Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators

Abstract

Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011), based on the very recent work by Dang et al. (2012), in this paper, we propose an inertial iterative algorithm for solving the split common fixed-point problem for quasi-nonexpansive operators in the Hilbert space. We also prove the asymptotical convergence of the algorithm under some suitable conditions. The results improve and develop previously discussed feasibility problems and related algorithms.

1. Introduction

The convex feasibility problem (CFP), as an important optimization problem [1], is to find a common point in the intersection of finitely many convex sets. It has been applied to many areas, for instance, approximation theory [2], image reconstruction from projections [3, 4], control [5], and so on. When there are only two sets and constraints are imposed on the solutions in the domain of a linear operator as well as in this operator’s ranges, the problem is said to be a split feasibility problem (SFP) which has the following formula: finding a point 𝑥 satisfying 𝑥∈𝐶,𝐴𝑥∈𝑄, (1) where 𝐶 is a closed convex subset of a Hilbert space 𝐻₁, 𝑄 is a closed convex subset of a Hilbert space 𝐻₂, and 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. The SFP was originally introduced in [6], and it has also broad applications in many fields, such as image reconstruction problem, signal processing, and radiation therapy. Many projection methods have also been developed for solving the SFP; see [7–9]. Denote by 𝑃_𝐶 the orthogonal projection onto 𝐶; that is, 𝑃_𝐶(𝑥)=argmin_𝑦∈𝐶‖𝑥−𝑦‖, over all 𝑥∈𝐶. Assuming that the SFP is consistent (i.e., (1) has a solution), it is not hard to see that 𝑥∈𝐶 solves (1) if and only if it solves the fixed-point equation: 𝑥=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴(𝑥), (2) where 0<𝛾 is any positive constant and 𝐴^∗ denotes the adjoint of 𝐴.

To solve (2), in [10], Byrne introduced the so-called CQ algorithm, which generates a sequence {𝑥^𝑘} by 𝑥^𝑘+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^𝑘, (3) where 0<𝛾<2/𝜌(𝐴^𝑇𝐴) and 𝜌(𝐴^𝑇𝐴) is the spectral radius of 𝐴^∗𝐴.

The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [11]. Our main purpose here is to give an extension of the results developed in [12] to the split common fixed-point problem for quasi-nonexpansive operators, and we will introduce weak symposium convergence result of the algorithm under some suitable conditions. This will be done in the context of general Hilbert spaces.

The paper is organized as follows. In Section 2, we recall some preliminaries. In Section 3, we present an inertial CQ algorithm and show its convergence.

2. Preliminaries

Throughout the rest of the paper, 𝐼 denotes the identity operator and Fix(𝑇) denotes the set of the fixed points of an operator 𝑇, that is, Fix(𝑇)∶={𝑥∣𝑥=𝑇(𝑥)}.

Recall that a mapping 𝑇 is said to be quasi-nonexpansive (𝜀_Q) if 𝑇𝑥−𝑞≤𝑥−𝑞,∀𝑥,𝑞∈𝐻×Fix(𝑇). (4) A mapping 𝑇 is called nonexpansive (𝜀_N) if 𝑇(𝑥)−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻×𝐻. (5) A mapping 𝑇 is called firmly nonexpansive (𝜀_FN) if 𝑇(𝑥)−𝑇(𝑦)²≤𝑥−𝑦²−𝑥−𝑦−𝑇(𝑥)−𝑇𝑦²,∀𝑥,𝑦∈𝐻×𝐻. (6) A mapping 𝑇 is called firmly quasi-nonexpansive (𝜀_FQ) if 𝑇(𝑥)−𝑞²≤𝑥−𝑞²−‖𝑥−𝑇(𝑥)‖²,∀𝑥,𝑞∈𝐻×Fix(𝐻). (7) It is easily observed that 𝜀_FN⊂𝜀_N⊂𝜀_Q and that 𝜀_FN⊂𝜀_FQ⊂𝜀_Q. Furthermore, 𝜀_FN is well known to include resolvents and projection operators, while 𝜀_FQ contains subgradient projection operators (see, e.g., [13], and the references therein).

Recently, Bauschke and Combettes [14] have considered a class of mappings satisfying the condition 𝑞−𝑇𝑥,𝑥−𝑇𝑥≤0,∀𝑥,𝑞∈𝐻×Fix(𝑇). (8) It can easily be seen that the class of mappings satisfying the latter condition coincides with that of firmly quasi-nonexpansive mappings.

Usually, the convergence of fixed-point algorithms requires some additional smoothness properties of the mapping 𝑇 such as demiclosedness.

Definition 1. A mapping 𝑇 is said to be demiclosed if for any sequence {𝑥^𝑘} which weakly converges to 𝑦 and if the sequence {𝑇(𝑥^𝑘)} strongly converges to 𝑧, then 𝑇(𝑦)=𝑧.
In what follows, only the particular case of demiclosedness at zero will be used, which is the particular case when 𝑧=0.
The following lemmas will be needed in the proof of the convergence of the algorithm.

Lemma 2. Let 𝑇 be a quasi-nonexpansive mapping. Set 𝑇_𝛼∶=(1−𝛼)𝐼+𝛼𝑇. Then, it is immediate that for all (𝑥,𝑞)∈𝐻×Fix(𝑇):(1)⟨𝑥−𝑇(𝑥),𝑥−𝑞⟩≥(1/2)‖𝑥−𝑇(𝑥)‖² and ⟨𝑥−𝑇(𝑥),𝑞−𝑇(𝑥)⟩≤(1/2)‖𝑥−𝑇(𝑥)‖²;(2)‖𝑇_𝛼(𝑥)−𝑞‖²≤‖𝑥−𝑞‖²−𝛼(1−𝛼)‖𝑥−𝑇(𝑥)‖;(3)⟨𝑥−𝑇_𝛼(𝑥),𝑥−𝑞⟩≥(𝛼/2)‖𝑥−𝑇(𝑥)‖².

Lemma 3 (see [8]). Assume 𝜑_𝑘∈[0,∞) and 𝛿_𝑘∈[0,∞) satisfy(1)𝜑_𝑘+1−𝜑_𝑘≤𝜃_𝑘(𝜑_𝑘−𝜑_𝑘−1)+𝛿_𝑘, (2)∑^+∞_𝑘=1𝛿_𝑘<∞, (3){𝜃_𝑘}⊂[0,𝜃],  where 𝜃∈[0,1).Then, the sequence  {𝜑_𝑘} is convergent with ∑^+∞_𝑘=1[𝜑_𝑘+1−𝜑_𝑘]₊<∞, where [𝑡]₊∶=max{𝑡,0}  (for any 𝑡∈𝑅).

3. The Inertial Algorithm and Its Asymptotic Convergence

In what follows, we will focus our attention on the following general two-operator split common fixed-point problem: find𝑥^∗∈𝐶suchthat𝐴𝑥^∗∈𝑄, (9) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator and 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ are two quasi-nonexpansive operators with nonempty fixed-point sets Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄, and denote the solution set of the two-operator SCFP by Γ=𝑦∈𝐶∶𝐴𝑦∈𝑄. (10)

3.1. The Inertial Algorithm

To solve (9), Moudafi [15] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: 𝑥^𝑘+1=𝑈_𝛼𝑘𝑥^𝑘+𝛾𝐴^∗𝑇_𝛽−𝐼𝐴𝑥^𝑘,𝑘∈𝑁, (11) where 𝛽∈(0,1), 𝛼_𝑘∈(0,1) are relaxation parameters and 𝛾>0. Inspired by the inertial technique, we introduce the following inertial method and then present its convergence analysis.

Algorithm 4.  Initialization: Let 𝑥⁰∈𝐻¹ be arbitrary. Iterative step: For 𝑘∈𝑁, set 𝑢=𝐼+𝛾𝜂𝐴^∗(𝑇−𝐼)𝐴, and let 𝑦^𝑘=𝑥^𝑘+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1𝑥^𝑘+1=1−𝛼_𝑘𝑢𝑦^𝑘+𝛼_𝑘𝑈𝑢𝑦^𝑘,𝑘∈𝑁, (12)where 𝜂∈(0,1), 𝛼_𝑘∈(0,1), and 𝛾∈(0,1/(𝜆𝜂)), with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴, 𝜃_𝑘∈[0,1).

3.2. Asymptotic Convergence of the Inertial Algorithm

In this subsection, we establish the asymptotic convergence of Algorithm 4.

Lemma 5 (Opial [16]). Let 𝐻 be a Hilbert space and let {𝑥^𝑘} be a sequence in 𝐻 such that there exists a nonempty set 𝑆⊂𝐻 satisfying(1)for every 𝑥^∗, lim_𝑘‖𝑥^𝑘−𝑥^∗‖ exists,(2)any weak cluster point of the sequence {𝑥^𝑘} belongs to 𝑆. Then, there exists 𝑧∈𝑆 such that {𝑥^𝑘} weakly converges to 𝑧.

Theorem 6. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ be a quasi-nonexpansive operator with nonempty Fix(𝑈)=𝐶 and let 𝑇∶𝐻₂→𝐻₂ be a quasi-nonexpansive operator with nonempty Fix(𝑇)=𝑄. Assume that 𝑈−𝐼 and 𝑇−𝐼 are demiclosed at 0. If Γ≠∅, then any sequence {𝑥^𝑘} generated by Algorithm 4 weakly converges to a split common fixed point, provided that we choose 𝜃_𝑘 satisfying 𝜃_𝑘∈[0,𝜃_𝑘] with 𝜃_𝑘∶=min{𝜃,1/(𝑘‖𝑥^𝑘−𝑥^𝑘−1‖)²}, 𝜃∈[0,1). 𝛾∈(0,1/(𝜆𝜂)) and 𝛼_𝑘∈(𝛿,1−𝛿) for a small enough 𝛿>0.

Proof. Taking 𝑧∈Γ, and using (2) in Lemma 2, we obtain 𝑥^𝑘+1−𝑧²=1−𝛼_𝑘𝑢𝑦^𝑘+𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑧²≤𝑢𝑦^𝑘−𝑧²−𝛼_𝑘1−𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑢𝑦^𝑘2. (13) On the other hand, we have 𝑢𝑦^𝑘−𝑧²=𝑦^𝑘+𝛾𝜂𝐴^∗(𝑇−𝐼)𝐴𝑦^𝑘−𝑧²=𝑦^𝑘−𝑧²+𝛾²𝜂²𝐴^∗(𝑇−𝐼)𝐴𝑦^𝑘2+2𝛾𝜂𝑦^𝑘−𝑧,𝐴^∗(𝑇−𝐼)𝐴𝑦^𝑘≤𝑦^𝑘−𝑧²+𝜆𝛾²𝜂²(𝑇−𝐼)𝐴𝑦^𝑘2+2𝛾𝜂𝐴𝑦^𝑘−𝐴𝑧,(𝑇−𝐼)𝐴𝑦^𝑘, (14) that is, 𝑢(𝑦^𝑘)−𝑧²≤𝑦^𝑘−𝑧²+𝜆𝛾²𝜂²(𝑇−𝐼)𝐴𝑦^𝑘2+2𝛾𝜂𝐴𝑦^𝑘−𝐴𝑧,(𝑇−𝐼)𝐴𝑦^𝑘. (15) Now, by setting 𝜐∶=2𝛾𝜂⟨𝐴𝑦^𝑘−𝐴𝑧,(𝑇−𝐼)(𝐴𝑦^𝑘)⟩ and using (1) of Lemma 2, we obtain 𝜐=2𝛾𝜂𝐴𝑦^𝑘−𝐴𝑧,(𝑇−𝐼)𝐴𝑦^𝑘=2𝛾𝜂𝐴𝑦^𝑘−𝐴𝑧+(𝑇−𝐼)𝐴𝑦^𝑘−(𝑇−𝐼)𝐴𝑦^𝑘,(𝑇−𝐼)𝐴𝑦^𝑘=2𝛾𝜂𝑇𝐴𝑦^𝑘−𝐴𝑧,(𝑇−𝐼)𝐴𝑦^𝑘−(𝑇−𝐼)𝐴𝑦^𝑘21≤2𝛾𝜂2(𝑇−𝐼)𝐴𝑦^𝑘2−(𝑇−𝐼)𝐴𝑦^𝑘2≤−𝛾𝜂(𝑇−𝐼)𝐴𝑦^𝑘2. (16) Combining the key inequality above with (15) yields 𝑥^𝑘+1−𝑧²≤𝑦^𝑘−𝑧²−𝛾𝜂1−𝜆𝛾𝜂(𝑇−𝐼)𝐴𝑦^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑢𝑦^𝑘2. (17) Define the auxiliary real sequence 𝜑_𝑘∶=(1/2)‖𝑥^𝑘−𝑧‖². Therefore, from (17), we have 𝜑_𝑘+1≤12𝑦^𝑘−𝑧²−12𝛾𝜂1−𝜆𝛾𝜂(𝑇−𝐼)𝐴𝑦^𝑘2−12𝛼_𝑘1−𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑢𝑦^𝑘2. (18) By deducing, we have 12𝑦^𝑘−𝑧²=12𝑥^𝑘+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1−𝑧²=12𝑥^𝑘−𝑧²+𝜃_𝑘𝑥^𝑘−𝑧,𝑥^𝑘−𝑥^𝑘−1+𝜃²_𝑘2𝑥^𝑘−𝑥^𝑘−12=𝜑_𝑘+𝜃_𝑘𝑥^𝑘−𝑧,𝑥^𝑘−𝑥^𝑘−1+𝜃²_𝑘2𝑥^𝑘−𝑥^𝑘−12. (19) It is easy to check that 𝜑_𝑘=𝜑_𝑘−1+⟨𝑥^𝑘−𝑧,𝑥^𝑘−𝑥^𝑘−1⟩−(1/2)‖𝑥^𝑘−𝑥^𝑘−1‖².
Hence, 12𝑦^𝑘−𝑧²=𝜑_𝑘+𝜃_𝑘𝜑_𝑘−𝜑_𝑘−1+𝜃_𝑘+𝜃²_𝑘2𝑥^𝑘−𝑥^𝑘−12. (20) Putting (20) into (18), we get 𝜑_𝑘+1≤𝜑_𝑘+𝜃_𝑘𝜑_𝑘−𝜑_𝑘−1+𝜃_𝑘+𝜃²_𝑘2𝑥^𝑘−𝑥^𝑘−12−12𝛾𝜂1−𝜆𝛾𝜂(𝑇−𝐼)𝐴𝑦^𝑘2−12𝛼_𝑘1−𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑢𝑦^𝑘2. (21) Since 𝛾∈(0,1/(𝜆𝜂)), according to 𝜃²_𝑘≤𝜃_𝑘, 𝛼_𝑘∈(0,1) and (21), we derive 𝜑_𝑘+1≤𝜑_𝑘+𝜃_𝑘𝜑_𝑘−𝜑_𝑘−1+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−12. (22) Evidently, ^+∞_𝑘=1𝜃_𝑘𝑥^𝑘−𝑥^𝑘−12<∞, (23) due to 𝜃_𝑘‖𝑥^𝑘−𝑥^𝑘−1‖²≤1/𝑘². Let 𝛿_𝑘∶=𝜃_𝑘‖𝑥^𝑘−𝑥^𝑘−1‖² in Lemma 3. We deduce that the sequence {‖𝑥^𝑘−𝑧‖} is convergent (hence, {𝑥^𝑘} is bounded). By (23) and Lemma 3, we obtain ∑^+∞_𝑘=1[‖𝑥^𝑘−𝑧‖²−‖𝑥^𝑘−1−𝑧‖²]₊<∞. By reason of (21), we have 12𝛾𝜂1−𝜆𝛾𝜂(𝑇−𝐼)𝐴𝑦^𝑘2≤𝜑_𝑘−𝜑_𝑘+1+𝜃_𝑘𝜑_𝑘−𝜑_𝑘−1+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−12,12𝛼_𝑘1−𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑢𝑦^𝑘2≤𝜑_𝑘−𝜑_𝑘+1+𝜃_𝑘𝜑_𝑘−𝜑_𝑘−1+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−12. (24) Hence, ^+∞_𝑘=112𝛾𝜂1−𝜆𝛾𝜂(𝑇−𝐼)𝐴𝑦^𝑘2<∞,_+∞𝑘=112𝛼_𝑘1−𝛼_𝑘𝑈𝑢𝑦^𝑘−𝑢𝑦^𝑘2<∞. (25) By 𝛾∈(0,1/(𝜆𝜂)) and the assumption on 𝛼_𝑘, we get (𝑇−𝐼)𝐴𝑦^𝑘2𝑈𝑢𝑦⟶0,_𝑘−𝑢𝑦^𝑘2⟶0. (26)(27) Denoting by 𝑥^∗ a weak-cluster point {𝑥^𝑘}, let {𝑥^𝑘𝜎} be a subsequence of {𝑥^𝑘}. Obviously, 𝑤−lim_𝜎𝑦^𝑘𝜎=𝑤−lim_𝜎𝑥^𝑘𝜎=𝑥^∗. (28) Then, from (26) and the demiclosedness of 𝑇−𝐼 at 0, we obtain 𝑇𝐴𝑥^∗=𝐴𝑥^∗, (29) it follows that 𝐴𝑥^∗∈𝑄.

Now, by setting 𝑢_𝑘=𝑦^𝑘+𝛾𝜂𝐴^∗(𝑇−𝐼)(𝐴𝑦^𝑘), it follows that 𝑤−lim_𝜎𝑢^𝑘𝜎=𝑥^∗. By the demiclosedness of 𝑈−𝐼 at 0, from (27), we have 𝑈𝑥^∗=𝑥^∗. (30) Hence, 𝑥^∗∈𝐶, and therefore 𝑥^∗∈Γ.

Since there is no more than one weak-cluster point, the weak convergence of the whole sequence {𝑥^𝑘} follows by applying Lemma 5 with 𝑆=Γ.

Remark 7. Since the current value of ‖𝑥^𝑘−𝑥^𝑘−1‖ is known before choosing the parameter 𝜃_𝑘, then 𝜃_𝑘 is well-defined in Theorem 6. In fact, from the process of proof for Theorem 6, we can get the following assert: the convergence result of Theorem 6 always holds provided that we take 𝜃_𝑘∈[0,𝜃], 𝜃∈[0,1),forall𝑘≥0, with ^+∞_𝑘=1𝜃_𝑘𝑥^𝑘−𝑥^𝑘−12<∞. (31)

To conclude, we have proposed an algorithm for solving the SCFP in the wide class of quasi-nonexpansive operators and proved its convergence in general Hilbert spaces. Next, we will improve the algorithm to assure the strong convergence in infinite Hilbert spaces.

Acknowledgments

This work was supported by the National Science Foundation of China (under Grant no. 11171221), Shanghai Municipal Committee of Science and Technology (under Grant no. 10550500800), Shanghai Leading Academic Discipline (under Grant no. XTKX 2012), Basic and Frontier Research Program of the Science and Technology Department of Henan Province (under Grant nos. 112300410277 and 082300440150), and China Coal Industry Association Scientific and Technical Guidance to Project (under Grant no. MTKJ-2011-403).