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Iterative algorithm for solving the multiplesets split equality problem with split selfadaptive step size in Hilbert spaces
Journal of Inequalities and Applications volumeÂ 2016, ArticleÂ number:Â 34 (2016)
Abstract
The split equality problem is a generalization of the split feasibility problem, meanwhile it is a special case of multiplesets split equality problems. In this paper, we propose an iterative algorithm for solving the multiplesets split equality problem whose iterative step size is split selfadaptive. The advantage of the split selfadaptive step size is that it could be obtained directly from the iterative procedure without needing to have any information of the spectral norm of the related operators. Under suitable conditions, we establish the theoretical convergence of the algorithm proposed in Hilbert spaces, and several numerical results confirm the effectiveness of the algorithm proposed.
1 Introduction
There arise various linear inverse problems in phase retrieval, radiation therapy treatment, signal processing, and medical image reconstruction etc. Censor and Elfving [1] summarized one of these classes of problems and proposed a new concept which is called the split feasibility problem (SFP), and the SFP can be characterized mathematically as
where C, Q are closed, convex, and nonempty subsets of the Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively, and \(A:H_{1}\mapsto H_{2}\) is a bounded and linear operator.
For solving it, Byrne [2, 3] presented the wellknown CQalgorithm, inspired by the idea of iterative scheme of fixed point theory. It is worth noting that the step size of the CQalgorithm is fixed, depending upon the norm of the operator A. Later, Qu and Xiu [4] and Lopez et al. [5] revised the CQalgorithm by using the Armijolike search method and the selfadaptive step size, respectively. Both of the methods need not know the spectral norm of operator A. For more information as regards algorithms for solving the SFP, see [6, 7]. In 2005, Censor et al. [8] made an extension upon the form of SFP, replaced the convex set C with an intersection of a family of closed and convex sets, which is the original multiplesets split feasibility problem (MSSFP), and introduced its applications for inverse problems.
Then Moudafi [9] generalized the content of SFP, to one called the split equality problem (SEP), which can be characterized mathematically as
where C, Q are closed, convex, and nonempty subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively, and \(H_{3}\) is also a Hilbert space, \(A:H_{1}\mapsto H_{3}\), \(B:H_{2}\mapsto H_{3}\) are two bounded and linear operators. When \(B=I\), the SEP is just the SFP. Later, Byrne and Moudafi [10] presented the following algorithms to solve it.
Alternating CQalgorithm:
Relaxed alternating CQalgorithm:
We know, in order to make the sequences generated above convergent, the values of step sizes \(\gamma_{k}\) depend upon the norms of operators A, B. Then Shi et al. [11] improved Moudafiâ€™s algorithms and obtained a strong convergent result:
But the defect is the same as Moudafiâ€™s: that the Î³ above also relies upon the norms of operators A, B. For more information as regards methods solving the split equality problem, see [12, 13].
In this paper, along with Censorâ€™s idea, we consider the multiplesets form of the split equality problem (MSSEP), which can be characterized mathematically as
where r, t are positive integers, \(\{C_{i}\}^{r}_{i=1}\) and \(\{Q_{j}\}^{t}_{j=1}\) are closed, convex, and nonempty subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively; and \(H_{3}\) is also a Hilbert space, \(A:H_{1}\mapsto H_{3}\), \(B:H_{2}\mapsto H_{3}\) are two bounded and linear operators. Obviously, when \(r=t=1\), the MSSEP reduces to the SEP. We propose an iterative algorithm with split selfadaptive step size where we need not calculate or estimate the spectral norms of related operators.
The general structure of this paper is as follows. In the next section, we provide some lemmas and definitions as well as their properties which will be useful in succedent processes. In SectionÂ 3, we present an iterative algorithm with split selfadaptive step size and provide the proof of its convergence. In SectionÂ 4, several numerical results are showed to confirm the effectiveness of our algorithm. In the last section, there are some conclusions.
2 Preliminaries
As a matter of convenience, we introduce several notations first. Let H be a real Hilbert space with inner product \(\langle\cdot,\cdot\rangle\) and norm \(\\cdot\\); I denotes the unit operator on H. \(x^{k}\rightarrow x\) and \(x^{k}\rightharpoonup x\) represent sequences \(\{x^{k}\}\) converging strongly and weakly to x, respectively. \(\omega_{w}(x^{k})\) denotes the weak cluster point set of sequence \(\{x^{k}\}\); \(\operatorname{Fix}(T)\) and \(T^{*}\) are the fixed points set and adjoint operator of operator T, respectively.
Next, there are several definitions and lemmas that will be available in the following proof process.
Definition 2.1
[14]
A mapping \(T:H\mapsto H\) goes by the name of

(i)
nonexpansive, if \(\TxTy\\leq\xy\\), \(\forall x,y\in H\);

(ii)
firmly nonexpansive, if \(\TxTy\^{2}\leq\langle xy,TxTy\rangle\), \(\forall x,y\in H\).
Review that \(P_{C}\) is a mapping from H onto a closed, convex, and nonempty subset C of H, if
then \(P_{C}\) is called the orthogonal projection from H onto C.
Bauschke et al. presented the following properties of the orthogonal projection operator.
Lemma 2.2
[14]
Let C be a closed, convex, and nonempty subset of H, then for any \(x,y\in H\) and \(z\in C\),

(i)
\(\langle xP_{C}x,zP_{C}x\rangle\leq0\);

(ii)
\(\P_{C}xP_{C}y\^{2}\leq\langle P_{C}xP_{C}y,xy\rangle\);

(iii)
\(\P_{C}xz\^{2}\leq\xz\^{2}\P_{C}xx\^{2}\).
Remark 2.2â€²
Based on the CauchySchwarz inequality, it is not hard to show that a firmly nonexpansive mapping is nonexpansive. It follows from LemmaÂ 2.2 that \(P_{C}\) is firmly nonexpansive and nonexpansive. It also can be deduced that \(IP_{C}\) is firmly nonexpansive and nonexpansive.
Definition 2.3
[14]
Let C be a nonempty subset of H, and let \(\{x^{k}\}\) be a sequence in H. Then \(\{x^{k}\}\) is FejÃ©r monotone with respect to C, if
Obviously, a FejÃ©r monotone sequence \(\{x^{k}\}\) is bounded and \(\lim_{k\rightarrow\infty}\x^{k}z\\) exists.
The demiclosedness principle is a perfect conclusion and plays a significant role in fixed point theory.
Lemma 2.4
Let X be a Banach space, C be a closed and convex subset of X, and \(T:C\mapsto C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset \). If \(\{x^{k}\}\rightharpoonup x\) and \(\{(IT)x^{k}\}\rightarrow y\), then \((IT)x=y\).
The following lemma is a primarily used tool in the proof of our main results.
Lemma 2.5
[17]
Let K be a closed, convex, and nonempty subset of H, and \(\{x^{k}\}\) be a sequence in H, if

(i)
\(\lim_{k\rightarrow\infty}\x^{k}x\\) exists for each \(x\in K\);

(ii)
\(\omega_{w}(x^{k})\subseteq K\),
then \(\{x^{k}\}\) converges weakly to a point in K.
3 Main results
Recall the multiplesets split equality problem (MSSEP), without loss of generality, we assume that \(t>r\) in (1.3), and make \(C_{r+1}=C_{r+2}=\cdots=C_{t}=H_{1}\), then the problem MSSEP (1.3) can be described equivalently as
Let \(S_{i}=C_{i}\times Q_{i}\subseteq H=H_{1}\times H_{2}\), \(i=1,2,\ldots,t\), \(G=[A,B]:H\mapsto H_{3}\), \(G^{*}\) be the adjoint operator of G, then the original problem (1.3) can be modified as
Theorem 3.1
Let \(\Omega\neq\emptyset\) be the solution set of the problem MSSEP (3.2). Choose an initial point \(w_{0}\in H\) arbitrarily, the iterative sequence \(\{w^{k}\}\) with split selfadaptive step size is obtained by the following:
or componentwise
where \(0<\underline{\rho}_{1}\leq\rho_{1}^{k}\leq\bar{\rho}_{1}<1\), \(0<\underline{\rho }_{2}\leq\rho_{2}^{k}\leq\bar{\rho}_{2}<1\), \(\{\alpha_{i}\}_{i=1}^{t}>0\), then \(\{w^{k}\}\) converges weakly to a solution of the problem MSSEP (3.2).
Proof
Since \(\Omega\neq\emptyset\), taking a \(\bar{w}\in\Omega\), \(G\bar{w}=0\),
Next, we prove that the iterative sequence \(\{w^{k}\}\) is FejÃ©r monotone with respect to Î©.
First, according to the property of the projection operator (LemmaÂ 2.2) and the definition of an adjoint operator, we obtain the following estimations:
and
Substituting (3.6) and (3.7) into (3.5), we get
Based on the assumptions of \(\{\rho_{1}^{k}\}\) and \(\{\rho_{2}^{k}\}\), it follows from (3.8) that
wÌ„ is taken arbitrarily in Î©, hence, the iterative sequence \(\{w^{k}\}\) is FejÃ©r monotone with respect to Î©. Therefore, \(\lim_{k\rightarrow\infty}\w^{k}\bar{w}\\) exists.
Since \(\rho_{1}^{k}\in[\underline{\rho}_{1},\bar{\rho}_{1}]\subset(0,1)\), from (3.8) we have
Letting \(k\rightarrow\infty\) on both sides of the above inequality (3.9), we obtain
The sequence \(\{w^{k}\}\) is bounded, there exists a real number \(M>0\) such that \(\\sum_{i=1}^{t}\alpha_{i}(P_{S_{i}}w^{k}w^{k})\\leq M\) for all \(k\geq0\), and \(P_{S_{i}}I\) is nonexpansive, consequently, it follows from (3.10) that
which is equal to
Analogously, due to the fact \(\rho_{2}^{k}\in[\underline{\rho}_{2},\bar {\rho}_{2}]\subset(0,1)\), from (3.8) one deduces that
letting \(k\rightarrow\infty\) in (3.12), we arrive at
Since \(\{w^{k}\}\) is bounded, and by the boundedness of A, B, we know that \(G^{*}G\) is also a bounded and linear operator, so there exists a real number \(L>0\) such that \(\G^{*}Gw^{k}\^{2}\leq L\), therefore,
Now, we prove that the weak cluster point set of the sequence \(\{w^{k}\}\) lies in Î©, i.e., \(\omega_{w}(w^{k})\subset\Omega\). In fact, \(\{w^{k}\}\) is bounded, then \(\omega_{w}(w^{k})\neq\emptyset\). Let \(\{ w^{k_{n}}\}\) be a subsequence of \(\{w^{k}\}\) which weakly converges to a point Åµ in \(\omega_{w}(w^{k})\). According to LemmaÂ 2.4, we infer from (3.11) that
Since \(w^{k_{n}}\rightharpoonup\hat{w}\), by the FrÃ©chetRiesz representation theorem, we have
By virtue of (3.13), it follows that \(\lim_{n\rightarrow\infty}\ Gw^{k_{n}}\=0\). Hence,
that is,
Combining (3.14) with (3.15), we conclude that \(\hat{w}\in\Omega\). Due to the fact that \(\hat{w},\bar{w}\in\Omega\) are taken arbitrarily in Î©, the conditions of LemmaÂ 2.5 are satisfied, it follows that the iterative sequence \(\{w^{k}\}\) weakly converges to a point in Î©. The proof of TheoremÂ 3.1 is completed.â€ƒâ–¡
When \(t=1\) in TheoremÂ 3.1, it is the iterative algorithm for solving the SEP (1.1).
Corollary 3.2
Assume that the solution set of SEP (1.1) is nonempty. For any initial point \(w^{0}\in H\), the iterative sequence \(\{w^{k}\}\) with split selfadaptive step size is obtained by
or componentwise
where \(0<\underline{\rho}_{1}\leq\rho_{1}^{k}\leq\bar{\rho}_{1}<1\), \(0<\underline {\rho}_{2}\leq\rho_{2}^{k}\leq\bar{\rho}_{2}<1\), the sequence \(\{w^{k}\}\) converges weakly to a solution of the SEP (1.1).
4 Numerical experiments
In this section, we provide several numerical results and compare with Byrneâ€™s algorithm (1.2) in [10] to confirm the effectiveness of our proposed algorithm. The whole program was written in Wolfram Mathematica (version 9.0). All the numerical results were carried out on a personal Lenovo Thinkpad computer with Intel(R) Core(TM) i54200M CPU 2.50 GHz and RAM 4.00 GB.
The SEP with \(A=(a_{ij})_{P\times M}\), \(B=(b_{ij})_{P\times N}\), \(C=\{x\in R^{M}\x\\leq1\}\), \(Q=\{y\in R^{N}\y\\leq2\}\), where \(a_{ij}\in [0,1]\), \(b_{ij}\in[0,1]\) are all generated randomly, P, M, N are positive integers. We take the initial points \(x_{0}=(1,1,\ldots,1)\in R^{M}\), \(y_{0}=(0,0,\ldots,0)\in R^{N}\), \(\rho_{1}^{k}=\rho_{2}^{k}=0.1\) in TheoremÂ 3.1, \(\gamma_{k}=0.01\) in (1.2), and \(\AxBy\<\epsilon\) as the termination condition. For P, M, N and error value Ïµ, we take two values, respectively. In TablesÂ 14, n and t are the iterative steps and CPU time, respectively.
From TablesÂ 14, we see that, under the same conditions, both the number of iterative steps and the CPU times of our algorithm are less than Byrneâ€™s. So to some extent, the numerical results indicate that our algorithm is better than Byrneâ€™s.
5 Conclusions
We propose a new iterative algorithm with split selfadaptive step size to solve the multiplesets split equality problem, which ensures that we can leave out much work on the calculation or estimation of the spectral norms of related operators. Under proper conditions, the theoretical convergence of the algorithm proposed is presented. Several numerical results confirm the effectiveness of the algorithm proposed.
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Acknowledgements
This research was supported by NSFC Grants No.Â 11226125, No.Â 11301379.
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The main idea of this paper was proposed by DT; LS and RC prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Tian, D., Shi, L. & Chen, R. Iterative algorithm for solving the multiplesets split equality problem with split selfadaptive step size in Hilbert spaces. J Inequal Appl 2016, 34 (2016). https://doi.org/10.1186/s1366001609827
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DOI: https://doi.org/10.1186/s1366001609827