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R E S E A R C H

Open Access

Iterative process for solving a multiple-set

split feasibility problem

Yazheng Dang

1,2*†

and Zhonghui Xue

3†

*Correspondence: jgdyz@163.com 1School of Management, University

of Shanghai for Science and Technology, Shanghai, 200093, China

2College of Computer Science and

Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China

Equal contributors

Full list of author information is available at the end of the article

Abstract

This paper deals with a variant relaxed CQ algorithm by using a new searching direction, which is not the gradient of a corresponding function. The strategy is to intend to improve the convergence. Its convergence is proved under some suitable conditions. Numerical results illustrate that our variant relaxed CQ algorithm converges more quickly than the existing algorithms.

MSC: 47H05; 47J05; 47J25

Keywords: multiple-set split feasibility problem; subgradient; accelerated iterative algorithm; convergence

1 Introduction

The multiple-set split feasibility problem (MSSFP) is to find a point contained in the in-tersection of a family of closed convex sets in one space such that its image under a linear transformation is contained in the intersection of another family of closed convex sets in the image space. Formally, given nonempty closed convex setsCiN,i= , , . . . ,t,

in theN-dimensional Euclidean spaceN and nonempty closed convex setsQjM, j= , , . . . ,r, and anM×Nreal matrixA, the MSSFP is to find a pointxsuch that

xC=

t

i=

Ci, AxQ= r

j=

Qj. (.)

Such MSSFP, formulated in [], arises in the field of intensity-modulated radiation therapy (IMRT) when one attempts to describe physical dose constrains and equivalent uniform dose (EUD) constraints within a single model, see [, ]. Specially, whent=r= , the problem reduces to the two-set split feasibility problem (abbreviated as SFP), which is to find a pointxCsuch thatAxQ(see [–]).

For solving the MSSFP, Censoret al.in [] introduced a proximity functionp(x) to mea-sure the aggregate distance of a point to all sets. The functionp(x) is defined as

p(x) := 

t

i=

αixPCi(x)

+ 

r

j=

βjAxPQj(Ax)

,

(2)

whereαi> ,βj>  for alliandj, respectively, and t

i=αi+ r

j=βj= . Then they

pro-posed a projection algorithm as follows:

xk+=Pxkγpxk,

where N is an auxiliary set, xk is the current iterative point.  <γ < /L with L=ti=αi+ρ(ATA)

r

j=βj andρ(ATA) is the spectral radius of ATA. Subsequently,

many methods have been developed for solving the MSSFP [–], while most of these algorithms aimed at minimizing the proximity functionp(x) and used its gradient∇p.

Different from most of the existing methods, in this paper, we construct a new searching direction, which is not the gradient∇p. And this difference causes a very different way of analysis. Moreover, some preliminary numerical experiments show that our new method converges faster than most existing methods.

The paper is organized as follows. Section  reviews some preliminaries. Section  gives a variant relaxed projection algorithm and shows its convergence. Section  gives some numerical experiments. Some conclusions are drawn in Section .

2 Preliminaries

Throughout the rest of the paper,Idenotes the identity operator,Fix(T) denotes the set of the fixed points of an operatorT,i.e.,Fix(T) :={x|x=T(x)}.

LetT be a mapping fromℵ ⊆ N intoN.T is called co-coercive onwith modulus

μ>  if

T(x) –T(y),xyμT(x) –T(y), ∀x,y∈ ℵ;

it is called Lipschitz continuous onℵfor constantL>  if

T(x) –T(y)≤Lxy, x,y∈ ℵ;

it is called monotone onℵif

T(x) –T(y),xy ≥, ∀x,y∈ ℵ.

It is obvious that the co-coercivity (with modulusμ) implies the Lipschitz continuity (with constant /μ) and monotonicity.

LetSbe a nonempty closed convex subset ofN. Denote byP

Sthe orthogonal projection

ontoS; that is,

PS(x) =arg min yNxy,

over allxS.

It is well known that the orthogonal projection operatorPS, for anyx,yN and any zS, is characterized by the inequalities []

(3)

and

PS(x) –z

xz–PS(x) –x

. (.)

Recall the notion of the subdifferential for an appropriate convex function.

Definition . Letf :N→ be convex. The subdifferential off atxis defined as

∂f(x) =ξN|f(y)≥f(x) +ξ,yx,∀yN. (.)

Evidently, an element of∂f(x) is said to be a subgradient.

Lemma .[] An operator T is co-coercive with modulusif and only if the operator IT is co-coercive with modulus,where I denotes the identity operator.

It is easy to see from the above lemmas that the orthogonal projection operators are monotone, co-coercive with modulus , and the operatorIPQis also co-coercive with

modulus .

3 Algorithm and its convergence

3.1 The variant relaxed-CQ algorithm

As in [], we suppose that the following conditions are satisfied:

() The solution set of the MSSFP is nonempty. () The setsCi,i= , , . . . ,t, are denoted as

Ci=

xN|ci(x)≤

, (.)

whereci:N→ ,i= , , . . . ,t, are appropriately convex andCi,i= , , . . . ,t, are nonempty.

The setQj,j= , , . . . ,r, is denoted as

Qj=

yM|qj(y)≤

, (.)

where qj :M → , j= , , . . . ,r, are appropriately convex and Qj, j = , , . . . ,r, are

nonempty.

() For anyxN, at least one subgradientξ

i∂ci(x)can be calculated.

For anyyM, at least one subgradientηj∂qj(y) can be computed.

Now, we define the following half-spaces at pointxk:

Cik=xN|ci

xk+ξik,xxk ≤, (.)

whereξikis an element in∂ci(xk) fori= , , . . . ,t, and

Qkj =yM|qj

Axk+ηkj,yAxk ≤, (.)

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By the definition of subgradient, it is clear that the half-spacesCik andQkj containCi

andQj,i= , , . . . ,r;j= , , . . . ,t, respectively. Due to the specific form ofCki andQkj,

the orthogonal projections ontoCk

i andQkj,i= , , . . . ,r;j= , , . . . ,t, may be computed

directly, see [].

Now, we give the variant relaxed CQ algorithm.

Algorithm . Givenαi>  andβj≥ such that t

i=αi= , r

j=βj= ,γ ∈(,ρ(ATA)),

tk∈(, ).

For an arbitrary initial point,xnis the current point. Define a mappingF

k:NN as

Fk(x) = r

j=

βjAT(IPQk

j)Ax. (.)

Fork= , , , . . . , compute

yk=

t

i=

αiPCik

xkγFk

xk. (.)

Let

dk=xkyk+γFk

ykFk

xk. (.)

Set

xk+=xktkdk. (.)

In this algorithm, we can takedk<εfor some given precision as the stopping

cri-terion. And we applyykandF

k to construct the searching directiondk. The choice of a

new searching direction leads to quite different in establishing the convergence result of Algorithm ..

By Lemma . in [], the operatorAT(IP

Qkj)Ais /ρ(ATA)-inverse strongly

mono-tone (/ρ(ATA)-ism) or co-coercive with modulus /ρ(ATA) and Lipschitz continuous

withρ(ATA).

3.2 Convergence of the variant relaxed-CQ algorithm

In this subsection, we establish the convergence of Algorithm ..

The following results will be needed in convergence analysis of the proposed algorithm.

Lemma .[, ] Suppose that f :N is convex.Then its subdifferential is uni-formly bounded on any bounded subsets ofN.

Lemma . Assume that z is an arbitrary solution of the MSSFP(i.e.,zSOL(MSSFP))

and uN,it holds that

Fk(u),uzr

j=

βj(IPQkj)(Au) 

(5)

Proof IfzSOL(MSSFP), thenAzQjQkj for allj= , . . . ,r, thus Fk(z) = , we have

known that the mappingsIPQk

j are co-coercive with modulus , it follows that

Fk(u),uz =

Fk(u) –Fk(z),uz

=

r

j=

βj

(AT(IPQk j)AuA

T(IP

Qkj)Az,uz

=

r

j=

βj

(IPQk

j)Au– (IPQkj)Az,AuAz

r

j=

βj(IPQkj)Au– (IPQkj)Az

=

r

j=

βj(IPQkj)Au

.

Now, we state the convergence of Algorithm ..

Theorem . Assume that the set of solutions of the constrained multiple-set split feasibil-ity problem is nonempty.Then any sequence{xk}

k=generated by Algorithm.converges to a solution of the multiple-set split feasibility problem.

Proof Letzbe a solution of MSSFP. SinceCiCi,k,QjQkj, thenz=PCiz=PCi,kzand

Az=PQjAz=PQj,kAzfor alliandjand thereforeFk(z) = . By Algorithm ., we have

xk+–z=xktkdkz

=xkz– tk

dk,xkz +tkdk,

hence

xk+z=xkz– t k

dk,ykz – t k

dk,xkyk +tkdk

. (.)

By (.) we have

dk,ykz =xkγFk

xkyk,ykz +γFk

yk,ykz. (.)

From Lemma ., we obtain

Fk

yk,ykzr

j=

βj(IPQkj)Ay k

≥. (.)

Letzk=xkγF

k(xk). For that t

i=αi= , we obtain from (.) that

xkγFk

xkyk,ykz =zkyk,ykz

=

t

i=

αi

zkPCk i

zk,

t

i=

αiPCk i

zkz

=

t

i= t

h=

αiαh

zkPCk i

zk,PCk

h

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Ifi=h, thenzkP

Cki(zk),PCkh(zk) –z ≥, sincezCiC k

i by Lemma .. Otherwise, if i=h, we have

αiαh

zkPCk i

zk,PCk

h

zkz +αhαi

zkPCk h

zk,PCk

i

zkz

=αiαh

zkPCk i

zk,PCk

i

zkz +zkPCk i

zk,PCk

h

zkPCk i

zk

+αhαi

zkPCk h

zk,PCk

h

zkz +zkPCk h

zk,PCk

i

zkPCk h

zk

αiαhPCki

zkP Ckh

zk.

It means

xkγ

kFk

xkyk,ykz i<h

αiαhPCki

zkP Chk

zk. (.)

By combining (.) and (.) with (.), we obtain

dk,ykzi<h

αiαhPCk i

zkPCk h

zk+γ

r

j=

βj(IPQk j)Ay

k

. (.)

On the other hand, by definition ofdkin (.), we have

dk,xkyk =dk,xkyk+γFk

ykγkFk

xk +γdk,Fk

xkFk

yk

=dk+γxkyk+γFk

ykγFk

xk,Fk

xkFk

yk

=dk+γxkyk,Fk

xkFk

ykγFk

xkFk

yk.

From Lemma ., we arrive atxkyk,F

k(xk) –Fk(yk) ≥/ρ(ATA)Fk(xk) –Fk(yk)for

allk, hence

dk,xkykdk+γγρATAxkyk,Fk

xkFk

yk

=dk+γ –γρATA r

j=

βj

AxkAyk,

(IPQk j)Ax

k– (IP Qkj)Ay

k.

Furthermore, from the -co-coercivity ofIPQk

j, we have

dk,xkykdk+γ –γρATA r

j=

βj(IPQkj)Ax

k– (IP Qkj)Ay

k

. (.)

From (.), (.) and (.), we have

xk+–z =xkz– tk

dk,ykz – tk

dk,xkyk +tkdk

,

(7)

– tkγ( –γρ

ATA r

j=

βj(IPQk j)Ax

k– (IP Qkj)Ay

k

– tk

i<h

αiαhPCki

zkPCk h

zk

+γ

r

j=

βj(IPQk j)Ay

k

. (.)

Sincetk∈(, ),γ ∈(,ρ(ATA)) in the algorithm, we conclude that the sequence{xk

z}is monotonously nonincreasing and convergent and{xk}is bounded. We have shown that the sequence {xkz}is monotonically decreasing and bounded, therefore there

exists the limit

lim k→∞

xkz=d, (.)

which combined with (.)-(.), (.) implies

lim k→∞d

k= lim k→∞x

kyk+γF k

ykFk

xk= , (.)

lim

k→∞(IPQkj)Ax

k– (IP Qkj)Ayk

= , ∀j, (.)

lim k→∞

PCk i

zkPCk h

zk= , ∀i=h, (.)

lim k→∞

(IPQk j)Ay

k= , j. (.)

Since the sequence{xk}is bounded, there exist a subsequence{xkl}of{xk}converging

to a pointx∗and a corresponding subsequence{Axkl}of{Axk}converging to a pointAx.

Now we will show that x∗∈SOL(MSFP), namely we will showlimkl→∞ci(x

kl) and

limkl→∞qi(x

kl) for alliandj.

First, sinceP QkljQ

kl

j , we have

qj

Axkl+ηkl

j ,PQkl

j

AxklAxkl.

We know from Lemma . that the subgradient sequence{ηjk}is bounded. By (.) we getP

Qklj (Ax

kl) –Axkl. Thus, we havelim

kl→∞qi(x

kl) for all andj.

Second, noting thatP CiklC

kl

i , we have

ci

xkl+ξkl

i ,PCkl

i

xklxkl.

Since{xk}is bounded, by Lemma . the sequence{ξk

i}is also bounded. Then all we need

is to show that P Ciklx

kl. We know from (.) and (.) that F

kl(y

kl) and

Fkl(xkl)→. It follows thatzkl =xklγFkl(xkl)→x∗, and then by (.),yklx∗.

Com-biningykl=t

i=αiPCkl i

(xklγF

kl(x

kl)) withF

kl(x

kl) andP

Ckli (x kl) –P

Cklh(x

(8)

i=hby (.), we conclude thatykl P

Ckli (x

kl) sincet

i=αi= . This leads to P

Cikl(x

kl) –xkl, and therebylim

kl→∞ci(x

kl) fori= , , . . . ,t.

Replacingzbyx∗in (.), we have

lim k→∞

xkx∗=d,

furthermore

lim k→∞

AxkAx∗=Ad,

on the other hand,

lim l→∞

xklx= lim

l→∞

AxklAx= .

Thus, limk→∞xkx∗=liml→∞AxkAx∗= . The proof of Theorem . is

com-plete.

4 Numerical experiments

In the numerical results listed in Tables  and , ‘Iter.’, ‘Sec.’ denote the number of iter-ations and the cpu time in seconds, respectively. We denote e= (, , . . . , )∈ N and e= (, , . . . , )∈ N. In the both numerical experiments, we take the weights /(r+t) for both Algorithm . and Censor’s algorithm. The stopping criterion isd<ε= –.

Example . The MSFP with

A=

⎡ ⎢ ⎢ ⎢ ⎣

 –   

    

    –

 –  – 

⎤ ⎥ ⎥ ⎥ ⎦;

C=x∈ |x+ x+x+x≤;

C=x∈ |x+ x+ x≤

and

Q=y∈ |y+y≤;

Q=y∈ |y+ y≤;

Q=

y∈ |y+ y≤

.

Consider the following three cases:

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[image:9.595.117.479.97.200.2]

Table 1 The numerical results of Example 4.1

Case Censor

γ= 1

Algo. 3.1

γ= 1 tk= 0.1

Censor

γ= 0.6

Algo. 3.1

γ= 0.6 tk= 0.1

Censor

γ= 1.8

Algo. 3.1

γ= 1.8 tk= 0.1

I Iter. = 1,051

Sec. = 1.043

Iter. = 146

Sec. = 0.401

Iter. = 1,867

Sec. = 1.480

Iter. = 224

Sec. = 0.334

Iter. = 832

Sec. = 0.700

Iter. = 89

Sec. = 0.062

II Iter. = 197

Sec. = 0.320

Iter. = 28

Sec. = 0.017

Iter. = 289

Sec. = 0.466

Iter. = 62

Sec. = 0.0751

Iter. = 87

Sec. = 0.068

Iter. = 9

Sec. = 0.010

III Iter. = 207

Sec. = 0.360

Iter. = 62

Sec. = 0.049

Iter. = 362

Sec. = 0.551

Iter. = 67

Sec. = 0.0728

Iter. = 139

Sec. = 0.217

Iter. = 17

Sec. = 0.020

Table 2 The numerical results of Example 4.2

N t,r Censor

γ= 1

Algo. 3.1

γ= 1 tk= 0.01

Censor

γ= 0.8

Algo. 3.1

γ= 0.8 tk= 0.01

Censor

γ= 1.6

Algo. 3.1

γ= 1.6 tk= 0.01

N= 20 t= 5

r= 5

Iter. = 181

Sec. = 0.268

Iter. = 16

Sec. = 0.021

Iter. = 288

Sec. = 0.499

Iter. = 20

Sec. = 0.022

Iter. = 147

Sec. = 0.213

Iter. = 9

Sec. = 0.017

N= 40 t= 10

r= 15

Iter. = 1,012

Sec. = 1.032

Iter. = 39

Sec. = 0.048

Iter. = 2,320

Sec. = 2.122

Iter. = 57

Sec. = 0.059

Iter. = 893

Sec. = 0.795

Iter. = 19

Sec. = 0.031

Example .[] In this example, because the step is related toρ(ATA), for easy control of the spectral radius, we take diagonal matricesAandaii∈(, ) generated randomly

Ci=

xN| xdi≤ri

, i= , , . . . ,t;

Qj=

xN |LjyUj

, j= , , . . . ,r;

wherediis thecenter of the ballCi,edi≤e, andri∈(, ) is the radius,diandri

are all generated randomly.LjandUjare the boundary of the boxQjand are also generated

randomly, satisfying eLj≤e, eUj≤e. In this test, we takeeas the

initial point.

In Tables -, the results showed that for most of the initial point, the number of iter-ative steps and the CPU time of Algorithm . are obviously less than those of Censoret al.’s algorithm. Moreover, when we takeN= ,, the number of iteration steps of Algo-rithm . is only hundreds of times. The numerical results also show that for large scale problems Algorithm . converges faster than Censor’s algorithm.

5 Conclusion

The multiple-set split feasibility problem arises in many practical applications in the real world. This paper constructed a new searching direction, which is not the gradient of a corresponding function. This different direction results in a very different way of analysis. And preliminary numerical results show that our new method converges faster, and this becomes more obvious while the dimension is increasing. Finally, the theoretically analysis is based on the assumption that the solution set of the MSSFP is nonempty.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

[image:9.595.119.479.235.314.2]
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Author details

1School of Management, University of Shanghai for Science and Technology, Shanghai, 200093, China.2College of

Computer Science and Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China. 3School of Physics and Chemistry, Henan Polytechnic University, Shiji Road, Jiaozuo, Henan 454000, China.

Acknowledgements

This work was supported by Natural Science Foundation of Shanghai (14ZR1429200), National Science Foundation of China (11171221), National Natural Science Foundation of China (61403255), Shanghai Leading Academic Discipline Project under Grant XTKX2012, Innovation Program of Shanghai Municipal Education Commission under Grant 14YZ094, Doctoral Program Foundation of Institutions of Higher Education of China under Grant 20123120110004, Doctoral Starting Projection of the University of Shanghai for Science and Technology under Grant ID-10-303-002, and Young Teacher Training Projection Program of Shanghai for Science and Technology.

Received: 2 November 2014 Accepted: 26 January 2015

References

1. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems21, 2071-2084 (2005)

2. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms8, 221-239 (1994)

3. Censor, Y, Bortfel, D, Martin, B, Trofimov, A: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol.51, 2353-2365 (2006)

4. Byrne, C: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl.18, 441-453 (2002)

5. Dang, Y, Gao, Y: The strong convergence of a KM-CQ-like algorithm for split feasibility problem. Inverse Problems27, 015007 (2011)

6. Dang, Y, Gao, Y: A perturbed projection algorithm with inertial technique for split feasibility problem. J. Appl. Math. (2012). doi:10.1155/2012/207323

7. Dang, Y, Gao, Y: An extrapolated iterative algorithm for multiple-set split feasibility problem. Abstr. Appl. Anal.2012, Article ID 149508 (2012)

8. Yang, Q: The relaxed CQ algorithm solving the split feasibility problem. Inverse Problems20, 1261-1266 (2004) 9. Xu, H: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems22,

2021-2034 (2006)

10. Masad, E, Reich, S: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal.8, 367-371 (2007)

11. Censor, Y, Alexander, S: On string-averaging for sparse problems and on the split common fixed point problem. Contemp. Math.513, 125-142 (2010)

12. Censor, Y, Alexander, S: The split common fixed point problem for directed operators. J. Convex Anal.16, 587-600 (2009)

13. Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl.327, 1244-1256 (2007)

14. Gao, Y: Piecewise smooth Lyapunov function for a nonlinear dynamical system. J. Convex Anal.19, 1009-1016 (2012) 15. Bauschke, HH, Jonathan, MB: On projection algorithms for solving convex feasibility problems. SIAM Rev.38, 367-426

(1996)

16. Bauschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)

17. Byne, C: An unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl.20, 103-120 (2004)

Figure

Table 1 The numerical results of Example 4.1

References

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