A Perturbed Projection Algorithm with Inertial Technique for Split Feasibility Problem

Abstract

This paper deals with the split feasibility problem that requires to find a point closest to a closed convex set in one space such that its image under a linear transformation will be closest to another closed convex set in the image space. By combining perturbed strategy with inertial technique, we construct an inertial perturbed projection algorithm for solving the split feasibility problem. Under some suitable conditions, we show the asymptotic convergence. The results improve and extend the algorithms presented in Byrne (2002) and in Zhao and Yang (2005) and the related convergence theorem.

1. Introduction

Let 𝐶⊂𝑅^𝑛 and 𝑄⊂𝑅^𝑚 be nonempty closed convex sets, and let 𝐴 be an 𝑚×𝑛 real matrix. The split feasibility problem (SFP) is to find a point𝑥∈𝐶suchthat𝐴𝑥∈𝑄. (1.1)

This problem was first presented and analyzed by Censor and Elfving [1] and appeared in signal processing, image reconstruction [2], and so on. Many well-known iterative algorithms for solving (1.1) were established, see the papers [3–5]. Denoted by 𝑃_𝑆, the orthogonal projection operator onto convex set 𝑆, that is𝑃_𝑆(𝑥)=argmin‖𝑐−𝑥‖,𝑐∈𝑆, (1.2) where ‖⋅‖ indicates the 2-norm. The CQ algorithm proposed by Byrne in [6] has the following iterative process: 𝑥^𝑘+1=𝑃_𝐶𝑥^𝑘+𝛾𝐴^𝑇𝑃_𝑄−𝐼𝐴𝑥^𝑘,𝑘≥0, (1.3) where 𝛾∈(0,2/𝐿), 𝐿 denotes the largest eigenvalue of the matrix 𝐴^𝑇𝐴, and 𝐼 is the identity operator.

In some cases, it is difficult or even impossible to compute orthogonal projection; to avoid computing projection, Zhao and Yang in [7] proposed the perturbed projections algorithm for the SFP. This development was based on results of Santos and Scheimberg [8] who suggested replacing each nonempty closed convex set of the convex feasibility problem by a convergent sequence of supersets. If such supersets can be constructed with reasonable efforts and projecting onto them is simpler than projecting onto the original convex sets, then a perturbed algorithm is favorable. The concrete iterative process of perturbed CQ algorithm [7] is as follows:𝑥^𝑘+1=1−𝛼_𝑘𝑥^𝑘+𝛼_𝑘𝑃_𝐶𝑘𝑥^𝑘+𝛾𝐴^𝑇𝑃_𝑄𝑘−𝐼𝐴𝑥^𝑘, (1.4) where 𝛼_𝑘∈[0,1], 𝛾∈(0,2/𝐿), 𝐿, 𝐼 defined as in algorithm (1.3), and 𝐶−_𝑘^𝑀→𝐶 and 𝑄−_𝑘^𝑀→𝑄 (see the definitions in the Section 2), while the perturbed projections algorithm sometime converges slowly by reason of using only the current point to get the next iterative point.

Many papers have studied the inertial-type extrapolation recently, see [9–12], which uses the term 𝜃_𝑘 and the two previous iterative points 𝑥^𝑘−1, 𝑥^𝑘 to get the next iterative point 𝑥^𝑘+1. As an acceleration process, it can considerably improve the speed of convergence for the following causes: one is that the vector 𝑥^𝑘−𝑥^𝑘−1 acts as an impulsion term, the other is that the parameter 𝜃_𝑘 acts as a speed regulator.

To the best of our knowledge, no publications deal with perturbed projection algorithm and inertial process simultaneously. In this paper, we apply the inertial technique to the perturbed projection algorithm to get a perturbed inertial projection algorithm for the split feasibility problem. The results improve and extend the algorithms presented in [6] and in [7] and the related convergence theorem.

The paper is organized as follows. In Section 2, some preliminaries are given. The inertial perturbed algorithm and the corresponding convergence theorem for the split feasibility problem are presented in Section 3.

2. Preliminaries

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

An operator 𝑇 is said to be nonexpansive (ne) if𝑇(𝑥)−𝑇𝑦≤𝑥−𝑦. (2.1) It is well known that the projection operator is nonexpansive.

Recall the following notions of the convergence and 𝜌-distance.

Definition 2.1. Let 𝑁 be an operator on a Hilbert space 𝐻; let 𝑁_𝑘,𝑘=0,1,2,… be a family of operators on a Hilbert space. {𝑁_𝑘} is said to be convergent to 𝑁 if ‖𝑁_𝑘(𝑥)−𝑁(𝑥)‖→0 as 𝑘→+∞ for all 𝑥∈𝐻.

Definition 2.2. The 𝜌-distance (𝜌≥0) for operators 𝑁₁ and 𝑁₂ on 𝐻 is given by 𝐷_𝜌𝑁₁,𝑁₂=sup_{‖𝑥‖≤𝜌}𝑁₁(𝑥)−𝑁₂(𝑥). (2.2)

Now we introduce the Mosco-convergence for sequences of sets in a reflexive Banach space.

Definition 2.3 (see [13]). Let 𝑋 be a reflexive Banach space and 𝐶 and {𝐶_𝑘}_𝑘∈𝑁 (𝑁 is a set of natural numbers) a sequence of subsets of 𝑋. The sequence {𝐶_𝑘}_𝑘∈𝑁 is Mosco-convergent to 𝐶, denoted by 𝐶−_𝑘^𝑀→𝐶, if (1)∀𝑥∈𝐶,∃𝑥^𝑘_𝑘∈𝑁with𝑥^𝑘∈𝐶_𝑘(𝑘∈𝑁)suchthat𝑥^𝑘𝑋𝑠(2)∀𝑘⟶𝑥,_{𝑗𝑗∈𝑁},∀𝑥^𝑗_𝑗∈𝑁,𝑥^𝑗∈𝐶_𝑘𝑗𝑗∈𝑁,𝑥⟶^𝑗𝑋𝑤𝑥⟹𝑥∈𝐶, (2.3) where 𝑋_𝑠 and 𝑋_𝑤 denote the strong and weak topologies, respectively. In particular, if {𝐶_𝑘} and 𝐶 are in 𝑅^𝑛, then 𝐶−_𝑘^𝑀→𝐶 is equivalent to (1)∀𝑥∈𝐶,∃𝑥^𝑘_𝑘∈𝑁,𝑥^𝑘∈𝐶_𝑘(𝑘∈𝑁)suchthat𝑥^𝑘(2)∀𝑘⟶𝑥,_{𝑗𝑗∈𝑁},∀𝑥^𝑗_𝑗∈𝑁,𝑥^𝑗∈𝐶_𝑘𝑗𝑗∈𝑁,𝑥^𝑗⟶𝑥⟹𝑥∈𝐶. (2.4)

Using the notation NCCS(𝑅^𝑛) for the family of nonempty closed convex subsets of 𝑅^𝑛, let 𝐶 and 𝐶_𝑘 be sets in NCCS(𝑅^𝑛), for 𝑘=0,1,…. It is easy to verify that if the sequence {𝐶_𝑘} converges to 𝐶 in the Mosco sense, then the operator sequence {𝑃_𝐶𝑘} converges to 𝑃_𝐶.

Definition 2.4. Let 𝐶₁ and 𝐶₂ be elements in NCCS(𝑅^𝑛). The 𝜌-distance is defined by 𝑑_𝜌𝐶₁,𝐶₂=sup_{‖𝑥‖2≤𝜌}𝑃_𝐶1(𝑥)−𝑃_𝐶2(𝑥)₂. (2.5)
Let 𝐶 and 𝐶_𝑘 be sets in NCCS(𝑅^𝑛), then 𝐶−_𝑘^𝑀→𝐶 if and only if 𝑑_𝜌(𝐶_𝑘,𝐶)→0 for all 𝜌≥0.

The following lemmas will be used in convergence analysis later on.

Lemma 2.5 (see [14]). Let {𝛿_𝑘} and {𝛾_𝑘} be nonnegative sequences satisfying ∑_𝑘𝛿_𝑘<+∞ and 𝛾_𝑘+1≤𝛾_𝑘+𝛿_𝑘, 𝑘=0,1,…. Then, {𝛾_𝑘} is a convergent sequence.

Lemma 2.6 (see [10]). Let 𝜙_𝑘∈[0,∞), 𝑘=1,2,…, and 𝛿_𝑘∈[0,∞), 𝑘=1,2,… satisfying(1)𝜙_𝑘+1−𝜙_𝑘≤𝜃_𝑘(𝜙_𝑘−𝜙_𝑘−1)+𝛿_𝑘, (2)∑_𝑘𝛿_𝑘<∞,(3)𝜃_𝑘∈[0,𝜃], 𝑘=1,2,…, where 𝜃∈[0,1).Then, {𝜙_𝑘} is a converging sequence and ∑_𝑘(𝜙_𝑘+1−𝜙_𝑘)₊<∞, where (𝑡)₊∶=max{𝑡,0} for any 𝑡∈𝑅.

Lemma 2.7 (Opial [15]). 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^∗∈𝑆,lim_𝑘‖𝑥^𝑘−𝑥^∗‖ exists,(2)any weak cluster point of {𝑥^𝑘} belongs to 𝑆.Then, there exists 𝑥^∗∈𝑆 such that {𝑥^𝑘} weakly converges to 𝑥^∗.

Lemma 2.8 (see [15]). Let 𝐻 be a Hilbert space, 𝑇∶𝐻→𝐻 a nonexpansive operator, and 𝑦 a weak cluster point of a sequence {𝑥^𝑘}, and let ‖𝑇(𝑥^𝑘)−𝑥^𝑘‖→0. Then 𝑦∈Fix(𝑇).

3. The Inertial Perturbed Algorithm and the Asymptotic Convergence for the SPF

Let 𝐶 and 𝐶_𝑘 be sets in NCCS(𝑅^𝑛), and let 𝑄 and 𝑄_𝑘 be sets in NCCS(𝑅^𝑚), for 𝑘=0,1,…, with 𝐶−_𝑘^𝑀→𝐶 and 𝑄−_𝑘^𝑀→𝑄. Then 𝑑_𝜌(𝐶_𝑘,𝐶)→0 and 𝑑_𝜌(𝑄_𝑘,𝑄)→0 for all 𝜌>0. We denote 𝑁=𝑃_𝐶𝑈,𝑈=𝐼−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝑁𝐴,_𝑘=𝑃_𝐶𝑘𝑈_𝑘,𝑈_𝑘=𝐼−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝑘𝐴. (3.1) From Lemma  3.1 in [5], we know that 𝑥^∗ solves the SFP (1.1) if and only if 𝑥^∗∈Fix(𝑁).

It is well known that the operator 𝐴^𝑇(𝐼−𝑃_𝑄)𝐴 is 𝜆-Lipschitz continuous with 𝜆=𝜌(𝐴^𝑇𝐴). The same is true for the operators 𝐴^𝑇(𝐼−𝑃_𝑄𝑘)𝐴 for 𝑘=0,1,…; it is easy to obtain the following conclusion.

Lemma 3.1 (see [7]). Let 𝐶 and 𝐶_𝑘 be sets in NCCS(𝑅^𝑛), and let 𝑄 and 𝑄_𝑘 be sets in NCCS(𝑅^𝑚), for 𝑘=0,1,…, with 𝐶−_𝑘^𝑀→𝐶 and 𝑄−_𝑘^𝑀→𝑄. Then, the operators 𝑁 and 𝑁_𝑘 defined in (3.1) are nonexpansive operators for 𝛾∈(0,2/𝜆). Moreover, the operator sequence {𝑁_𝑘} converges to 𝑁.

Now we give the perturbed inertial KM-type algorithm for SFP.

Algorithm 3.2. Given arbitrary elements in 𝑅^𝑛 for 𝑘=0,1,…, let 𝑥^𝑘+1=1−𝛼_𝑘𝐼+𝛼_𝑘𝑁_𝑘𝑦^𝑘,𝑦^𝑘=𝑥^𝑘+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1, (3.2) where 𝑁_𝑘=𝑃_𝐶𝑘[(𝐼−𝛾𝐴^𝑇(𝐼−𝑃_𝑄𝑘)𝐴), 𝛼_𝑘∈(0,1), for any 𝑘 and 𝛾∈(0,2/𝜆) with 𝜆=𝜌(𝐴^𝑇𝐴), {𝜃_𝑘}⊂[0,𝜃], 𝜃∈[0.1).

The following theorem is necessary for the convergence analysis of Algorithm 3.2.

Theorem 3.3. Let 𝑁 and 𝑁_𝑘 for 𝑘=0,1,2,… be nonexpansive (ne) operators in finite-dimensional Hilbert space, with 𝑁_𝑘→𝑁, and let {𝛼_𝑘} be a sequence in (0,1) satisfying ^+∞_𝑘=0𝛼_𝑘1−𝛼_𝑘=+∞,_+∞𝑘=0𝛼_𝑘𝐷_𝜌𝑁_𝑘,𝑁<+∞ (3.3)(3.4) for all 𝜌>0. Then, the sequence {𝑥^𝑘} defined by the iterative step 𝑥^𝑘+1=1−𝛼_𝑘𝑦^𝑘+𝛼_𝑘𝑁_𝑘𝑦^𝑘,𝑦^𝑘=𝑥^𝑘+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1 (3.5)(3.6) converges to a fixed point of 𝑁 provided that we choose parameter 𝜃_𝑘 satisfying 𝜃_𝑘⊂0,𝜃_𝑘,with𝜃_𝑘1∶=min𝜃,max𝑘²𝑥^𝑘−𝑥^𝑘−12,𝑘²𝑥^𝑘−𝑥^𝑘−1,𝜃∈[0,1), (3.7) whenever such fixed points exist.

Proof. We first prove that the sequence {𝑥^𝑘} is bounded and {‖𝑥^𝑘−𝑧‖} is convergent for all 𝑧∈Fix(𝑁), where Fix(𝑁) denotes the set of the fixed points of the operator 𝑁, that is, 𝑁(𝑧)=𝑧. Since 𝑁 and 𝑁_𝑘 are ne operators, we have 𝑥^𝑘+1−𝑧=1−𝛼_𝑘𝑦^𝑘+𝛼_𝑘𝑁_𝑘𝑦^𝑘−1−𝛼_𝑘𝑧−𝛼_𝑘𝑁(𝑧≤)1−𝛼_𝑘𝑦^𝑘−𝑧+𝛼_𝑘𝑁_𝑘𝑦^𝑘(𝑧≤−𝑁)1−𝛼_𝑘𝑦^𝑘−𝑧+𝛼_𝑘𝑁_𝑘𝑦^𝑘−𝑁_𝑘(𝑧)+𝛼_𝑘𝑁_𝑘(𝑧)(𝑧≤𝑦−𝑁)_𝑘−𝑧+𝛼_𝑘𝐷_𝜌𝑁_𝑘=𝑥,𝑁_𝑘+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1−𝑧+𝛼_𝑘𝐷_𝜌𝑁_𝑘≤𝑥,𝑁_𝑘−𝑧+𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1+𝛼_𝑘𝐷_𝜌𝑁_𝑘,𝑁, (3.8) where 𝜌≥‖𝑧‖. From the selection of parameter 𝜃_𝑘, we have 𝜃_𝑘𝑥^𝑘−𝑥^𝑘−1≤𝑥^𝑘−𝑥^𝑘−1max𝑘²𝑥^𝑘−𝑥^𝑘−12,𝑘²𝑥^𝑘−𝑥^𝑘−1≤1𝑘². (3.9) It is easy to get _𝑘≥0𝑥^𝑘−𝑥^𝑘−1<∞. (3.10) By (3.4), (3.8)-(3.10), we obtain from Lemma 2.5 that the sequence {‖𝑥^𝑘−𝑧‖} is convergent and hence the sequence {𝑥^𝑘} is bounded.
We next prove that liminf_𝑘→∞‖𝑦^𝑘−𝑁(𝑦^𝑘)‖=0. Let 𝑒^𝑘=𝑁_𝑘(𝑦^𝑘)−𝑁(𝑦^𝑘) and notice the fact that 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦². (3.11) Then, we have 𝑥^𝑘+1−𝑧²=1−𝛼_𝑘𝑦^𝑘−𝑧+𝛼_𝑘𝑁_𝑘𝑦^𝑘−𝑁𝑦^𝑘+𝛼_𝑘𝑁𝑦^𝑘−𝑁(𝑧)²=1−𝛼_𝑘𝑦^𝑘−𝑧+𝛼_𝑘𝑒^𝑘+𝛼_𝑘𝑁𝑦^𝑘−𝑁(𝑧)+𝛼_𝑘𝑒^𝑘2=1−𝛼_𝑘𝑦^𝑘−𝑧+𝛼_𝑘𝑒^𝑘2+𝛼_𝑘𝑁𝑦^𝑘−𝑁(𝑧)+𝛼_𝑘𝑒^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2≤1−𝛼_𝑘𝑦^𝑘−𝑧²+2𝛼_𝑘𝑦^𝑘−𝑧𝑒^𝑘+𝛼_𝑘²𝑒^𝑘2+𝛼_𝑘𝑁𝑦^𝑘−𝑁(𝑧)²+2𝛼_𝑘𝑁𝑦^𝑘−𝑁(𝑧)𝑒^𝑘+𝛼_𝑘²𝑒^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2≤𝑦^𝑘−𝑧²+2𝛼_𝑘𝑦^𝑘−𝑧𝑒^𝑘+𝛼_𝑘²𝑒^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2. (3.12) From (3.6), we obtain 𝑦^𝑘−𝑧²=𝑥^𝑘−𝑧−𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘2=𝑥^𝑘−𝑧²−2𝜃_𝑘𝑥^𝑘−𝑧,𝑥^𝑘−1−𝑥^𝑘+𝜃²_𝑘𝑥^𝑘−1−𝑥^𝑘2. (3.13) By (3.12) and observing that 𝜃²_𝑘≤𝜃_𝑘 (since 𝜃_𝑘∈[0,1]), we have 𝑥^𝑘+1−𝑧²≤𝑥^𝑘−𝑧²−2𝜃_𝑘𝑥^𝑘−𝑧,𝑥^𝑘−1−𝑥^𝑘+𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘2+2𝛼_𝑘𝑦^𝑘−𝑧𝑒^𝑘+𝛼²_𝑘𝑒^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2. (3.14) Combining ⟨𝑎,𝑏⟩=−(1/2)‖𝑎−𝑏‖²+(1/2)‖𝑎‖²+(1/2)‖𝑏‖² with (3.14), we get 𝑥^𝑘+1−𝑧²−𝑥^𝑘−𝑧²≤𝜃_𝑘𝑥^𝑘−𝑧²−𝑥^𝑘−1−𝑧²+2𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘2+2𝛼_𝑘𝑦^𝑘−𝑧𝑒^𝑘+𝛼²_𝑘𝑒^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2≤𝜃_𝑘𝑥^𝑘−𝑧²−𝑥^𝑘−1−𝑧²+2𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘2+2𝛼_𝑘𝑥^𝑘−𝑧+𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘𝑒^𝑘+𝛼²_𝑘𝑒^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2. (3.15) We have known that the sequence {𝑥^𝑘} is bounded and {‖𝑥^𝑘−𝑧‖} is convergent; hence, there exist 𝜌≥𝜌>0 and 𝐺>0 such that ‖𝑥^𝑘‖≤𝜌 and ‖𝑥^𝑘−𝑧‖≤𝐺 for all 𝑘.
Thus 𝑒^𝑘=𝑁_𝑘𝑦^𝑘−𝑁𝑦^𝑘≤𝐷_𝜌𝑁_𝑘,𝑁. (3.16) Moreover, one has that 𝑥^𝑘+1−𝑧²−𝑥^𝑘−𝑧²≤𝜃_𝑘𝑥^𝑘−𝑧²−𝑥^𝑘−1−𝑧²+2𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘2+2𝛼_𝑘𝐺𝐷_𝜌𝑁_𝑘,𝑁+2𝛼_𝑘𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘𝐷_𝜌𝑁_𝑘,𝑁+𝛼²_𝑘𝐷_𝜌𝑁_𝑘,𝑁²−𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2. (3.17) Denoting 𝜎_𝑘=2𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘2+2𝛼_𝑘𝐺𝐷_𝜌𝑁_𝑘,𝑁+2𝛼_𝑘𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘𝐷_𝜌𝑁_𝑘,𝑁+𝛼²_𝑘𝐷_𝜌𝑁_𝑘,𝑁², (3.18) we get 𝑥^𝑘+1−𝑧²−𝑥^𝑘−𝑧²≤𝜃_𝑘𝑥^𝑘−𝑧²−𝑥^𝑘−1−𝑧²+𝜎_𝑘. (3.19) Similarly, from the selection of parameter 𝜃_𝑘, we have 𝜃_𝑘𝑥^𝑘−𝑥^𝑘−12≤𝑥^𝑘−𝑥^𝑘−12max𝑘²𝑥^𝑘−𝑥^𝑘−12,𝑘²𝑥^𝑘−𝑥^𝑘−1≤1𝑘². (3.20) It is easy to get _𝑘≥0𝑥^𝑘−𝑥^𝑘−12<∞. (3.21) Both (3.10) and (3.21) manifest _𝑘≥0𝛼_𝑘𝜃_𝑘𝑥^𝑘−1−𝑥^𝑘𝐷_𝜌𝑁_𝑘,𝑁<∞. (3.22) Then, from (3.4), (3.21), and (3.22), we get ^+∞_𝑘=0𝜎_𝑘<+∞. (3.23) According to Lemma 2.6, we obtain ∑_𝑘≥0[‖𝑥^𝑘−𝑧‖²−‖𝑥^𝑘−1−𝑧‖²]₊<∞.
From (3.17), it follows that ^+∞_𝑘=0𝛼_𝑘1−𝛼_𝑘𝑦^𝑘−𝑁𝑦^𝑘2<+∞. (3.24) Because of (3.3) ∑^+∞_𝑘=0𝛼_𝑘(1−𝛼_𝑘)=+∞, we conclude that liminf_𝑘→∞‖𝑦^𝑘−𝑁(𝑦^𝑘)‖=0.
Finally, we prove that {𝑥^𝑘} converges to a fixed point of 𝑁. From the above computation, we know that the sequence {𝑦^𝑘} is also bounded; hence there exist 𝑥^∗ and a subsequence of {𝑦^𝑘} (denoted {𝑦^𝑘𝑙}) such that lim_𝑙→∞𝑦^𝑘𝑙−𝑥^∗=0,lim_𝑙→∞𝑦^𝑘𝑙−𝑁𝑦^𝑘𝑙=liminf_𝑘→∞𝑦^𝑘−𝑁𝑦^𝑘=0. (3.25) From Lemmas 2.7 and 2.8, we have 𝑥^∗∈Fix(𝑁). It is easy to obtain that lim_𝑙→∞‖𝑥^𝑘𝑙−𝑥^∗‖=0, because ‖𝑦^𝑘−𝑥^𝑘‖=𝜃_𝑘‖𝑥^𝑘−𝑥^𝑘−1‖→0 by (3.10). Since {‖𝑥^𝑘−𝑥^∗‖} is convergent, it follows that lim_𝑘→∞‖𝑥^𝑘−𝑥^∗‖=lim_𝑙→∞‖𝑥^𝑘𝑙−𝑥^∗‖=0. The proof is completed.

Remark 1. Since the current value of ‖𝑥^𝑘−𝑥^𝑘−1‖ is known when choosing the parameter 𝜃_𝑘, then 𝜃_𝑘 is well defined in Theorem 3.3. In fact, from the process of proof for the Theorem 3.3, we can get the following assert: the convergence result of Theorem 3.3 always holds provided that we select 𝜃_𝑘∈[0,𝜃], 𝜃∈[0,1), forall𝑘≥0 with _𝑘≥0𝑥^𝑘−𝑥^𝑘−1<∞,_𝑘≥0𝑥^𝑘−𝑥^𝑘−12<∞. (3.26)

Now let us return to the convergence analysis of Algorithm 3.2.

Theorem 3.4. Let the hypotheses in Lemma 3.1 be satisfied. Then the sequence {𝑥^𝑘} generated by (3.2) converges to a fixed point of 𝑁 if ^+∞_𝑘=0𝛼_𝑘1−𝛼_𝑘=+∞,^+∞_𝑘=0𝛼_𝑘𝑑_𝜌𝐶_𝑘,𝐶+2𝜆^−1/2𝑑_𝜌𝑄_𝑘,𝑄<+∞,𝜌>0, (3.27) where parameter 𝜃_𝑘 is satisfying (3.10) and (3.21).

Proof. For any 𝑦∈𝑅^𝑛, ‖𝑦‖≤𝜌, by reason of the nonexpansive properties of projection, we have 𝑁_𝑘𝑦−𝑁𝑦=𝑃_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝑘𝐴𝑦−𝑃_𝐶𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄≤𝑃𝐴𝑦_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝑘𝐴𝑦−𝑃_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄+𝑃𝐴𝑦_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦−𝑃_𝐶𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝐴𝑦≤𝛾^𝑇𝐼−𝑃_𝑄𝑘𝐴𝑦−𝐴^𝑇𝐼−𝑃_𝑄+𝑃𝐴𝑦_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦−𝑃_𝐶𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦≤𝛾𝜆^1/2𝑃_𝑄𝑘𝐴𝑦−𝑃_𝑄+𝑃𝐴𝑦_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦−𝑃_𝐶𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦≤2𝜆^−1/2𝑃_𝑄𝑘𝐴𝑦−𝑃_𝑄+𝑃𝐴𝑦_𝐶𝑘𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦−𝑃_𝐶𝑦−𝛾𝐴^𝑇𝐼−𝑃_𝑄𝐴𝑦. (3.28) Obviously, 𝐷_𝜌𝑁_𝑘,𝑁≤𝑑_𝜌𝐶_𝑘,𝐶+2𝜆^−1/2𝑑_𝜌𝑄_𝑘,𝑄, (3.29) where 𝜌≥max{‖𝐴𝑦‖,‖𝑦−𝛾𝐴^𝑇(𝐼−𝑃_𝑄)𝐴𝑦‖}.
Since ∑^+∞_𝑘=0𝛼_𝑘(𝑑_𝜌(𝐶_𝑘,𝐶)+2𝜆^−1/2𝑑_𝜌(𝑄_𝑘,𝑄))<+∞ for any given 𝜌>0, the result of this theorem can be obtained using Theorem 3.3.

Acknowledgments

This paper is supported by National Science Foundation of China (under Grant no. 11171221), Shanghai Municipal Committee of Science and Technology (under Grant no. 10550500800), Shanghai Municipal Government (under Grant no. S30501), Basic and Frontier Research Program of Science and Technology Department of Henan Province (under Grant nos. 112300410277, 082300440150), and China Coal Industry Association Scientific and Technical Guidance to Project (under Grant no. MTKJ-2011-403).