- Research
- Open Access

# A preconditioning method of the CQ algorithm for solving an extended split feasibility problem

- Peiyuan Wang
^{1}Email author and - Haiyun Zhou
^{1, 2}

**2014**:163

https://doi.org/10.1186/1029-242X-2014-163

© Wang and Zhou; licensee Springer. 2014

**Received: **27 October 2013

**Accepted: **3 April 2014

**Published: **6 May 2014

## Abstract

In virtue of preconditioning technology, we propose a preconditioning CQ algorithm for an extend split feasibility problem (ESFP). Comparing with the others, the proposed algorithm can get faster convergence without considering to adjust the stepsize. The convergence is also established under mild conditions. Several extensions of the preconditioning CQ algorithm are presented. Moreover, we present an approximate variable preconditioner which does not compute the matrix inverse. Finally, some numerical experiments show the better behaviors of the proposed methods.

**MSC:**47J10, 47J20, 65B05.

## Keywords

## 1 Introduction

The problem to find $x\in C$ with $Ax\in Q$, if such *x* exists, was called the split feasibility problem (SFP) by Censor and Elfving [1], where $C\in {R}^{N}$ and $Q\in {R}^{M}$ are nonempty closed convex sets, and *A* is an *M* by *N* matrix. This problem plays an important role in the study of signal processing, image reconstruction, and so on [2, 3]. Censor and Elfving’s algorithm in [1], as well as others obtained later [4, 5] involve matrix inverses at each step. Byrne [6] presented a method called the CQ algorithm for solving the SFP that does not involve matrix inverses.

**The CQ algorithm**Let ${x}^{0}$ be arbitrary. For $k=0,1,\dots $ , calculate

where $\gamma \in (0,2/L)$ and *L* denotes the largest eigenvalue of the matrix ${A}^{T}A$, *I* is the identical matrix. ${P}_{C}$ and ${P}_{Q}$ are the orthogonal projections onto *C* and *Q*, respectively.

In recent years, how to modify the CQ algorithm so that it can easily be implemented and converge faster is the hot topic. The typical modifications are as follows: Yang [7] presented a relaxed CQ algorithm for solving the SFP, then the orthogonal projections onto halfspaces ${C}_{k}$ and ${Q}_{k}$ can be executed exactly. Qu and Xiu [8] proposed the modified relaxed algorithm which does not need to compute the largest eigenvalue of the matrix ${A}^{T}A$ and can get an adaptive stepsize by adopting an Armijo-like search. The paper [9] extended the algorithm in [10] and proposed a relaxed inexact projection method for the SFP. Xu [11] extended the problem into infinite-dimensional Hilbert spaces, and modified the CQ algorithm with Mann’s iteration. In [12], López *et al.* presented a variable stepsize, and improved the algorithm with a Halpern-type iteration.

However, using preconditioning technology to accelerate the CQ algorithm not only has not been taken into account, but also one will obtain a special effect. In this paper, we consider to modify the CQ algorithm from the views of fixed point and variational inequality. Combining with the appropriate preconditioner, the SFP can be transformed into an extended split feasibility problem (ESFP). Naturally, a preconditioning CQ algorithm for solving the ESFP can also solve the SFP indirectly.

The rest of the paper is organized as follows. In Section 2, we review some concepts and existing results. In Section 3, we propose a preconditioning CQ algorithm for solving ESFP and establish its convergence. Several extensions are presented in Section 4. In Section 5, we discuses the methods how to estimate the approximate inverse preconditioner. In Section 6, we report some computational results with the proposed algorithm and methods. Finally, Section 7 gives some concluding remarks.

## 2 Preliminaries

Our argument mainly depends on monotone operators, nonexpansive mappings, and the metric projections.

**Definition 2.1** [13]

*T*be a mapping from a set $C\subset {R}^{N}$ into itself. Then

- (i)
*T*is said to be monotone on*C*, if$\u3008Tx-Ty,x-y\u3009\ge 0,\phantom{\rule{1em}{0ex}}\text{for all}x,y\in C,$ - (ii)a mapping $T:C\to C$ is nonexpansive if$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\text{for all}x,y\in C.$

We denote by $Fix(T)$ the set of fixed points of *T*; that is, $Fix(T)=\{x\in C:Tx=x\}$. Note that $Fix(T)$ is always closed and convex (but maybe empty).

*C*is the mapping ${P}_{C}:{R}^{N}\to C$, which assigns to each point $x\in C$ the unique point ${P}_{C}x\in C$ satisfying the property

where $\parallel \cdot \parallel $ is the 2-norm.

The following properties of projections are useful and pertinent to our purpose.

**Lemma 2.1** [13]

*Given*$x\in {R}^{N}$

*Consequently*, ${P}_{C}$

*is nonexpansive and monotone*,

*and*$I-{P}_{C}$

*is also nonexpansive*,

*then*

**Lemma 2.2**

*For*$\mathrm{\forall}x,y\in {R}^{N}$

**Lemma 2.3** [14]

*Let*$U=I-\gamma {A}^{T}(I-{P}_{Q})A$,

*where*$\gamma \in (0,2/L)$.

- (i)
*U**is an averaged operator*;*i*.*e*.*there exist some*$\beta \in (0,1)$*and a nonexpansive operator**V*, $U=(1-\beta )U+\beta V$. - (ii)
$Fix(U)={A}^{-1}(Q)$,

*then*$Fix({P}_{C}U)=C\cap {A}^{-1}(Q)$.

**Proposition 2.1** [15]

*For every*$k\ge 0$,

*let*${x}^{k}\in {R}^{N}$, ${C}_{k}$

*and*${Q}_{k}$

*be defined as in*[7].

*Then for any*$x\in {R}^{N}$

*and*$y\in {R}^{M}$

*we have*

*and*

## 3 The preconditioning CQ algorithm

Stand [16] and Piana and Bertero [17] have applied the preconditioning matrix technologies to improve the Landweber and projected Landweber algorithms. The analyses deal with the operators *A* and ${A}^{\ast}A$, which are based on the singular value decomposition and a more general spectrum, respectively. We can also extend the technologies to improve the CQ algorithm.

*i.e.*the SFP has a nonempty solution set, and ${x}^{\ast}$ is the solution of SFP. Thus, we have

*U*in Lemma 2.3, and we can obtain

Now we present a new algorithm, which is named a preconditioning CQ algorithm (PCQ).

**Algorithm 3.1**Let $D:C\to C$ be a $N\times N$ symmetrical positive definite matrix, ${x}^{0}\in C$ be arbitrary. For $k=0,1,\dots $ , calculate

where $\gamma \in (0,2/L)$, $L={\parallel D{A}^{T}\parallel}^{2}$.

Algorithm 3.1 is to solve an extended SFP (ESFP), which can be represented as follows.

**Definition 3.1** Let *C* and *Q* be nonempty closed convex sets in ${R}^{N}$ and ${R}^{M}$, respectively, and *A* is an *M* by *N* matrix, *D* is an *N* by *N* symmetrical positive definite matrix, the ESFP is to find $x\in C$ with $ADx\in Q$. We denote the solution set of ESFP by *G*.

**Remark 3.1** If we set $\tilde{x}=Dx\in D(C)=\tilde{C}$, then $A\tilde{x}\in Q$, the problem in Definition 3.1 is transformed into SFP.

**Remark 3.2** If we set *D* is an unit matrix, then to find $x\in C$ with $Ax\in Q$, the problem in Definition 3.1 is transformed into SFP.

The following immediately follows.

**Theorem 3.1**

*Assume*$G\ne \mathrm{\varnothing}$,

*then*${x}^{\ast}\in G$,

*if and only if*${x}^{\ast}=argmin\{f(x)|x\in C\}$,

*if and only if*$\u3008\mathrm{\nabla}f({x}^{\ast}),x-{x}^{\ast}\u3009\ge 0$, $\mathrm{\forall}x\in C$,

*where*

In order to establish the convergence of Algorithm 3.1, we need the following theorem.

**Theorem 3.2**

*Assume that*$G=C\cap {(AD)}^{-1}(Q)\ne \mathrm{\varnothing}$,

*then*

- (i)
$Fix({U}_{D})={(AD)}^{-1}(Q)=\{x\in {R}^{N}|ADx\in Q\}$;

- (ii)
$Fix({P}_{C}{U}_{D})=G$.

*Proof* As ${D}^{T}=D$, ${(AD)}^{T}=D{A}^{T}$, we have ${U}_{D}=I-\gamma {(AD)}^{T}(I-{P}_{Q})(AD)$.

Therefore, $x\in Fix({U}_{D})$.

Secondly, we prove $Fix({U}_{D})\subset {(AD)}^{-1}(Q)$.

As $G=C\cap {(AD)}^{-1}(Q)\ne \mathrm{\varnothing}$, we choose $z\in G$, then $z\in C$ and $z\in {(AD)}^{-1}(Q)$, so $(AD)z\in Q$.

then $(I-{P}_{Q})ADx=0$, and $ADx={P}_{Q}(AD)x\in Q$.

We obtain $x\in {(AD)}^{-1}(Q)$, thus (i) is proved.

We can also deduce that $Fix({P}_{C}{U}_{D})=Fix({P}_{C})\cap Fix({U}_{D})\triangleq G=C\cap {(AD)}^{-1}(Q)\ne \mathrm{\varnothing}$. □

**Theorem 3.3** *Assume* $G\ne \mathrm{\varnothing}$, $0<\gamma <2/L$, $L={\parallel D{A}^{T}\parallel}^{2}$, *the sequence* $\{{x}^{k}\}$ *is generated by* (15), *there exists* ${lim}_{k\to \mathrm{\infty}}{x}^{k}\to {x}^{\ast}\in G$.

*Proof*Firstly, we show that if $\gamma =2/L$, the operator

is nonexpansive.

and *V* is nonexpansive, hence, while $0<\gamma <2/L$, ${U}_{D}$ is an averaged nonexpansive operator.

which implies that $\{{\parallel {x}^{k}-p\parallel}^{2}\}$ is monotonically decreasing and hence ${lim}_{k\to \mathrm{\infty}}{\parallel p-{x}^{k}\parallel}^{2}=d\ge 0$. Specially, $\{{x}^{k}\}$ is bounded.

as ${\parallel {x}^{k}-p\parallel}^{2}-{\parallel {x}^{k+1}-p\parallel}^{2}\to 0$, we have ${x}^{k}-V{x}^{k}\to 0$.

Let ${x}^{\ast}$ be an arbitrary cluster point of the sequence $\{{x}^{k}\}$. Then there exists a subsequence $\{{x}^{{k}_{j}}\}\subset \{{x}^{k}\}$, then ${x}^{{k}_{j}}\to {x}^{\ast}$ ($j\to \mathrm{\infty}$). As $\{{x}^{k}\}\subset C$, ${x}^{\ast}\in C$, and ${x}^{\ast}={P}_{C}{x}^{\ast}$. Because *V* is nonexpansive and continuous, then $V{x}^{{k}_{j}}\to V{x}^{\ast}$ ($j\to \mathrm{\infty}$).

As $\parallel {x}^{\ast}-V{x}^{\ast}\parallel \le \parallel {x}^{\ast}-{x}^{{k}_{j}}\parallel +\parallel {x}^{{k}_{j}}-V{x}^{{k}_{j}}\parallel +\parallel V{x}^{{k}_{j}}-V{x}^{\ast}\parallel \to 0$, we have ${x}^{\ast}=V{x}^{\ast}$, then ${U}_{D}{x}^{\ast}=(1-\beta ){x}^{\ast}+\beta V{x}^{\ast}={x}^{\ast}$. Therefore, ${x}^{\ast}={P}_{C}{x}^{\ast}={P}_{C}{U}_{D}{x}^{\ast}$, from Theorem 3.2, we have ${x}^{\ast}\in G$. However, ${lim}_{n\to \mathrm{\infty}}\parallel {x}^{k}-{x}^{\ast}\parallel =d\ge 0$ exists, and there exists a subsequence $\{{x}^{{k}_{j}}\}$ of $\{{x}^{k}\}$ s.t. ${x}^{{k}_{j}}\to {x}^{\ast}$ ($j\to \mathrm{\infty}$), therefore, there must be ${x}^{k}\to {x}^{\ast}$ ($j\to \mathrm{\infty}$). □

## 4 Several extensions of the preconditioning CQ algorithm

In virtue of kinds of CQ-like algorithms for solving the SFP, we can also deduce the following meaningful results for solving the ESFP without proof.

According to the relaxed CQ algorithm [7], we firstly obtain the relaxed projection method.

**Algorithm 4.1**Let $D:{C}_{k}\to {C}_{k}$ be a $N\times N$ symmetrical positive definite matrix, ${x}^{0}$ be arbitrary. For $k=0,1,\dots $ , calculate

where $\gamma \in (0,2/L)$, $L={\parallel D{A}^{T}\parallel}^{2}$.

**Theorem 4.1** *Let* $\{{x}^{k}\}$ *be a sequence generated by the relaxed preconditioning CQ algorithm*. *Then* $\{{x}^{k}\}$ *converges to a solution of ESFP*.

and we can obtain an adaptive algorithm with strong convergence.

**Algorithm 4.2**Let $D:{C}_{k}\to {C}_{k}$ be a $N\times N$ symmetrical positive definite matrix, given constants $\lambda >0$, $l\in (0,1)$, $\mu \in (0,1)$. Let ${x}^{0}$ be arbitrary, for $k=0,1,\dots $ , let

*m*such that

where $\{{\alpha}_{k}\}$ is a real sequence in $(0,1)$ that satisfies conditions (C_{1}) ${lim}_{k\to \mathrm{\infty}}{\alpha}_{k}=0$; and (C_{2}) ${\sum}_{k=1}^{\mathrm{\infty}}{\alpha}_{k}=\mathrm{\infty}$.

**Lemma 4.1** *For all* $k=0,1,\dots $ , $\mathrm{\nabla}{f}_{k}$ *is Lipschitz continuous on* ${R}^{N}$ *with constant* *L* *and co*-*coercive on* ${R}^{N}$ *with modulus* $1/L$, *where* *L* *is the largest eigenvalue of the matrix* ${A}^{T}A$. *Therefore*, *the Armijo*-*like search rule* (22) *is well defined*.

**Lemma 4.2** *For all* $k=0,1,\dots $ , $\frac{\mu l}{L}<{\rho}_{k}\le \gamma $.

**Theorem 4.2** *Let* $\{{x}^{k}\}$ *be a sequence generated by Algorithm* 4.2. *If the solution set of the SFP is nonempty*, *then* $\{{x}^{k}\}$ *converges strongly to a solution of the ESFP*.

As there exists an Armijo-like search step in Algorithm 4.2, the complexity of the implementation will be increased. Next, we propose a new variable stepsize to improve Algorithm 3.1.

**Algorithm 4.3**Let ${D}^{k}:C\to C$ be a variable $N\times N$ symmetrical positive definite matrix, $k=0,1,\dots $ . For $\mathrm{\forall}{x}^{0}\in C$, calculate

where ${\gamma}^{k}\in (0,2/(L\ast {M}_{D}))$, *L* is the largest eigenvalue of ${A}^{T}A$, ${M}_{D}$ is the minimum value of all the largest eigenvalues of ${D}^{k}$, for $k=0,1,\dots $ . Specially, set ${\gamma}^{k}=1/(L\ast {M}_{D})$.

## 5 Approximating a variable preconditioner

*D*is continuous, positive definite and bounded so that it has a continuous inverse [17]. According to the preconditioning CQ algorithm, we set

*D*commutes with the operator ${A}^{T}A$. Therefore, we set a matrix function

*F*with positive value, and its dimension should be consistent with $dim({A}^{T}A)$. Moreover,

*F*should have a positive lower bound in order to satisfy the existing inverse. Strand [16] also assumes that

*F*is a polynomial or a rational function. Thus, the operator

*D*is given by

Assume ${({A}^{T}A)}^{-1}$ is existed, the product $D{A}^{T}AD$ is without restrictions in (10), we can deduced that sometimes *D* should be chose close to ${({A}^{T}A)}^{-1/2}$ as more as possible.

The best condition is to calculate ${({A}^{T}A)}^{-1}$ exactly, but it is always hard with the signal and image reconstruction problems. If *F* be a polynomial function, Stand [16] have provided an example with seventh-order, and Neumann’s series approach can also express ${({A}^{T}A)}^{-1}$. However, the polynomial method needs to calculate the high order matrix multiplication, therefore, it can not be implemented easily.

*F*be a rational function, a simple example, closely related to the Tikhonov regularization method has been used in [17]. According to the example, the approximate inverse preconditioner

*D*can be given by

where Re denotes the real part, *α* is a positive real parameter, and good choices of *α* may be much smaller than the values provided by the methods used for estimating optimum values of the Tikhonov regularization parameter.

As (26) involves the matrix inverse, we next propose a diagonal format of *D* that does not calculate matrix inverses. Furthermore, the choice of *D* is related to the convergence properties of the algorithm. If *D* is evolutive following the iterations, the convergence rate of algorithm will also be accelerated.

*D*should be chosen closely to ${({A}^{T}A)}^{-1/2}$, we assume

*λ*is the approximate eigenvalue matrix of ${A}^{T}A$, and then we set

*k*th iteration, we have ${M}_{{D}^{k}}=min\{{L}_{{D}^{n}}|n=0,1,\dots ,k\}$. Then set ${l}_{{\overline{D}}^{k}}$, the minimum eigenvalue of ${\overline{D}}^{k}$, $k=0,1,\dots $ , on the

*k*th iteration, we have ${L}_{k}=max\{1/{l}_{{\overline{D}}^{n}}|n=0,1,\dots ,k\}$. Therefore, a variable stepsize with respect to Algorithm 4.3 can be approximated by

## 6 Numerical results

In this section, we present some numerical results for the proposed method. The following three examples are taken from the test problems in [15]. For Examples 6.2 and 6.3, we should first transform into the ESFP. The stopping criterion is $\parallel {x}^{k+1}-{x}^{k}\parallel <\epsilon $, and we took $\epsilon ={10}^{-10}$, $\gamma =1/{\parallel D{A}^{T}\parallel}^{2}$, $\alpha =0.1$. The projections are computed by Proposition 2.1.

**Numerical results for Example 6.1**

Starting point | The method (26) | The method (30) | ||||
---|---|---|---|---|---|---|

k | CPU (s) | Approximate solution | k | CPU (s) | Approximate solution | |

${(1,2,3)}^{\prime}$ | 6 | 0.0156 | ${(1.4000,1.1864,1.6101)}^{\prime}$ | 6 | 0.0262 | ${(1.1798,1.1380,1.6447)}^{\prime}$ |

${(1,1,1)}^{\prime}$ | 1 | 0.0149 | ${(1,1,1)}^{\prime}$ | 1 | 0.0252 | ${(1,1,1)}^{\prime}$ |

$rand{(3,1)}^{\prime}\ast 10$ | 7 | 0.0153 | ${(9.5717,1.0372,1.7101)}^{\prime}$ | 7 | 0.0270 | ${(9.5717,1.0372,1.7101)}^{\prime}$ |

**Numerical results for Example 6.2**

Starting point | The method (26) | The method (30) | ||||
---|---|---|---|---|---|---|

k | CPU (s) | Approximate solution | k | CPU (s) | Approximate solution | |

${(1,2,3)}^{\prime}$ | 19 | 0.0166 | ${(-0.3097,-0.1638,0.1067)}^{\prime}$ | 5 | 0.0264 | ${(-0.6409,-0.0027,0.3205)}^{\prime}$ |

${(1,1,1)}^{\prime}$ | 5 | 0.0155 | ${(0.3988,0.0763,-0.2023)}^{\prime}$ | 10 | 0.0271 | ${(0.1828,0.0493,-0.1644)}^{\prime}$ |

$rand{(3,1)}^{\prime}\ast 10$ | 7 | 0.0159 | ${(6.0095,-0.0363,-3.0054)}^{\prime}$ | 7 | 0.0280 | ${(3.7284,0.4046,-2.6730)}^{\prime}$ |

**Numerical results for Example 6.3**

Starting point | The method (26) | The method (30) | ||||
---|---|---|---|---|---|---|

k | CPU (s) | Approximate solution | k | CPU (s) | Approximate solution | |

${(1,2,3)}^{\prime}$ | 6 | 0.0156 | ${(1.4000,1.1864,1.6101)}^{\prime}$ | 8 | 0.0274 | ${(-0.0500,0.0480,0.0234)}^{\prime}$ |

${(1,1,1)}^{\prime}$ | 1 | 0.0149 | ${(1,1,1)}^{\prime}$ | 4 | 0.0269 | ${(0.3412,0.0403,-0.1714)}^{\prime}$ |

$rand{(3,1)}^{\prime}\ast 10$ | 7 | 0.0153 | ${(9.5717,1.0372,1.7101)}^{\prime}$ | 7 | 0.0273 | ${(0.9290,-0.2692,-0.5000)}^{\prime}$ |

**Example 6.1** (A convex feasibility problem, CFP)

Let $C=\{x\in {R}^{3}|{x}_{2}^{2}-{x}_{3}^{2}-4\le 0\}$, $Q=\{y\in {R}^{3}|{y}_{3}-1-{y}_{1}^{2}\le 0\}$. Find some point *x* in $C\cap Q$.

**Example 6.2** (A split feasibility problem)

Let $A=\left(\begin{array}{ccc}2& -1& 3\\ 4& 2& 5\\ 2& 0& 2\end{array}\right)$, $C=\{x\in {R}^{3}|{x}_{1}+{x}_{2}^{2}+2{x}_{3}\le 0\}$, $Q=\{y\in {R}^{3}|{y}_{1}^{2}+{y}_{2}-{y}_{3}\le 0\}$. Find $x\in C$ with $Ax\in Q$.

**Example 6.3** (A split feasibility problem)

Let $A=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 2& 0& 1\end{array}\right)$, $C=\{x\in {R}^{3}|{x}_{1}+{x}_{2}^{2}+2{x}_{3}\le 0\}$, $Q=\{y\in {R}^{3}|{y}_{1}^{2}+{y}_{2}-{y}_{3}\le 0\}$. Find $x\in C$ with $Ax\in Q$.

## 7 Conclusions

In this paper, by adopting the preconditioning techniques, a modified CQ algorithm is named preconditioning CQ algorithm, and its extensions for solving the ESFP have been presented. The approximate methods for how to estimate the preconditioner *D* are also discussed; the approximate diagonal preconditioner method does not need to compute the matrix inverses and the largest eigenvalue of the matrix ${A}^{T}A$. Thus, the algorithm can be implemented easily. Moreover, the corresponding convergence property has been established in the feasible case of ESFP. The numerical results showed that the proposed algorithms and methods are effective to solve some problems.

## Declarations

### Acknowledgements

The authors would like to thank the associate editor and the referees for their comments and suggestions. This research was supported by the National Natural Science Foundation of China (11071053).

## Authors’ Affiliations

## References

- Censor Y, Elfving T:
**A multiprojection algorithm using Bregman projections in a product space.***Numer. Algorithms*1994,**8:**221-239. 10.1007/BF02142692MATHMathSciNetView ArticleGoogle Scholar - Bauschke HH, Borwein JM:
**On projection algorithms for solving convex feasibility problems.***SIAM Rev.*1996,**38:**367-426. 10.1137/S0036144593251710MATHMathSciNetView ArticleGoogle Scholar - Byrne CL:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Probl.*2004,**20:**103-120. 10.1088/0266-5611/20/1/006MATHMathSciNetView ArticleGoogle Scholar - Byrne CL:
**Iterative projection onto convex sets using multiple Bregman distances.***Inverse Probl.*1999,**15:**1295-1313. 10.1088/0266-5611/15/5/313MATHMathSciNetView ArticleGoogle Scholar - Byrne CL:
**Bregman-Legendre multidistances projection algorithm for convex feasibility and optimization.**In*Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications*. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, Amsterdam; 2001:87-100.View ArticleGoogle Scholar - Byrne CL:
**Iterative oblique projection onto convex sets and the split feasibility problem.***Inverse Probl.*2002,**18:**441-453. 10.1088/0266-5611/18/2/310MATHMathSciNetView ArticleGoogle Scholar - Yang QZ:
**The relaxed CQ algorithm solving the split feasibility problem.***Inverse Probl.*2004,**20:**1261-1266. 10.1088/0266-5611/20/4/014MATHView ArticleGoogle Scholar - Qu B, Xiu N:
**A note on the CQ algorithm for the split feasibility problem.***Inverse Probl.*2005,**21:**1655-1665. 10.1088/0266-5611/21/5/009MATHMathSciNetView ArticleGoogle Scholar - Wang Z, Yang Q:
**The relaxed inexact projection methods for the split feasibility problem.***Appl. Math. Comput.*2011,**217:**5347-5359. 10.1016/j.amc.2010.11.058MATHMathSciNetView ArticleGoogle Scholar - Yang Q, Zhao J:
**The projection-type methods for solving the split feasibility problem.***Math. Numer. Sin.*2006,**28:**121-132. (in Chinese)MathSciNetGoogle Scholar - Xu HK:
**Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces.***Inverse Probl.*2010,**26:**1-17.MATHView ArticleGoogle Scholar - López G, Martín-Márquez V, Wang F, Xu H-K:
**Solving the split feasibility problem without prior knowledge of matrix norms.***Inverse Probl.*2012,**28:**1-18.View ArticleGoogle Scholar - Zarantonello EH:
**Projections on convex sets in Hilbert space and spectral theory.**In*Contributions to Nonlinear Functional Analysis*. Edited by: Zarantonello EH. Academic Press, New York; 1971.Google Scholar - Wang F, Xu HK:
**Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem.***J. Inequal. Appl.*2010.,**2010:**Article ID 102085Google Scholar - Abdellah B, Muhammad AN:
**On descent-projection method for solving the split feasibility problems.***J. Glob. Optim.*2012,**54:**627-639. 10.1007/s10898-011-9782-2MATHView ArticleGoogle Scholar - Strand ON:
**Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind.***SIAM J. Numer. Anal.*1974,**11:**798-825. 10.1137/0711066MATHMathSciNetView ArticleGoogle Scholar - Piana M, Bertero M:
**Projected Landweber method and preconditioning.***Inverse Probl.*1997,**13:**441-463. 10.1088/0266-5611/13/2/016MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.