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Convergence rate analysis of an iterative algorithm for solving the multiplesets split equality problem
Journal of Inequalities and Applications volume 2019, Article number: 277 (2019)
Abstract
This paper considers an iterative algorithm of solving the multiplesets split equality problem (MSSEP) whose step size is independent of the norm of the related operators, and investigates its sublinear and linear convergence rate. In particular, we present a notion of bounded Hölder regularity property for the MSSEP, which is a generalization of the wellknown concept of bounded linear regularity property, and give several sufficient conditions to ensure it. Then we use this property to conclude the sublinear and linear convergence rate of the algorithm. In the end, some numerical experiments are provided to verify the validity of our consequences.
Introduction
Set \(H_{1}\), \(H_{2}\) and \(H_{3}\) be three real Hilbert spaces, \(C\subseteq H_{1}\) and \(Q\subseteq H_{2}\) be two closed, convex and nonempty sets. And set two operators \(A:H_{1}\rightarrow H_{3}\) and \(B:H_{2}\rightarrow H_{3}\) be bounded and linear. Moudafi [1] proposed the split equality problem (SEP) for the first time, which can be formulated as
This kind of problem attracted many authors’ attention because of its widespread applications in many areas of applied mathematics such as intensitymodulated radiation therapy and decomposition methods for partial differential equations. In order to solve the split equality problem, various algorithms were introduced. One of the most significant algorithms is the alternating CQalgorithm (ACQA), and it was presented by Moudafi [1]. The iterative form of the ACQA is
He also proved that this algorithm converges weakly to a solution of the SEP (1.1).
The ACQA is related to \(P_{C}\) and \(P_{Q}\). If \(P_{C}\) or \(P_{Q}\) does not have an analytical expression, it might be difficult to implement. Then Moudafi [2] presented the relaxed alternating CQalgorithm (RACQA) to solve this problem:
The above algorithm also converges weakly to a solution of the SEP (1.1). Afterwards, for getting a strong convergence result, Shi et al. [3] proposed the following algorithm:
For more information with respect to the algorithms of solving the split equality problem; see [4, 5] and the references therein. But all these papers did not consider the convergence rate of the algorithms.
In this paper, we think about the multiplesets split equality problem (MSSEP), which generalizes the split equality problem. It can be characterized mathematically as
where r and t are two positive integers, \(\{C_{i}\}^{t}_{i=1}\) and \(\{Q_{j}\}^{r}_{j=1}\) are closed, convex and nonempty sets in Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively, \(H_{3}\) is also a Hilbert space, and two operators \(A:H_{1}\rightarrow H_{3}\) and \(B:H_{2}\rightarrow H_{3}\) are bounded and linear. Obviously, when \(t=r=1\), the MSSEP (1.2) becomes the SEP (1.1). Without loss of generality, set \(t>r\) and take \(Q_{r+1}=Q_{r+2}=\cdots =Q_{t}=H_{2}\). Set \(S_{i}=C_{i}\times Q_{i}\subseteq H=H_{1}\times H_{2}\), \(i=1,2,\ldots ,t\), \(S=\bigcap^{t}_{i=1}S_{i}\), \(G=[A,B]:H \rightarrow H_{3}\) and \(G^{\ast }\) be the adjoint operator of G. Then the MSSEP (1.2) can be restated as
To solve the multiplesets split equality problem, Tian et al. [6] gave the following algorithm and obtained a weak convergence result:
The step size of the algorithm is split selfadaptive, namely, it does not need any information about the relevant operators, which can save much time for our calculation. The main purpose of this paper is to investigate the sublinear and linear convergence rate of algorithm (1.4).
The rest of this paper is organized as follows. In Sect. 2, we recall some definitions and lemmas which are useful for our convergence analysis later. We also introduce a concept of bounded Hölder regularity property for the MSSEP and provide some conditions to guarantee this property. In Sect. 3, under a bounded Hölder regularity assumption, we study the sublinear and linear convergence of algorithm (1.4) and conclude its convergence rate. In Sect. 4, we perform some numerical experiments and clarify the effectiveness of our results.
Preliminaries
Set H be a real Hilbert space which has inner product \(\langle \cdot ,\cdot \rangle \) and norm \(\\cdot \\). For a point \(w\in {H}\) and a set \(S\subseteq H\), we denote the classical metric projection of w onto S and the distance of w from S by using \(P_{S}(w)\) and \(d_{S}(w)\), respectively, and they are defined by
Bauschke et al. [7] listed several basic properties of the projection operator. These properties are as follows.
Lemma 2.1
([7])
Let S be a closed, convex and nonempty subset of H, then for any \(x,y\in H\) and \(z\in S\),

(i)
\(\langle xP_{S}x,zP_{S}x\rangle \leq 0\);

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

(iii)
\(\P_{S}xz\^{2}\leq \xz\^{2}\P_{S}xx\^{2}\).
Set operator \(G :H\rightarrow H_{3}\) be bounded and linear. We utilize \(\ker G=\{x\in {H}:Gx=0\}\) to denote the kernel of G. The orthogonal complement of kerG is represented by \((\ker G)^{\bot }=\{y\in {H}:\langle x,y\rangle =0, \forall x\in {\ker G}\}\). As is well known, kerG and \((\ker G)^{\bot }\) are both closed subspaces of H. Throughout this paper, we denote the solution set of the MSSEP (1.3) by using Γ, which is defined by
We assume that the MSSEP is consistent, then Γ is a closed, convex and nonempty set.
Next, we shift our attention to the bounded Hölder regularity property for a collection of closed and convex subsets of a Hilbert space.
Definition 2.2
([8])
Let \(\{S_{i}\}_{i\in {I}}\) be a collection of closed convex subsets in a Hilbert space H and \(S=\bigcap_{i\in {I}}S_{i}\neq \emptyset \). The collection \(\{S_{i}\}_{i\in {I}}\) has a bounded Hölder regular intersection if for each bounded set K, there exist an exponent \(\gamma \in {(0,1]}\) and a scalar \(\beta >0\) such that
Furthermore, if the exponent γ is independent of the set K, we say the collection \(\{S_{i}\}_{i\in {I}}\) is bounded Hölder regular with uniform exponent γ.
It is obvious that any collection including only a set has a bounded Hölder regular intersection whose uniform exponent γ is equal to 1. The above definition with \(\gamma =1\) is the bounded linear regularity property, which was introduced in [9]. Then we provide a significant notion of bounded Hölder regularity property for the MSSEP (1.3) on the basis of Definition 2.2.
Definition 2.3
The MSSEP is said to satisfy the bounded Hölder regularity property if for each bounded set K, there exist an exponent \(\gamma \in {(0,1]}\) and a scalar \(\beta >0\) such that
Furthermore, if the exponent γ is independent of the set K, we say the MSSEP is bounded Hölder regular with uniform exponent γ.
It is worth noting that when \(\gamma =1\), the MSSEP satisfies the bounded linear regularity property [10].
Lemma 2.4
([11])
Let \(G : H\rightarrow H_{3}\) be a bounded linear operator. Then G is injective and has closed range if and only if G is bounded below, i.e., there exists a positive constant γ such that \(\Gw\\ge \gamma \w\\) for all \(w\in {H}\).
The following lemma gives some conditions which make the bounded Hölder regularity property for the MSSEP (1.3) hold.
Lemma 2.5
\(\{S, \ker G\}\) has a bounded Hölder regular intersection and the range of G is closed, then the MSSEP (1.3) satisfies the bounded Hölder regularity property.
Proof. \(\{S, \ker G\}\) has a bounded Hölder regular intersection, so for any bounded set K, there exist an exponent \(\gamma \in {(0,1]}\) and a scalar \(\beta >0\) such that
Since G restricted to \((\ker G)^{\bot }\) is injective and its range is closed, by Lemma 2.4, we know that there exists \(v>0\) such that
Hence,
Combining (2.2) and (2.3), we have
Then the proof is split into two cases:
Case 1: when \(\frac{1}{v}<1\), we have
Case 2: when \(\frac{1}{v}\geq 1\), we have
The proof is finished.
Lemma 2.6
([10])
Let \(\{S, \ker G\}\) be boundedly linearly regular and G has closed range. Then the MSSEP (1.3) satisfies the bounded linear regularity property.
In order to complete the convergence rate analysis of algorithm (1.4), the following definition and lemmas are also essential tools.
Definition 2.7
([7])
Let C be a nonempty subset of H, and \(\{x_{k}\}\) be a sequence in H. \(\{x_{k}\}\) is called Fejér monotone with respect to C, if
Clearly, a Fejér monotone sequence \(\{x_{k}\}\) is bounded and \(\lim_{k\rightarrow \infty }\x_{k}z\\) exists.
Lemma 2.8
([8])
Let C be a closed, convex and nonempty set of a Hilbert space H, and s be a positive integer. Suppose that the sequence \(\{w_{k}\}\) is Fejér monotone with respect to C and satisfies
for some \(\delta >0\) and \(\theta \geq 1\). Then \(w_{k}\rightarrow w ^{\ast }\) for some \(w^{\ast }\in {C}\) and there exist constants \(M_{1},M_{2}\geq 0\) and \(r\in {[0,1)}\) such that
Furthermore, the constants may be chosen to be
and δ necessarily lies in \((0,1]\) whenever \(\theta =1\).
Lemma 2.9
([12])
Let \(p>0\) and \(\{\delta _{k}\}_{k\epsilon \mathbf{N}}\) and \(\{\beta _{k}\}_{k\epsilon \mathbf{N}}\) be two sequences of nonnegative numbers such that
Then
where the convention that \(\frac{1}{0}=+\infty \) is adopted.
Finally, we end this section by reviewing algorithm (1.4) in detail.
Algorithm 2.10
([6])
For an arbitrary initial point \(w_{0}=(x_{0},y_{0}) \in {H}\), the sequence \(\{w_{k}\}\) is generated by
or componentwise
where \(0<\underline{\rho}_{1}\leq \rho _{1,k}\leq \overline{\rho } _{1}<1\), \(0<\underline{\rho}_{2}\leq \rho _{2,k}\leq \overline{\rho } _{2}<1\) and \(\{\alpha _{i}\}_{i=1}^{t}>0\).
Main results
In this section, we conclude the sublinear and linear convergence rate of Algorithm 2.10 under a bounded Hölder regularity assumption. Now, we give the most important theorem in this paper and prove it.
Theorem 3.1
The MSSEP (1.3) satisfies the bounded Hölder regularity property, the sequence \(\{w_{k}\}\) is defined by Algorithm 2.10, and \(\{S_{i}\} _{i=1}^{t}\) has a bounded Hölder regular intersection, then \(\{w_{k}\}\) converges to a solution \(w^{\ast }\) of the MSSEP (1.3) at least with a sublinear rate \(O(k^{\iota })\) for some \(\iota >0\).
In particular, if the MSSEP satisfies the bounded Hölder regularity property with uniform exponent \(q\in {(0,1]}\) and \(\{S_{i}\}_{i=1} ^{t}\) has a bounded Hölder regular intersection with uniform exponent \(p\in {(0,1]}\), then there exist constants \(M_{1},M_{2},F \geq 0\) and \(r\in {[0,1)}\) such that when \(k\geq F\),
where \(\theta =\frac{1}{pq}\).
Proof
Set \(\beta _{k}:=\frac{\rho _{1,k}\sum_{i=1}^{t}\alpha _{i}\P _{S_{i}}w_{k}w_{k}\^{2}}{\\sum_{i=1}^{t}\alpha _{i}(P_{S_{i}}w_{k}w _{k})\^{2}}\) and \(\gamma _{k}:=\frac{\rho _{2,k}\Gw_{k}\^{2}}{\G ^{*}Gw_{k}\^{2}}\). For the first assertion, we will firstly prove that the sequence \(\{w_{k}\}\) is Fejér monotone with respect to Γ.
Since \(\varGamma \neq \emptyset \), take \(\bar{w}\in \varGamma \), then \(G\bar{w}=0\), and
We get the following formulas by using the properties of the projection operator and the definition of the adjoint operator:
and
Substituting (3.2) and (3.3) into (3.1), we get
According to the assumptions of \(\{\rho _{1,k}\}\) and \(\{\rho _{2,k}\}\), it follows from (3.4) that
That is, the sequence \(\{w_{k}\}\) is Fejér monotone with respect to Γ. Hence, \(\{w_{k}\}\) is bounded and \(\lim_{k\rightarrow \infty }\w_{k}\bar{w}\\) exists.
For getting a better conclusion, we need to prove that \(d_{\varGamma }(w _{k})<1\) when k is enough large.
Assume that the following inequality with \(\delta >0\) and \(\theta \geq 1\) is true:
And assume that \(w_{0}\notin \varGamma \) and set \(\lambda _{k}:=d_{\varGamma }^{2}(w_{k})\) and \(j:=\theta 1\geq 0\), then the inequality (3.5) reduces to
Then the proof is split into two cases based on the value of θ:
Case 1: when \(\theta >1\), we know that \(\frac{1}{\theta 1}>0\) and by Lemma 2.9, we have
that is,
So we can find a positive integer \(T_{1}\) such that \(d_{\varGamma }(w _{k})<1\) when \(k\geq T_{1}\).
Case 2: when \(\theta =1\), by (3.6), we have
where \(\delta \in (0,1]\). Then
So we can find a positive integer \(T_{2}\) such that \(d_{\varGamma }(w _{k})<1\) when \(k\geq T_{2}\).
Set \(T:=\max \{T_{1},T_{2}\}\), we have \(d_{\varGamma }(w_{k})<1\) when \(k\geq T\).
Next, we will prove that the sequence \(\{w_{k}\}\) satisfies the inequality (3.5) for some \(\delta >0\) and \(\theta \geq 1\).
Since w̄ is arbitrary in Γ, we have
On the one hand, by the assumptions of \(\{\rho _{1,k}\}\) and \(\{\rho _{2,k}\}\), we get
and
Hence, we can find two positive integers N and M such that
and
Set \(L:=\max \{N,M\}\), then the inequality (3.7) reduces to
On the other hand, set K be a bounded set such that \(\{w_{k}:k \in \mathbf{N}\}\subseteq K\), since \(\{S_{i}\}_{i=1}^{t}\) has a bounded Hölder regular intersection, there exist an exponent \(p\in {(0,1]}\) and a scalar \(\mu >0\) such that
that is,
And since the MSSEP satisfies the bounded Hölder regularity property, there exist an exponent \(q\in {(0,1]}\) and a scalar \(\nu >0\) such that
that is,
Substituting (3.9) and (3.10) into (3.8), we get
where \(\alpha =\min \{\alpha _{i},~i=1,2,\ldots ,t\}\), \(\eta =\min \{a _{1}\alpha \mu ^{\frac{2}{p}},a_{2}\}\). Then the proof is split into two cases:
Case 1: when \(\max \{d_{S}(w_{k}),\Gw_{k}\\}=d_{S}(w_{k})\), we have
So the inequality (3.5) is true with \(\delta =2\eta ({\frac{1}{ \nu }})^{\frac{2}{pq}}\) and \(\theta =\frac{1}{pq}\).
Case 2: when \(\max \{d_{S}(w_{k}),\Gw_{k}\\}=\Gw_{k}\\), we have
So the inequality (3.5) is true with \(\delta =2\eta ({\frac{1}{ \nu }})^{\frac{2}{q}}\) and \(\theta =\frac{1}{q}\). Set \(F:=\max \{L,T \}\). When \(k\geq F\), we have
In conclusion, we get the inequality (3.5) where \(\theta = \frac{1}{pq}\) and
By Lemma 2.8, we see that the first assertion is true.
For the second assertion, the proof is the same as the above proof. And we notice that p and q is independent of K. Then the second assertion can be obtained. The proof is finished. □
The SEP is a special case of the MSSEP. When \(t=1\), Algorithm 2.10 reduces to an iterative algorithm for solving the SEP (1.1) [6]. Thus Theorem 3.1 becomes the following form.
Corollary 3.2
The SEP (1.1) satisfies the bounded Hölder regularity property and the sequence \(\{w_{k}\}\) is defined by
or componentwise
where \(0<\underline{\rho}_{1}\leq \rho _{1,k}\leq \overline{\rho} _{1}<1\), \(0<\underline{\rho}_{2}\leq \rho _{2,k}\leq \overline{\rho} _{2}<1\), then \(\{w_{k}\}\) converges to a solution \(w^{\ast }\) of the SEP (1.1) at least with a sublinear rate \(O(k^{\iota })\) for some \(\iota >0\).
In particular, if the SEP satisfies the bounded Hölder regularity property with uniform exponent \(q\in {(0,1]}\), then there exist constants \(M_{1},M_{2}\geq 0\) and \(r\in {[0,1)}\) such that
where \(\theta =\frac{1}{q}\).
Its proof is similar to the proof of Theorem 3.1.
Numerical experiments
Set \(H_{1}=R\), \(H_{2}=R^{2}\) and \(H_{3}=R^{3}\). We consider the SEP which has two subsets \(C=\{x\in H_{1}:\x\\leq 15\}\) and \(Q=\{x \in H_{2}:\x\\leq 15\}\). The two operators \(A:H_{1}\rightarrow H _{3}\) and \(B:H_{2}\rightarrow H_{3}\) are defined by
respectively. Set \(S=C\times Q\subseteq H_{3}\) and \(G=[A,B]:H_{3} \rightarrow H_{3}\). G is defined by
Then \(\ker G=\{(x,x,0),x\in {R}\}\neq \emptyset \), the range of G is closed and the solution set of this SEP is \(\varGamma =S\cap \ker G=\{(x,x,0),x \in {C}\}\). It is easy to know that the SEP satisfies the bounded linear regularity property by Lemma 2.6, namely, it satisfies the bounded Hölder regularity property.
Take \(w_{0}=(x_{0},y_{0},z_{0})\in S\). In consideration of algorithm (3.11), we have
In this algorithm, we take \(\rho _{2,k}=0.3\), 0.5, respectively. Then we get some numerical experiments which were run on a personal Dell computer with Intel(R)Core(TM)i54210U CPU 1.70 GHz and RAM 4.00 GB. And we wrote all the programs in Wolfram Mathematica (version 9.0).
We take the initial value \(w_{0}=(6,10,2)\). Set the error to be 10^{−5}, 10^{−10}, respectively. Note that we denote the number of iterations and the logarithm of the error by using the xcoordinate and the ycoordinate of the figures, respectively.
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Acknowledgements
The authors would like to express their sincere thanks to the editors and reviewers for their noteworthy comments, suggestions, and ideas, which helped to improve this paper.
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This research was supported by NSFC Grants No:11301379; No:11226125; No:11671167.
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The main idea of this paper was proposed by SS, MF and LS 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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Sun, S., Feng, M. & Shi, L. Convergence rate analysis of an iterative algorithm for solving the multiplesets split equality problem. J Inequal Appl 2019, 277 (2019). https://doi.org/10.1186/s136600192219z
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DOI: https://doi.org/10.1186/s136600192219z
Keywords
 Convergent rate
 Bounded Hölder regularity
 Multiplesets split equality problem