• No results found

Strong convergence of an extragradient type algorithm for the multiple sets split equality problem

N/A
N/A
Protected

Academic year: 2020

Share "Strong convergence of an extragradient type algorithm for the multiple sets split equality problem"

Copied!
11
0
0

Loading.... (view fulltext now)

Full text

(1)

R E S E A R C H

Open Access

Strong convergence of an

extragradient-type algorithm for the

multiple-sets split equality problem

Ying Zhao

and Luoyi Shi

*†

*Correspondence:

shiluoyi@tjpu.edu.cn

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China

Equal contributors

Abstract

This paper introduces a new extragradient-type method to solve the multiple-sets split equality problem (MSSEP). Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Moreover, several numerical results are given to show the effectiveness of our algorithm.

Keywords: strong convergence; extragradient-type; multiple-sets split equality problem

1 Introduction

The split feasibility problem (SFP) was first presented by Censoret al.[]; it is an inverse problem that arises in medical image reconstruction, phase retrieval, radiation therapy treatment, signal processingetc. The SFP can be mathematically characterized by finding a pointxthat satisfies the property

xC, AxQ, (.)

if such a point exists, whereCandQare nonempty closed convex subsets of Hilbert spaces HandH, respectively, andA:H→His a bounded and linear operator.

There are various algorithms proposed to solve the SFP, see [–] and the references therein. In particular, Byrne [, ] introduced the CQ-algorithm motivated by the idea of an iterative scheme of fixed point theory. Moreover, Censoret al.[] introduced an extension upon the form of SFP in  with an intersection of a family of closed and convex sets instead of the convex setC, which is the original of the multiple-sets split feasibility problem (MSSFP).

Subsequently, an important extension, which goes by the name of split equality problem (SEP), was made by Moudafi []. It can be mathematically characterized by finding points xCandyQthat satisfy the property

Ax=By, (.)

(2)

if such points exist, whereCandQare nonempty closed convex subsets of Hilbert spaces HandH, respectively,His also a Hilbert space,A:H→HandB:H→Hare two

bounded and linear operators. WhenB=I, the SEP reduces to SFP. For more information about the methods for solving SEP, see [, ].

This paper considers the multiple-sets split equality problem (MSSEP) which general-izes the MSSFP and SEP and can be mathematically characterized by finding pointsxand ythat satisfy the property

xt

i=

Ci and yr

j=

Qj such thatAx=By, (.)

where r,tare positive integers, {Ci}ti=∈H and{Qj}rj=∈H are nonempty, closed and

convex subsets of Hilbert spacesH andH, respectively,His also a Hilbert space,A: H →H, B:H→H are two bounded and linear operators. Obviously, ifB=I, the

MSSEP is just right MSSFP; ift=r= , the MSSEP changes into the SEP. Moreover, when B=Iandt=r= , the MSSEP reduces to the SFP.

One of the most important methods for computing the solution of a variational inequal-ity and showing the quick convergence is an extragradient algorithm, which was first in-troduced by Korpelevich []. Moreover, this method was applied for finding a common element of the set of solutions for a variational inequality and the set of fixed points of a nonexpansive mapping, see Nadezhkina et al.[]. Subsequently, Cenget al.in [] presented an extragradient method, and Yaoet al.in [] proposed a subgradient extra-gradient method to solve the SFP. However, all these methods to solve the problem have only weak convergence in a Hilbert space. On the other hand, a variant extragradient-type method and a subgradient extragradient method introduced by Censoret al.[, ] possess strong convergence for solving the variational inequality.

Motivated and inspired by the above works, we introduce an extragradient-type method to solve the MSSEP in this paper. Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Finally, several numerical results are given to show the feasibility of our algorithm.

2 Preliminaries

LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetIdenote the identity operator onH.

Next, we recall several definitions and basic results that will be available later.

Definition . A mappingT:HHgoes by the name of

(i) nonexpansive if

TxTyxy, ∀x,yH;

(ii) firmly nonexpansive if

TxTyxy,TxTy, ∀x,yH;

(iii) contractive onxif there exists <α< such that

(3)

(iv) monotone if

TxTy,xy ≥, ∀x,yH;

(v) β-inverse strongly monotone if there existsβ> such that

TxTy,xyβTxTy, ∀x,yH.

The following properties of an orthogonal projection operator were introduced by Bauschkeet al.in [], and they will be powerful tools in our analysis.

Proposition .([]) Let PCbe a mapping from H onto a closed,convex and nonempty subset C of H if

PC(x) =arg min

yCxy, ∀xH,

then PCis called an orthogonal projection from H onto C.Furthermore,for any x,yH and zC,

(i) xPCx,zPCx ≤;

(ii) PCxPCy≤ PCxPCy,xy;

(iii) PCxz≤ xz–PCxx.

The following lemmas provide the main mathematical results in the sequel.

Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H,let

T:CH beα-inverse strongly monotone,and let r> be a constant.Then,for anyx,yC,

(IrT)x– (IrT)y≤ xy+r(r– α)T(x) –T(y).

Moreover, when  <r< α,IrTis nonexpansive.

Lemma .([]) Let{xk}and{yk}be bounded sequences in a Hilbert space H,and let{β k} be a sequence in[, ]which satisfies the condition <lim infk→∞βk≤lim supk→∞βk< . Suppose that xk+= (–βk)yk

kxkfor all k≥andlim supk→∞(yk+–ykxk+–xk)≤ .Thenlimk→∞ykxk= .

The lemma below will be a powerful tool in our analysis.

Lemma .([]) Let{ak}be a sequence of nonnegative real numbers satisfying the condi-tion ak+≤( –mk)ak+mkδk,∀k≥,where{mk},{δk}are sequences of real numbers such that

(i) {mk} ∈[, ]and

k=mk=∞or,equivalently, ∞

k=

( –mk) = lim k→∞

k

j=

(4)

(ii) lim supk→∞δk≤or

(ii)’ ∞k=δkmkis convergent.Thenlimk→∞ak= .

3 Main results

In this section, we propose a formal statement of our present algorithm. Review the multiple-sets split equality problem (MSSEP), without loss of generality, suppose t>r in (.) and defineQr+=Qr+=· · ·=Qt=H. Hence, MSSEP (.) is equivalent to the

following problem:

find xt

i=

Ci and yt

j=

Qj such thatAx=By. (.)

Moreover, setSi=Ci×QiH=H×H(i= , , . . . ,t),S=

t

i=Si,G= [A, –B] :HH, the adjoint operator ofGis denoted byG∗, then the original problem (.) reduces to

finding w= (x,y)∈S such thatGw= . (.)

Theorem . Let=∅be the solution set of MSSEP(.).For an arbitrary initial point w∈S,the iterative sequence{wn}can be given as follows:

⎧ ⎨ ⎩

vn=PS{( –αn)wnγnGGwn}, wn+=PS{wnμnGGvn+λn(vnwn)},

(.)

where {αn}∞n= is a sequence in [, ] such that limn→∞αn = ,and

n=αn =∞, and

{γn}∞n=,{λn}∞n=,{μn}∞n=are sequences in H satisfying the following conditions:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

γn∈(,ρ(G∗G)), limn→∞(γn+–γn) = ;

λn∈(, ), limn→∞(λn+–λn) = ;

μnρ(G∗G)λn, limn→∞(μn+–μn) = ;

n=(

γn λn) <∞.

(.)

Then{wn}converges strongly to a solution of MSSEP(.).

Proof In view of the property of the projection, we inferwˆ=PS(wˆ –tGGwˆ) for anyt> . Further, from the condition in (.), we get thatμnρ(G∗G)λn, it follows thatI

μn λnG

G

is nonexpansive. Hence,

wn+–wˆ

=PS

wnμnGGvn+λn(vnwn)

PS

ˆ

wtGGwˆ =PS

( –λn)wn+λn

Iμn λn

GG

vn

PS

( –λn)wˆ+λn

Iμn λn

GG

ˆ

w

≤( –λn)wnwˆ+λn Iμn

λn GG

vn

Iμn

λn GG

ˆ

w

(5)

Sinceαn→ asn→ ∞and from the condition in (.),γn∈(,ρ(G∗G)), it follows that

αn≤ –γnρ(G

G)

 asn→ ∞, that is,

γn –αn ∈(,

ρ(GG)). We deduce that

vnwˆ =PS

( –αn)wnγnGGwn

PS

ˆ

wtGGwˆ

≤( –αn)

wnγn  –αn

GGwn

αnwˆ + ( –αn)

ˆ

wγn  –αn

GGwˆ

≤–αnwˆ+ ( –αn)

wnγn  –αn

GGwnwˆ + γn  –αn

GGwˆ, (.)

which is equivalent to

vnwˆ ≤αnwˆ+ ( –αn)wnwˆ. (.)

Substituting (.) in (.), we obtain

wnwˆ ≤( –λn)wnwˆ+λn

αnwˆ+ ( –αn)wnwˆ

≤( –λnαn)wnwˆ+λnαnwˆ

≤maxwnwˆ,–wˆ

.

By induction,

wnwˆ ≤max

w–wˆ,–wˆ

.

Consequently,{wn}is bounded, and so is{vn}.

LetT= PSI. From Proposition ., one can know that the projection operatorPSis monotone and nonexpansive, and PSIis nonexpansive.

Therefore,

wn+= I+T

( –λn)wn+λn

 –μn λn

GG

vn

=Iλnwn+

λn

Iμn

λn GG

vn+

T

( –λn)wn+λn

Iμn λn

GG

vn

,

that is,

wn+=

 –λnwn+

 +λn

bn, (.)

wherebn=

λn(IμλnnGG)vn+T[(–λn)wn+λn(I

μn

λnGG)vn]

+λn .

Indeed,

bn+–bn

λn+

 +λn+

Iμn+ λn+

GG

vn+–

Iμn

λn GG

vn

+ λn+  +λn+

λn  +λn

(6)

×

Iμn

λn GG

vn

+ T

 +λn+

( –λn+)wn++λn+

Iμn+

λn+ GG

vn+

( –λn)wn+λn

Iμn λnG

Gv n

+ 

 +λn+

– 

 +λn

×T

( –λn)wn+λn

Iμn λn

GG

vn

. (.)

For convenience, letcn= (IμλnnG

G)v

n. By Lemma . in Shiet al.[], it follows that (Iμn

λnG

G) is nonexpansive and averaged. Hence,

bn+–bn

λn+

 +λn+

cn+–cn+ λn+

 +λn+

λn  +λn

cn

+ T

 +λn+

( –λn+)wn++λn+cn+–

( –λn)wn+λncn

+   +λn+

– 

 +λn

T( –λn)wn+λncn

λn+

 +λn+

cn+–cn+ λn+

 +λn+

λn  +λn

cn

+ –λn+  +λn+

wn+–wn+ λn+

 +λn+

cn+–cn+

λnλn+

 +λn+ wn

+λn+–λn  +λn+

cn+

 +λn+ –   +λn

T( –λn)wn+λncn. (.)

Moreover,

cn+–cn =

Iμn+

λn+ GG

vn+–

Iμn

λn GG

vn

vn+–vn =PS

( –αn+)wn+–γnGGwn+

PS

( –αn)wnγnGGwn

Iγn+GG

wn+–

Iγn+GG

wn+ (γnγn+)GGwn

+αn+–wn++αnwn

wn+–wn+|γnγn+|GGwn+αn+–wn++αnwn. (.)

Substituting (.) in (.), we infer that

bn+–bn

λn+

 +λn+

λn  +λn

cn+

λnλn+

 +λn+

wn+

λn+–λn  +λn+

cn

+wn+–wn+

 +λn+ –   +λn

T( –λn)wn+λncn

(7)

By virtue oflimn→∞(λn+–λn) = , it follows thatlimn→∞|+λλn+n+–

λn

+λn|= . Moreover,

{wn}and{vn}are bounded, and so is{cn}. Therefore, (.) reduces to

lim sup

n→∞

bn+–bnwn+–wn

≤. (.)

Applying (.) and Lemma ., we get

lim

n→∞bnwn= . (.)

Combining (.) with (.), we obtain

lim

n→∞xn+–xn= .

Using the convexity of the norm and (.), we deduce that

wn+–wˆ

≤( –λn)wnwˆ+λnvnwˆ

≤( –λn)wnwˆ+λn –αnwˆ

+ ( –αn)

wnγn  –αn

GGwn

ˆ

wγn  –αn

GGwˆ 

≤( –λn)wnwˆ+λnαnwˆ + ( –αn)λn

wnwˆ+ γn  –αn

γn  –αn

ρ(GG)

GGwnGGwˆ

wnwˆ+λnαnwˆ+λnγn

γn  –αn

– 

ρ(GG)

GGwnGGwˆ

,

which implies that

λnγn

ρ(GG)–

γn  –αn

GGwnGGwˆ

wnwˆ–wn+–wˆ+λnαnwˆ

wn+–wn

wnwˆ+wn+–wˆ

+λnαnwˆ.

Sincelim infn→∞λnγn(ρ(G∗G)–

γn

–αn) > ,limn→∞αn=  andlimn→∞wn+–wn= , we infer that

lim

n→∞GGw

nGGwˆ= . (.)

Applying Proposition . and the property of the projectionPS, one can easily show that

vnwˆ =PS

( –αn)wnγnGGwn

PS

ˆ

wγnGGwˆ

(8)

≤( –αn)wnγnGGwn

ˆ

wγnGGwˆ

,vnwˆ

=

wnγnGGw

n

ˆ

wγnGGwˆ

αnwn

+vnwˆ –( –αn)wnγnGGwn

ˆ

wγnGGwˆ

vn+wˆ

≤

wnwˆ+ αnwnwnγnGGwn

ˆ

wγnGGwˆ

αnwn

+vnwˆ–wnvnγnGG(wnwˆ) –αnwn

≤

wnwˆ+αnM+vnwˆ–wnvn + γn

wnvn,GG(wnwˆ)

+ αnwn,wnvnγnGG(wnwˆ) +αnwn

≤

wnwˆ+αnM+vnwˆ

wnvn+ γnwnvnGG(wnwˆ) + αnwnwnvn

wnwˆ+αnMwnvn+ γnwnvnGG(wnwˆ)

+ αnwnwnvn, (.)

whereM>  satisfies

M≥sup

k

–wnwnγnGGwn

ˆ

wγnGGwˆ

αnwn.

From (.) and (.), we get

wn+–wˆ

≤( –λn)wnwˆ+λnvnwˆ

wnwˆ–λnwnvn+αnM+ γnwnvnγnGG(wnwˆ) + αnwnwnvn,

which means that

λnwnvn≤ wn+–wn

wnwˆ+wn+–wˆ

+αnM

+ γnwnvnγnGG(wnwˆ) + αnwnwnvn.

Sincelimn→∞αn= ,limn→∞wn+–wn=  andlimn→∞GGwnGGwˆ= , we infer that

lim

(9)

Finally, we show thatwn→ ˆw. Using the property of the projectionPS, we derive

vnwˆ =PS

( –αn)

wn

γn  –αnG

Gw n

PS

αnwˆ+ ( –αn)

ˆ

wγn  –αn

GGwˆ 

( –αn)

Iγn

 –αn GG

(wnwˆ) –αnwˆ,vnwˆ

≤( –αn)wnwˆvnwˆ+αn ˆw,wˆ–vn

≤ –αn

wnwˆ+vnwˆ

+αn ˆw,wˆ–vn, which equals

vnwˆ≤  –αn  +αn

wnwˆ+ αn  –αn ˆ

w,wˆ –vn. (.)

It follows from (.) and (.) that

wn+–wˆ

≤( –λn)wnwˆ+λnvnwˆ

≤( –λn)wnwˆ+λn

 –αn  +αn

wnwˆ+ αn  –αn

ˆw,wˆ–vn

 –αnλn  +αn

wnwˆ+ αnλn  –αn ˆ

w,wˆ–vn. (.)

Since γn –αn∈(,

ρ(GG)), we observe thatαn∈(,  –

γnρ(GG)  ), then

αnλn  –αn

,λn( –γnρ(GG))

γnρ(GG)

,

that is to say,

αnλn  –αn ˆ

w,wˆ–vn

λn( –γnρ(GG)) γnρ(GG) ˆ

w,wˆ–vn.

By virtue of ∞n=(λn

γn) < ∞, γn ∈ (, 

ρ(GG)) and ˆw,wˆ –vn is bounded, we obtain

n=(

λn(–γnρ(GG))

γnρn(GG) ) ˆw,wˆ–vn<∞, which implies that

n=

αnλn  –αn

ˆw,wˆ –vn ≤ ∞.

Moreover,

n=

αnλn  –αn ˆ

w,wˆ –vn= ∞

n=

αnλn  +αn

 +αn  –αn ˆ

(10)
[image:10.595.111.353.69.394.2]

Table 1 = 10–5,P= 3,M= 3,N= 3

n t

Sequence (3.3) 60 0.078 Tian’s (3.15)’ 117 0.093 Byrne’s (1.2) 1, 845 1.125

Table 2 = 10–10,P= 3,M= 3,N= 3

n t

Sequence (3.3) 120 0.156 Tian’s (3.15) 294 0.29 Byrne’s (1.2) 8, 533 2.734

Table 3 = 10–5,P= 10,M= 10,N= 10

n t

Sequence (3.3) 63 0.093 Tian’s (3.15) 426 0.469 Byrne’s (1.2) 2, 287 1.313

Table 4 = 10–10,P= 10,M= 10,N= 10

n t

Sequence (3.3) 123 0.25 Tian’s (3.15) 948 0.906 Byrne’s (1.2) 13, 496 2.437

it follows that all the conditions of Lemma . are satisfied. Combining (.), (.) and Lemma ., we can show thatwn→ ˆw. This completes the proof.

4 Numerical experiments

In this section, we provide several numerical results and compare them with Tian’s [] algorithm (.)’ and Byrne’s [] algorithm (.) to show the effectiveness of our pro-posed algorithm. Moreover, the sequence given by our algorithm in this paper has strong convergence for the multiple-sets split equality problem. The whole program was writ-ten in Wolfram Mathematica (version .). All the numerical results were carried out on a personal Lenovo computer with Intel(R)Pentium(R) N CPU . GHz and RAM . GB.

In the numerical results,A= (aij)P×N,B= (bij)P×M, whereaij∈[, ],bij∈[, ] are all given randomly,P,M,N are positive integers. The initial pointx= (, , . . . , ), andy=

(, , . . . , ),αn= .,λn= .,γn= ρ(G.∗G),μn= ρ(G.∗G) in Theorem .,ρn=ρn = . in

Tian’s (.)’ andγn= . in Byrne’s (.). The termination condition isAxBy< . In Tables -, the iterative steps and CPU are denoted bynandt, respectively.

Competing interests

The authors declare that there are no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

(11)

Received: 23 December 2016 Accepted: 20 February 2017 References

1. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2-4), 221-239 (1994)

2. Xu, HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl.22(6), 2021-2034 (2006)

3. Lopez, G, Martin-Marqnez, V, Wang, F, Xu, HK: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl.28, 085004 (2012)

4. Yang, Q: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl.20(4), 1261-1266 (2004) 5. Byrne, C: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl.18(2), 441-453

(2002)

6. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl.20(1), 103-120 (2004)

7. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl.21(6), 2071-2084 (2005)

8. Moudafi, A: Alternating CQ algorithm for convex feasibility and split fixed point problem. J. Nonlinear Convex Anal. 15(4), 809-818 (2013)

9. Shi, LY, Chen, RD, Wu, YJ: Strong convergence of iterative algorithms for solving the split equality problems. J. Inequal. Appl.2014, Article ID 478 (2014)

10. Dong, QL, He, SN, Zhao, J: Solving the split equality problem without prior knowledge of operator norms. Optimization64(9), 1887-1906 (2015)

11. Korpelevich, GM: An extragradient method for finding saddle points and for other problems. Ekonomikai Matematicheskie Metody12(4), 747-756 (1976)

12. Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.128(1), 191-201 (2006)

13. Ceng, LC, Ansari, QH, Yao, JC: An extragradient method for solving split feasibility and fixed point problems. Comput. Math. Appl.64(4), 633-642 (2012)

14. Yao, Y, Postolache, M, Liou, YC: Variant extragradient-type method for monotone variational inequalities. Fixed Point Theory Appl.2013, Article ID 185 (2013)

15. Censor, Y, Gibali, A, Reich, S: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl.148(2), 318-335 (2011)

16. Censor, Y, Gibali, A, Reich, S: Strong convergence of subgradient extragradient method for the variational inequalities in Hilbert space. Optim. Methods Softw.26(4-5), 827-845 (2011)

17. Bauschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Space. Springer, London (2011)

18. Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.118(2), 417-428 (2003)

19. Suzuki, T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl.2005685918 (2005)

20. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.66(1), 240-256 (2002)

21. Tian, D, Shi, L, Chen, R: Iterative algorithm for solving the multiple-sets split equality problem with split self-adaptive step size in Hilbert spaces. Arch. Inequal. Appl.2016(1), 1-9 (2016)

Figure

Table 1 ϵ = 10–5,P = 3,M = 3,N = 3

References

Related documents

It has been confirmed by a large number of engineering practice that a series of widely-adopted classical data processing algorithms (e.g., least-squared

It aimed to verify the self-care of patients undergoing dialysis with Central Venous Catheter (CVC), identifying the social profile and listing the actions of

This study was therefore designed to evaluate and provide data on the prevalence of some of the major intestinal helminth infections reported at The Komfo Anokye

tomical varia ti ons (Table I), of which 79 specimens showed variations in the anterior (Fig. 2) and posterior portion of the circle of Willis (Fig. 3); and a rare finding

Spatial Interpolation of the Determination Coefficients (R2) resulting from the Spatial Regression between Levels of Education (Independent Variable Q22), Paid Work (Independent

The clinical data included the diagnosis of patients, dosages of paliperidone and risperidone, history of menstrual irregularities, galactorrhea, sexual dysfunction and