=0. Since {xn} is bounded, it has a subsequence {xnk} such that {xnk} converges weakly to some zH and (30)limsupnu-p,xn+1-p=limku-p,xnk-p=u-p,z-p0, where the last inequality follows from (15) since zΩ by Proposition 8 of , (29), and Lemmas 2 and 6. Moreover, from (27), we have (31)xn+1-p2(1-an)xn-p2+2anu-p,xn+1-p.

Accordingly, applying Lemma 4 to inequality (31), we conclude that (32)limnxn=p.

Case II. Suppose that {xn-p} is not eventually decreasing. In this case, by Lemma 5, there exists a nondecreasing sequence {mk} in N such that mk and (33)max{xmk-p,xk-p}xmk+1-p,kN. Then it follows from (27) and (33) that (34)xmk-p2xmk+1-p2(1-amk)[xmk-p2kkkkkkkkkkk-bmk(1-bmk)Gxmk-xmk2]+2amku-p,xmk+1-p. Therefore, (35)0(1-amk)bmk(1-bmk)Gxmk-xmk2-amkxmk-p2+2amku-p,xmk+1-p, which implies that (36)limkxmk-Gxmk=0, and then it follows that (37)limsupku-p,xmk+1-p0. From (35), we obtain (38)xmk-p22u-p,xmk+1-p, and thus, letting k, we obtain (39)limkxmk-p=0. Also, since (40)xmk+1-xmkamku-xmk+(1-amk)(1-bmk)Gxmk-xmk, which together with (36) and conditions (i) and (ii) implies that limkxmk+1-xmk=0, (41)limkxmk+1-p=0 by virtue of (39). Consequently, we conclude that limkxk-p=0 via (33) and (41). This completes the proof.

This theorem says that the sequence {xn} converges strongly to a point of Ω which is nearest to u. In particular, if u is taken to be 0, then the limit point p of the sequence {xn} is the unique minimum solution of two-operator SCFP (6).

Corollary 8.

Let C and Q be nonempty closed convex subsets of two Hilbert spaces H1 and H2, respectively. Assume that the solution set Ω of the SFP is nonempty. For any uH1, start with any x1H1 and define a sequence {xn} iteratively by (42)yn=xn-γA*(I-PQ)Axnxn+1=anu+(1-an)[bnxn+(1-bn)PCyn], where γ(0,1/A2) and {an} and {bn} are sequences in (0,1] satisfying that

limnan=0 and n=1an=;

liminfnbn(1-bn)>0.

Then the sequence {xn} converges strongly to p=PΩu.

Proof.

Putting S=PC and T=PQ in (20), the conclusion follows from Theorem 7.

Corollary 9.

Suppose that M and N are two maximal monotone operators on H1 and H2, respectively. Assume that the solution set Ω of problem (43)Findx*H1sothatx*M-10,Ax*N-10 is nonempty. Let α,β(0,). For any uH1, start with any x1H1 and define a sequence {xn} iteratively by (44)yn=xn-γA*(I-JβN)Axn,xn+1=anu+(1-an)[bnxn+(1-bn)JαMyn], where γ(0,1/A2) and {an} and {bn} are sequences in (0,1] satisfying that

limnan=0 and n=1an=;

liminfnbn(1-bn)>0.

Then the sequence {xn} converges strongly to p=PΩu.

Proof.

By Lemma 3, a resolvent of a maximal monotone operator is firmly nonexpansive. Hence, we may put S=JαM and T=JβN in (20) to get the conclusion which follows from Theorem 7.

Corollary 10.

Let M be a maximal monotone operator on H with M-10, and let α,β(0,). For any uH1, start with any x1H1 and define a sequence {xn} iteratively by (45)yn=xn-γ(I-JβM)xnxn+1=anu+(1-an)[bnxn+(1-bn)JαMyn], where γ(0,1) and {an} and {bn} are sequences in (0,1] satisfying that

limnan=0 and n=1an=;

liminfnbn(1-bn)>0.

Then, the sequence {xn} converges strongly to p=PM-10(u).

Proof.

Putting H1=H2=H, A=I, M=N, and S=JαM,T=JβM in Corollary 9, the result follows immediately.

4. Numerical Results

There are four examples in this section provided to demonstrate our presented algorithm. The first three examples are the SFP, while the fourth example is the common zero point problem of two maximal monotone operators. The performance of the presented algorithm to solve the three examples of SFP is compared with that of the recent damped projection method . The result shows that the presented algorithm is more efficient and more consistent than the damped algorithm. In the first three examples, we assign the parameters in both algorithms to be u=(0,0), an=1/(n+1), bn=0.5, and γn=γ=0.01. Let xn+1-xn10-10 be their stop criterion. All codes were written in Matlab R2011a and ran on laptop ASUS ZenbookUX31E with i7-2677M CPU.

Example 11.

Let C={(x,y)(x-1)2+(y-1)21}, Q={(x,y,z)(x-1)2+(y-1)2+(z-1)29}, and (46)A=. The metric projections for C and Q are (47)PC(x,y)={(x,y),if(x,y)C;(x-1,y-1)(x-1)2+(y-1)2+(1,1),if(x,y)C,PQ(x,y,z)={(x,y,z),if(x,y,z)Q;3(x-1,y-1,z-1)(x-1)2+(y-1)2+(z-1)2+(1,1,1),if(x,y,z)Q. Then, we can use both the presented algorithm and the damped projection algorithm to approach a point such that (48)x*Fix(PC),Ax*Fix(PQ). From Table 1, we observe that the presented algorithm is more efficient than the damped projection algorithm.

Numerical results for Example 11.

x 1 The damped projection method in  The presented method
CPU (sec.) n x n CPU (sec.) n x n
( 0,0 ) 67.4971 157248 ( 0.2929,0.2929 ) 34.8173 91018 ( 0.2929,0.2929 )
( 1,1 ) 125.352 328067 ( 0.2929,0.2929 ) 35.0441 91018 ( 0.2929,0.2929 )
( 10,10 ) 411.5836 1052792 ( 0.2929,0.2929 ) 38.6464 91018 ( 0.2929,0.2929 )
Example 12.

Let all conditions be the same with those in Example 11 except to (49)A=[2-14220]. The result for solving Example 12 is shown in Table 2. We observe that the presented algorithm is still more efficient than the damped algorithm. From the columns for the runtime (CPU) and the approximate solution (xn), the result of the presented algorithm is consistent although it starts from different initial points.

Numerical results for Example 12.

x 1 The damped projection method in  The presented method
CPU (sec.) n x n CPU (sec.) n x n
( 0,0 ) 31.3902 84818 ( 0.2929,0.2929 ) 35.5024 91018 ( 0.2929,0.2929 )
( 1,1 ) 142.6763 362480 ( 0.2929,0.2929 ) 37.0838 91018 ( 0.2929,0.2929 )
( 10,10 ) 448.3774 1042364 ( 0.2928,0.2930 ) 33.8532 91031 ( 0.2929,0.2929 )
Example 13.

In this example, we use A in Example 11 but change its C and Q. Let C={(x,y)(x-1)2+(y-3)29} and Q={(x,y,z)(x-6)2+(y-15)2+(z-22)29}. The metric projections for C and Q are (50)PC(x,y)={(x,y),if(x,y)C;3(x-1,y-3)(x-1)2+(y-3)2+(1,3),if(x,y)C,PQ(x,y,z)={(x,y,z),if(x,y,z)Q;3(x-6,y-15,z-22)(x-6)2+(y-15)2+(z-22)2+(6,15,22),if(x,y,z)Q.

The result is shown in Table 3. We also observe that the presented algorithm is more efficient and more consistent than the damped projection algorithm.

Numerical results for Example 13.

x 1 The damped projection method in  The presented method
CPU (sec.) n x n CPU (sec.) n x n
( 0,0 ) 382.487 933580 ( 1.5845,2.0122 ) 100.475 247651 ( 1.5844,2.0123 )
( 1,1 ) 581.5485 1438799 ( 1.5846,2.0121 ) 101.4875 247960 ( 1.5844,2.0123 )
( 10,10 ) >1000 100.0661 252832 ( 1.5844,2.0123 )

The presented algorithm contains an arbitrary point u and that is an advantage of the algorithm. Knowing any information about the solution of two-operator SCFP of interest, we can choose a better u to enhance the performance of the presented algorithm. For instance, let u=(3,3) which is different with u=(0,0) related to the result in Table 3. From Table 4, we observe that the runtime of the presented algorithm is reduced by one-third.

Numerical results for Example 13 with u=(3,3).

x 1 The presented method
CPU (sec.) n x n
( 0,0 ) 64.0093 159081 ( 2.3455,2.1812 )
( 1,1 ) 66.2390 159477 ( 2.3455,2.1812 )
( 10,10 ) 66.0244 172465 ( 2.3455,2.1812 )
Example 14.

Minimizing a convex function is called a convex minimization problem. This example shows that the presented algorithm can be used to search the common optimal solutions of two convex minimization problems. Let f and g be two functions from R2 to R and define f(x1,x2)=x12+x22+1-2x1x2-2x1+2x2 and g(x1,x2)=x12+x22+1+2x1x2-2x1-2x2. We know that both f and g are convex functions. Now, we would like to search a common minimal point of the two convex functions.

Let f/xi denote the partial derivative of function f with respect to xi. Define two operators M and N from R2 to R by (51)M=[fx1fx2]=[2-2-22][x1x2]+[-22],N=[gx1gx2]=[x1x2]+[-2-2]. Since f and g are convex functions, M and N are maximal monotone operators and any one of their common zero points is the common minimal point of f and g. The resolvents of M and N are (52)JαM[x1x2]=[2+α-2-22+α]-1([x1x2]-α[-22]),JαN[x1x2]=[2+α222+α]-1([x1x2]+α). According to Corollary 9, our presented algorithm can be used to search a common zero point of M and N. Let α=1, u=(1,1), an=1/(n+1), bn=0.5, and γ=0.5 in the algorithm, and let xn+1-xn10-6 be the stop criterion. We ran the algorithm and started from point x1=(0,0). The algorithm stopped at point x~=(1.0006,0.0019) after 1,988 iterations. We know that Mx~0 and Nx~0. Finally, we use Figure 1 to show the behavior of sequence {xn} which converges to the common minimal point of f and g.

The behavior of our presented algorithm to search the common minimal point of two convex minimization problems. The star sign marks the stop point, x~, of the algorithm.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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