Regularized Methods for the Split Feasibility Problem

Abstract

Many applied problems such as image reconstructions and signal processing can be formulated as the split feasibility problem (SFP). Some algorithms have been introduced in the literature for solving the (SFP). In this paper, we will continue to consider the convergence analysis of the regularized methods for the (SFP). Two regularized methods are presented in the present paper. Under some different control conditions, we prove that the suggested algorithms strongly converge to the minimum norm solution of the (SFP).

1. Introduction

The well-known convex feasibility problem is to find a point 𝑥^∗ satisfying the following:𝑥^∗∈^𝑚_𝑖=1𝐶_𝑖, (1.1) where 𝑚≥1 is an integer, and each 𝐶_𝑖 is a nonempty closed convex subset of a Hilbert space 𝐻. Note that the convex feasibility problem has received a lot of attention due to its extensive applications in many applied disciplines as diverse as approximation theory, image recovery and signal processing, control theory, biomedical engineering, communications, and geophysics (see [1–3] and the references therein).

A special case of the convex feasibility problem is the split feasibility problem (SFP) which is to find a point 𝑥^∗ such that𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄, (1.2) where 𝐶 and 𝑄 are two closed convex subsets of two Hilbert spaces 𝐻₁ and 𝐻₂, respectively, and 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. We use Γ to denote the solution set of the (SFP), that is,Γ={𝑥∈𝐶∶𝐴𝑥∈𝑄}. (1.3) Assume that the (SFP) is consistent. A special case of the (SFP) is the convexly constrained linear inverse problem ([4]) in the finite dimensional Hilbert spaces𝑥^∗∈𝐶,𝐴𝑥^∗=𝑏 (1.4) which has extensively been investigated by using the Landweber iterative method ([5]):With𝑥₀𝑥arbitraryand𝑛=0,1,…,let_𝑛+1=𝑥_𝑛+𝛾𝐴^𝑇𝑏−𝐴𝑥_𝑛. (1.5)

The (SFP) in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [6] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. The original algorithm introduced in [6] involves the computation of the inverse 𝐴⁻¹:𝑥_𝑘+1=𝐴⁻¹𝑃_𝑄𝑃_𝐴(𝐶)𝐴𝑥_𝑘,𝑘≥0, (1.6) where 𝐶,𝑄⊂𝑅^𝑛 are closed convex sets, 𝐴 a full rank 𝑛×𝑛 matrix, and 𝐴(𝐶)={𝑦∈𝑅^𝑛∣𝑦=𝐴𝑥,𝑥∈𝐶} and thus does not become popular. A more popular algorithm that solves the (SFP) seems to be the 𝐶𝑄 algorithm of Byrne ([7, 8]). The 𝐶𝑄 algorithm only involves the computations of the projections 𝑃_𝐶 and 𝑃_𝑄 onto the sets 𝐶 and 𝑄, respectively, and is therefore implementable in the case where 𝑃_𝐶 and 𝑃_𝑄 have closed-form expressions (e.g., 𝐶 and 𝑄 are the closed balls or half-spaces). There are a large number of references on the 𝐶𝑄 method for the (SFP) in the literature, see, for instance, [9–19]. It remains, however; a challenge how to implement the 𝐶𝑄 algorithm in the case where the projections 𝑃_𝐶 and/or 𝑃_𝑄 fail to have closed-form expressions though theoretically we can prove (weak) convergence of the algorithm.

Note that 𝑥∈Γ means that there is an 𝑥∈𝐶 such that 𝐴𝑥−𝑥^∗=0 for some 𝑥^∗∈𝑄. This motivates us to consider the distance function 𝑑(𝐴𝑥,𝑥^∗)=‖𝐴𝑥−𝑥^∗‖ and the minimization problemmin_{𝑥∈𝐶,𝑥∗∈𝑄}12𝐴𝑥−𝑥^∗2. (1.7) Minimizing with respect to 𝑥^∗∈𝑄 first makes us consider the minimization:min_𝑥∈𝐶𝑓(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥². (1.8) However, (1.8) is, in general, ill posed. So regularization is needed. We consider Tikhonov’s regularizationmin_𝑥∈𝐶𝑓_𝛼1∶=2𝐼−𝑃_𝑄𝐴𝑥²+12𝛼‖𝑥‖², (1.9) where 𝛼>0 is the regularization parameter. We can compute the gradient ∇𝑓_𝛼 of 𝑓_𝛼 as∇𝑓_𝛼=∇𝑓(𝑥)+𝛼𝐼=𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼. (1.10) Define a Picard iterates𝑥^𝛼_𝑛+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼𝑥^𝛼_𝑛 (1.11) Xu [20] shown that if the (SFP) (1.2) is consistent, then as 𝑛→∞,𝑥^𝛼_𝑛→𝑥_𝛼, and consequently the strong lim_𝛼→0𝑥_𝛼 exists and is the minimum-norm solution of the (SFP). Note that (1.11) is a double-step iteration. Xu [20] further suggested a single step-regularized method:𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓_𝛼𝑛𝑥_𝑛=𝑃_𝐶1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛. (1.12) Xu proved that the sequence {𝑥_𝑛} converges in norm to the minimum-norm solution of the (SFP) provided that the parameters {𝛼_𝑛} and {𝛾_𝑛} satisfy the following conditions:(i)𝛼_𝑛→0 and 0<𝛾_𝑛≤𝛼_𝑛/(‖𝐴‖²+𝛼_𝑛),(ii)∑_𝑛𝛼_𝑛𝛾_𝑛=∞,(iii)(|𝛾_𝑛+1−𝛾_𝑛|+𝛾_𝑛|𝛼_𝑛+1−𝛼_𝑛|)/(𝛼_𝑛+1𝛾_𝑛+1)²→0.

Recently, the minimum-norm solution and the minimization problems have been considered extensively in the literature. For related works, please see [21–29]. The main purpose of this paper is to further investigate the regularized method (1.12). Under some different control conditions, we prove that this algorithm strongly converges to the minimum norm solution of the (SFP). We also consider an implicit method for finding the minimum norm solution of the (SFP).

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.1) We will use Fix(𝑇) to denote the set of fixed points of 𝑇, that is, Fix(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}. A mapping 𝑇∶𝐶→𝐶 is said to be 𝜈-inverse strongly monotone (𝜈-ism) if there exists a constant 𝜈>0 such that𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶. (2.2) Recall that the (nearest point or metric) projection from 𝐻 onto 𝐶, denoted 𝑃_𝐶, assigns, to each 𝑥∈𝐻, the unique point 𝑃_𝐶(𝑥)∈𝐶 with the property𝑥−𝑃_𝐶(𝑥)=inf𝑥−𝑦∶𝑦∈𝐶. (2.3) It is well known that the metric projection 𝑃_𝐶 of 𝐻 onto 𝐶 has the following basic properties:(a)‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻,(b)⟨𝑥−𝑦,𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)⟩≥‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖² for every 𝑥,𝑦∈𝐻,(c)⟨𝑥−𝑃_𝐶(𝑥),𝑦−𝑃_𝐶(𝑥)⟩≤0 for all 𝑥∈𝐻, 𝑦∈𝐶.

Next we adopt the following notation:(i)𝑥_𝑛→𝑥 means that 𝑥_𝑛 converges strongly to 𝑥,(ii)𝑥_𝑛⇀𝑥 means that 𝑥_𝑛 converges weakly to 𝑥,(iii)𝜔_𝑤(𝑥_𝑛)∶={𝑥∶∃𝑥_𝑛𝑗⇀𝑥} is the weak 𝜔-limit set of the sequence {𝑥_𝑛}.

Lemma 2.1 (see [20]). Given that 𝑥^∗∈𝐻₁. 𝑥^∗ solves the (SFP) if and only if 𝑥^∗ solves the fixed point equation 𝑥^∗=𝑃_𝐶𝑥^∗−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗. (2.4)

Lemma 2.2 (see [8, 20]). We have the following assertions.(a)𝑇 is nonexpansive if and only if the complement 𝐼−𝑇 is 1/2-ism.(b)If 𝑆 is 𝜈-ism, then for 𝛾>0, 𝛾𝑆 is 𝜈/𝛾-ism.(c)𝑆 is averaged if and only if the complement 𝐼−𝑆 is 𝜈-ism for some 𝜈>1/2.(d)If 𝑆 and 𝑇 are both averaged, then the product (composite) 𝑆𝑇 is averaged.

Lemma 2.3 (see [30] Demiclosedness Principle). Let 𝐶 be a closed and convex subset of a Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦. (2.5) In particular, if 𝑦=0, then 𝑥∈Fix(𝑇).

Lemma 2.4 (see [31]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. (2.6) Suppose that 𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛 (2.7) for all 𝑛≥0 and limsup_𝑛→∞𝑦_n+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.8) Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.5 (see [32]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (2.9) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑛=∞, (2)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

In this section, we will state and prove our main results.

Theorem 3.1. Assume that the (SFP) (1.2) is consistent. Let {𝑥_𝑛} be a sequence generated by the following algorithm: 𝑥_𝑛+1=𝑃_𝐶1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛≥0, (3.1) where the sequences {𝛼_𝑛}⊂(0,1) and {𝛾_𝑛}⊂(0,(2)/(‖𝐴‖²+2𝛼_𝑛) satisfy the following conditions:(𝐶1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞,(𝐶2)liminf_𝑛→∞𝛾_𝑛>0 and lim_𝑛→∞(𝛾_𝑛+1−𝛾_𝑛)=0.Then the sequence {𝑥_𝑛} generated by (3.1) strongly converges to the minimum norm solution 𝑥 of the (SFP) (1.2).

Proof. It is known that 𝐴^∗(𝐼−𝑃_𝑄)𝐴 is 1/‖𝐴‖²-ism. Then, we have 𝑃_𝐶1−𝛼𝛾𝑥−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝑃_𝐶1−𝛼𝛾𝑦−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑦²≤1−𝛼𝛾𝑥−𝑦−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝐴^∗𝐼−𝑃_𝑄𝐴𝑦²=1−𝛼𝛾²𝑥−𝑦²−21−𝛼𝛾𝛾𝑥−𝑦,𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝐴^∗𝐼−𝑃_𝑄𝐴𝑦+𝛾²𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝐴^∗𝐼−𝑃_𝑄𝐴𝑦²≤1−𝛼𝛾²𝑥−𝑦²𝛾1−21−𝛼𝛾‖𝐴‖²𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝐴^∗𝐼−𝑃_𝑄𝐴𝑦²+𝛾²𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝐴^∗𝐼−𝑃_𝑄𝐴𝑦². (3.2) If 𝛾∈[0,(2)/‖𝐴‖²+2𝛼], then 2(1−𝛼𝛾)𝛾(1/‖𝐴‖²)≥𝛾². It follows that 𝑃_𝐶1−𝛼𝛾𝑥−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥−𝑃_𝐶1−𝛼𝛾𝑦−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑦²≤1−𝛼𝛾²𝑥−𝑦². (3.3) Thus, 𝑃_𝐶(𝐼−𝛾(𝐴^∗(𝐼−𝑃_𝑄)𝐴+𝛼𝐼)) is a contractive mapping with coefficient 𝜌≤1−𝛼𝛾.
Pick up any 𝑥^∗∈Γ. From Lemma 2.1, 𝑥^∗∈𝐶 solves the (SFP) if and only if 𝑥^∗=𝑃_𝐶(𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴)𝑥^∗ for any fixed positive number 𝛾. So, we have 𝑥^∗=𝑃_𝐶(𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑃_𝑄)𝐴)𝑥^∗ for all 𝑛≥0. From (3.1), we get 𝑥_𝑛+1−𝑥^∗=𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛−𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗≤𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛−𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥^∗+𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥^∗−𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗≤1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾_𝑛𝑥^∗≤max𝑥_𝑛−𝑥^∗,𝑥^∗. (3.4) By induction, we deduce 𝑥_𝑛−𝑥^∗≤max𝑥₀−𝑥^∗,𝑥^∗. (3.5) This indicates that the sequence {𝑥_𝑛} is bounded.
Since 𝐴^∗(𝐼−𝑃_𝑄)𝐴 is ‖𝐴‖²‖-Lipschitz, 𝐴^∗(𝐼−𝑃_𝑄)𝐴 is 1/‖𝐴‖²-ism, which then implies that 𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴 is 1/𝛾‖𝐴‖²-ism. So by Lemma 2.1, 𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑃_𝑄)𝐴 is 𝛾_𝑛‖𝐴‖²/2 averaged. That is, 𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑃_𝑄)𝐴=(1−(𝛾_𝑛‖𝐴‖²/2))𝐼+(𝛾_𝑛‖𝐴‖²/2)𝑇 for some nonexpansive mapping 𝑇. Since 𝑃_𝐶 is 1/2 averaged, 𝑃_𝐶=(𝐼+𝑆)/2 for some nonexpansive mapping 𝑆. Then, we can rewrite 𝑥_𝑛+1 as 𝑥_𝑛+1=12𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛=12𝑥_𝑛−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛=2−𝛾_𝑛‖𝐴‖²4𝑥_𝑛+𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛=2−𝛾_𝑛‖𝐴‖²4𝑥_𝑛+2+𝛾_𝑛‖𝐴‖²4𝑦_𝑛, (3.6) where 𝑦_𝑛=42+𝛾_𝑛‖𝐴‖²𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛. (3.7) It follows that ‖𝑦_𝑛+1−𝑦_𝑛‖=42+𝛾_𝑛+1‖𝐴‖²𝛾_𝑛+1‖𝐴‖²4𝑇𝑥_𝑛+1−12𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+12𝑆𝐼−𝛾_𝑛+1𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛+1𝐼𝑥_𝑛+1−42+𝛾_𝑛‖𝐴‖²𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛≤42+𝛾_𝑛+1‖𝐴‖²𝛾_𝑛+1‖𝐴‖²4𝑇𝑥_𝑛+1−12𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+12𝑆𝐼−𝛾_𝑛+1𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛+1𝐼𝑥_𝑛+1−𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛+42+𝛾_𝑛+1‖𝐴‖²−42+𝛾_𝑛‖𝐴‖²𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛≤42+𝛾_𝑛+1‖𝐴‖²𝛾_𝑛+1‖𝐴‖²4𝑇𝑥_𝑛+1−𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛+12𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+12𝛾_𝑛𝛼_𝑛𝑥_𝑛+22+𝛾_𝑛+1‖𝐴‖²𝐼−𝛾_𝑛+1𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛+1𝐼𝑥_𝑛+1−𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛+42+𝛾_𝑛+1‖𝐴‖²−42+𝛾_𝑛‖𝐴‖²𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛 (3.8) Now we choose a constant 𝑀 such that sup_𝑛𝑥_𝑛,‖𝐴‖²𝑇𝑥_𝑛,𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛≤𝑀. (3.9) We have the following estimates: 𝛾_𝑛+1‖𝐴‖²4𝑇𝑥_𝑛+1−𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛=𝛾_𝑛+1‖𝐴‖²4𝑇𝑥_𝑛+1−𝑇𝑥_𝑛+𝛾_𝑛+1‖𝐴‖²4−𝛾_𝑛‖𝐴‖²4𝑇𝑥_𝑛≤𝛾_𝑛+1‖𝐴‖²4𝑇𝑥_𝑛+1−𝑇𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛‖𝐴‖²𝑇𝑥_𝑛4≤𝛾_𝑛+1‖𝐴‖²4𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑀,𝐼−𝛾_𝑛+1𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛+1𝐼𝑥_𝑛+1−𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛≤𝐼−𝛾_𝑛+1𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+1−𝐼−𝛾_𝑛+1𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+𝛾_𝑛𝛼_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀. (3.10) Thus, we deduce that 𝑦_𝑛+1−𝑦_𝑛≤42+𝛾_𝑛+1‖𝐴‖²𝛾_𝑛+1‖𝐴‖²4𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑀+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀+22+𝛾_𝑛+1‖𝐴‖²𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀+42+𝛾_𝑛+1‖𝐴‖²−42+𝛾_𝑛‖𝐴‖²𝑀≤𝑥_𝑛+1−𝑥_𝑛+62+𝛾_𝑛+1‖𝐴‖²𝛾_𝑛+1−𝛾_𝑛+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀+4‖𝐴‖²2+𝛾_𝑛+1‖𝐴‖²2+𝛾_𝑛‖𝐴‖²𝛾_𝑛+1−𝛾_𝑛𝑀. (3.11) Note that 𝛼_𝑛→0 and 𝛾_𝑛+1−𝛾_𝑛→0. Hence, by Lemma 2.3, we get the following: limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.12) It follows that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.13) Consequently, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞2+𝛾_𝑛‖𝐴‖²4𝑦_𝑛−𝑥_𝑛=0. (3.14) Now we show that the weak limit set 𝜔_𝑤(𝑥_𝑛)⊂Γ. Choose any 𝑥∈𝜔_𝑤(𝑥_𝑛). Since {𝑥_𝑛} is bounded, there must exist a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥. At the same time, the real number sequence {𝛾_𝑛𝑗} is bounded. Thus, there exists a subsequence {𝛾_𝑛𝑗𝑖} of {𝛾_𝑛𝑗} which converges to 𝛾. Without loss of generality, we may assume that 𝛾_𝑛𝑗→𝛾. Note that 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/‖𝐴‖². So, 𝛾∈(0,2/‖𝐴‖²). That is, 𝛾_𝑛𝑗→𝛾∈(0,2/‖𝐴‖²) as 𝑗→∞. Next, we only need to show that 𝑥∈Γ. First, from (3.14) we have that ‖𝑥_𝑛𝑗+1−𝑥_𝑛𝑗‖→0. Then, we have the following: 𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗≤𝑥_𝑛𝑗−𝑥_𝑛𝑗+1+𝑥_𝑛𝑗+1−𝑃_𝐶𝐼−𝛾_𝑛𝑗𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗+𝑃_𝐶𝐼−𝛾_𝑛𝑗𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗=𝑃_𝐶𝐼−𝛾_𝑛𝑗𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝑗𝐼𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾_𝑛𝑗𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗+𝑃_𝐶𝐼−𝛾_𝑛𝑗𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥_𝑛𝑗+1≤𝛼_𝑛𝑗𝛾_𝑛𝑗𝑥_𝑛𝑗+𝛾_𝑛𝑗−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥_𝑛𝑗+1⟶0. (3.15) Since 𝛾∈(0,2/‖𝐴‖²),𝑃_𝐶(𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴) is nonexpansive. It then follows from Lemma 2.4 (demiclosedness principle) that 𝑥∈Fix(𝑃_𝐶(𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴)). Hence, 𝑥∈Γ because Ω=Fix(𝑃_𝐶(𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴)). So, 𝜔_𝑤(𝑥_𝑛)⊂Γ.
Finally, we prove that 𝑥_𝑛→𝑥, where 𝑥 is the minimum norm solution of (1.2). First, we show that limsup_𝑛→∞⟨𝑥,𝑥_𝑛−𝑥⟩≥0. Observe that there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} satisfying that limsup_𝑛→∞𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝑥,𝑥_𝑛𝑗−𝑥. (3.16) Since {x_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑖} of {𝑥_𝑛𝑗} such that 𝑥_𝑛𝑗𝑖⇀𝑥. Without loss of generality, we assume that 𝑥_𝑛𝑗⇀𝑥. Then, we obtain the following: limsup_𝑛→∞𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝑥,𝑥_𝑛𝑗−𝑥=⟨𝑥,𝑥−𝑥⟩≥0. (3.17) Since 𝛾_𝑛<2/(‖𝐴‖²+2𝛼_𝑛),𝛾_𝑛/(1−𝛼_𝑛𝛾_𝑛)<2/‖𝐴‖². So, 𝐼−(𝛾_𝑛/(1−𝛼_𝑛𝛾_𝑛))𝐴^∗(𝐼−𝑃_𝑄)𝐴 is nonexpansive. By using the property (b) of 𝑃_𝐶, we have the following: 𝑥_𝑛+1−𝑥²=𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛−𝑃_𝐶𝑥−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥²≤𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛−𝑥−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,𝑥_𝑛+1=−𝑥1−𝛼_𝑛𝛾_𝑛𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,𝑥_𝑛+1−𝑥−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1≤−𝑥1−𝛼_𝑛𝛾_𝑛𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥𝑥_𝑛+1−𝑥−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1≤−𝑥1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1≤−𝑥1−𝛼_𝑛𝛾_𝑛2𝑥_𝑛−𝑥²+12𝑥_𝑛+1−𝑥²−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1−𝑥. (3.18) It follows that 𝑥_𝑛+1−𝑥²≤1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥²+𝛼_𝑛𝛾_𝑛−𝑥,𝑥_𝑛+1−𝑥. (3.19) From Lemma 2.5, (3.17), and (3.19), we deduce that 𝑥_𝑛→𝑥. This completes the proof.

Remark 3.2. We obtain the strong convergence of the regularized method (3.1) under control conditions (𝐶1) and (𝐶2). In Xu’s [20] result, 𝛾_𝑛→0. However, in our result, liminf_𝑛→0𝛾_𝑛>0.
Finally, we introduce an implicit method for the (SFP).
Take a constant 𝛾 such that 0<𝛾<2/‖𝐴‖². For 𝑡∈(0,1), we define a mapping𝑊_𝑡∶=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄‖𝐴‖𝐴+𝑡𝐼,𝑡∈0,2−𝛾².2𝛾 (3.20) For 𝑡∈(0,1), we know that 𝐴^∗(𝐼−𝑃_𝑄)𝐴+𝑡𝐼 is (𝑡+‖𝐴‖²)-Lipschitz and 𝑡-strongly monotone. Thus, 𝑊_𝑡=𝑃_𝐶(𝐼−𝛾(𝐴^∗(𝐼−𝑃_𝑄)𝐴+𝑡𝐼)) is a contractive. So, 𝑊_𝑡 has a unique fixed point in 𝐶, denoted by 𝑥_𝑡, that is, 𝑥_𝑡=𝑃_𝐶𝑥_𝑡−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡+𝑡𝑥_𝑡‖𝐴‖,𝑡∈0,2−𝛾².2𝛾 (3.21) Next, we show the convergence of the net {𝑥_𝑡} defined by (3.21).

Theorem 3.3. Assume that the (SFP) (1.2) is consistent. As 𝑡→0+, the net {𝑥_𝑡} defined by (3.21) converges to the minimum norm solution of the (SFP).

Proof. Let 𝑥 be any a point in Γ. We can rewrite (3.21) as 𝑥_𝑡=𝑃_𝐶1−𝑡𝛾𝑥_𝑡−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡‖𝐴‖,𝑡∈0,2−𝛾².2𝛾 (3.22) Since 𝑡∈(0,(2−𝛾‖𝐴‖²)/2𝛾),𝐼−(𝛾/(1−𝑡𝛾))𝐴^∗(𝐼−𝑃_𝑄)𝐴 is nonexpansive. It follows that 𝑥_𝑡−𝑥=𝑃_𝐶1−𝑡𝛾𝑥_𝑡−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡−𝑃_𝐶𝑥−𝛾𝐴^∗𝐼−𝑃_𝑄≤𝑥𝐴𝑥1−𝑡𝛾_𝑡−𝛾𝐴1−𝑡𝛾^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡−𝛾1−𝑡𝛾𝑥−𝐴1−𝑡𝛾^∗𝐼−𝑃_𝑄≤𝑥𝐴𝑥−𝑡𝛾𝑥1−𝑡𝛾_𝑡−𝛾𝐴1−𝑡𝛾^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡−𝛾𝑥−𝐴1−𝑡𝛾^∗𝐼−𝑃_𝑄‖‖≤𝑥𝐴𝑥+𝑡𝛾𝑥1−𝑡𝛾_𝑡−𝑥+𝑡𝛾‖𝑥‖. (3.23) Hence, 𝑥_𝑡−𝑥≤‖𝑥‖. (3.24) Then, {𝑥_𝑡} is bounded.
From (3.21), we have the following: 𝑥_𝑡−𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_t≤𝑡𝛾𝑥_𝑡⟶0. (3.25) Next we show that {𝑥_𝑡} is relatively norm compact as 𝑡→0+. Assume that {𝑡_𝑛}⊂(0,(2−𝛾‖𝐴‖²)/2𝛾) is such that 𝑡_𝑛→0+ as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛. From (3.25), we have the following: lim_𝑛→∞𝑥_𝑛−𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡=0. (3.26) By using the property of the projection, we get the following: 𝑥_𝑡−𝑥²=𝑃_𝐶1−𝑡𝛾𝑥_𝑡−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡−𝑃_𝐶𝑥−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥²≤1−𝑡𝛾𝑥_𝑡−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡−𝑥−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,𝑥_𝑡=𝑥−𝑥1−𝑡𝛾_𝑡−𝛾𝐴1−𝑡𝛾^∗𝐼−𝑃_𝑄𝐴𝑥_𝑡−𝛾𝑥−1−𝑡𝛾𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,𝑥_𝑡−𝑥−𝑡𝛾𝑥,𝑥_𝑡≤𝑥−𝑥1−𝑡𝛾_𝑡−𝑥²−𝑡𝛾𝑥,𝑥_𝑡−𝑥. (3.27) Hence, 𝑥_𝑡−𝑥²≤−𝑥,𝑥_𝑡−𝑥. (3.28) In particular, 𝑥_𝑛−𝑥²≤−𝑥,𝑥_𝑛−𝑥,𝑥∈Γ. (3.29) Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} which converges weakly to a point 𝑥^∗. Without loss of generality, we may assume that {𝑥_𝑛} converges weakly to 𝑥^∗. Noticing (3.26) we can use Lemma 2.3 to get 𝑥^∗∈Γ. Therefore, we can substitute 𝑥^∗ for 𝑥 in (3.29) to get the following: 𝑥_𝑛−𝑥^∗2≤−𝑥^∗,𝑥_𝑛−𝑥^∗. (3.30) Consequently, 𝑥_𝑛⇀𝑥^∗ actually implies that 𝑥_𝑛→𝑥^∗. This has proved the relative norm-compactness of the net {𝑥_𝑡} as 𝑡→0+. Letting 𝑛→∞ in (3.29), we have 𝑥^∗−𝑥²≤−𝑥,𝑥^∗−𝑥,𝑥∈Γ. (3.31) This implies that −𝑥,𝑥−𝑥^∗≤0,𝑥∈Γ, (3.32) which is equivalent to the following −𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈Γ. (3.33) Hence, 𝑥^∗=𝑃_Γ(0). Therefore, each cluster point of {𝑥_𝑡} (as 𝑡→0+) equals 𝑥^∗. So, 𝑥_𝑡→𝑥^∗. This completes the proof.

Acknowledgments

Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105. W. Jigang is supported in part by NSFC 61173032. Y.-C. Liou was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Sub-Projects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012.