A preconditioning method of the CQ algorithm for solving an extended split feasibility problem
© Wang and Zhou; licensee Springer. 2014
Received: 27 October 2013
Accepted: 3 April 2014
Published: 6 May 2014
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.
Keywordspreconditioning method extended split feasibility problem relaxed method approximate variable preconditioner
The problem to find with , if such x exists, was called the split feasibility problem (SFP) by Censor and Elfving , where and 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 , as well as others obtained later [4, 5] involve matrix inverses at each step. Byrne  presented a method called the CQ algorithm for solving the SFP that does not involve matrix inverses.
where and L denotes the largest eigenvalue of the matrix , I is the identical matrix. and 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  presented a relaxed CQ algorithm for solving the SFP, then the orthogonal projections onto halfspaces and can be executed exactly. Qu and Xiu  proposed the modified relaxed algorithm which does not need to compute the largest eigenvalue of the matrix and can get an adaptive stepsize by adopting an Armijo-like search. The paper  extended the algorithm in  and proposed a relaxed inexact projection method for the SFP. Xu  extended the problem into infinite-dimensional Hilbert spaces, and modified the CQ algorithm with Mann’s iteration. In , 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.
Our argument mainly depends on monotone operators, nonexpansive mappings, and the metric projections.
Definition 2.1 
- (i)T is said to be monotone on C, if
- (ii)a mapping is nonexpansive if
We denote by the set of fixed points of T; that is, . Note that is always closed and convex (but maybe empty).
where is the 2-norm.
The following properties of projections are useful and pertinent to our purpose.
Lemma 2.1 
Lemma 2.3 
U is an averaged operator; i.e. there exist some and a nonexpansive operator V, .
, then .
Proposition 2.1 
3 The preconditioning CQ algorithm
Stand  and Piana and Bertero  have applied the preconditioning matrix technologies to improve the Landweber and projected Landweber algorithms. The analyses deal with the operators A and , 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.
Now we present a new algorithm, which is named a preconditioning CQ algorithm (PCQ).
where , .
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 and , respectively, and A is an M by N matrix, D is an N by N symmetrical positive definite matrix, the ESFP is to find with . We denote the solution set of ESFP by G.
Remark 3.1 If we set , then , the problem in Definition 3.1 is transformed into SFP.
Remark 3.2 If we set D is an unit matrix, then to find with , the problem in Definition 3.1 is transformed into SFP.
The following immediately follows.
In order to establish the convergence of Algorithm 3.1, we need the following theorem.
Proof As , , we have .
Secondly, we prove .
As , we choose , then and , so .
then , and .
We obtain , thus (i) is proved.
We can also deduce that . □
Theorem 3.3 Assume , , , the sequence is generated by (15), there exists .
and V is nonexpansive, hence, while , is an averaged nonexpansive operator.
which implies that is monotonically decreasing and hence . Specially, is bounded.
as , we have .
Let be an arbitrary cluster point of the sequence . Then there exists a subsequence , then (). As , , and . Because V is nonexpansive and continuous, then ().
As , we have , then . Therefore, , from Theorem 3.2, we have . However, exists, and there exists a subsequence of s.t. (), therefore, there must be (). □
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 , we firstly obtain the relaxed projection method.
where , .
Theorem 4.1 Let be a sequence generated by the relaxed preconditioning CQ algorithm. Then converges to a solution of ESFP.
and we can obtain an adaptive algorithm with strong convergence.
where is a real sequence in that satisfies conditions (C1) ; and (C2) .
Lemma 4.1 For all , is Lipschitz continuous on with constant L and co-coercive on with modulus , where L is the largest eigenvalue of the matrix . Therefore, the Armijo-like search rule (22) is well defined.
Lemma 4.2 For all , .
Theorem 4.2 Let be a sequence generated by Algorithm 4.2. If the solution set of the SFP is nonempty, then 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.
where , L is the largest eigenvalue of , is the minimum value of all the largest eigenvalues of , for . Specially, set .
5 Approximating a variable preconditioner
Assume is existed, the product is without restrictions in (10), we can deduced that sometimes D should be chose close to as more as possible.
The best condition is to calculate exactly, but it is always hard with the signal and image reconstruction problems. If F be a polynomial function, Stand  have provided an example with seventh-order, and Neumann’s series approach can also express . However, the polynomial method needs to calculate the high order matrix multiplication, therefore, it can not be implemented easily.
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.
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 . For Examples 6.2 and 6.3, we should first transform into the ESFP. The stopping criterion is , and we took , , . The projections are computed by Proposition 2.1.
Numerical results for Example 6.1
The method (26)
The method (30)
Numerical results for Example 6.2
The method (26)
The method (30)
Numerical results for Example 6.3
The method (26)
The method (30)
Example 6.1 (A convex feasibility problem, CFP)
Let , . Find some point x in .
Example 6.2 (A split feasibility problem)
Let , , . Find with .
Example 6.3 (A split feasibility problem)
Let , , . Find with .
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 . 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.
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).
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