Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces

Abstract

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let 𝐻₁, 𝐻₂, and  𝐻₃ be infinite-dimensional real Hilbert spaces, let 𝐶⊂𝐻₁ and  𝑄⊂𝐻₂ be two nonempty closed convex sets, and let 𝐴∶𝐻₁→𝐻₃ and  𝐵∶𝐻₂→𝐻₃ be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function 𝑓(𝑥,𝑦)=(1/2)‖𝐴𝑥−𝐵𝑦‖²₂ over 𝑥∈𝐶 and 𝑦∈𝑄. Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum-norm solution of the ASEP. Note that, by taking 𝐵=𝐼 in the ASEP, we recover the approximate split feasibility problem (ASFP).

1. Introduction

Let 𝐶⊆𝑅^𝑁 and 𝑄⊆𝑅^𝑀 be closed, nonempty convex sets, and let 𝐴 and 𝐵 be 𝐽 by 𝑁 and 𝐽 by 𝑀 real matrices, respectively. The split equality problem (SEP) in finite-dimensional Hilbert spaces is to find 𝑥∈𝐶 and 𝑦∈𝑄 such that 𝐴𝑥=𝐵𝑦; the approximate split equality problem (ASEP) in finite-dimensional Hilbert spaces is to minimize the function 𝑓(𝑥,𝑦)=(1/2)‖𝐴𝑥−𝐵𝑦‖²₂ over 𝑥∈𝐶 and 𝑦∈𝑄. When 𝐽=𝑀 and 𝐵=𝐼, the SEP reduces to the well-known split feasibility problem (SFP) and the ASEP becomes the approximate split feasibility problem (ASFP). For information on the split feasibility problem, please see [1–9].

In this paper, we work in the framework of infinite-dimensional Hilbert spaces. Let 𝐻₁, 𝐻₂, and 𝐻₃ be infinite-dimensional real Hilbert spaces, let 𝐶⊂𝐻₁ and 𝑄⊂𝐻₂ be two nonempty closed convex sets, and let 𝐴∶𝐻₁→𝐻₃ and 𝐵∶𝐻₂→𝐻₃ be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is =1tominimizethefunction𝑓𝑥,𝑦2𝐴𝑥−𝐵𝑦²₂ (1) over 𝑥∈𝐶 and 𝑦∈𝑄.

Very recently, for solving the SEP, Moudafi introduced the following alternating CQ-algorithms (ACQA) in [10]: 𝑥_𝑘+1=𝑃_𝐶𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝑃_𝑄𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘+1−𝐵𝑦_𝑘. (2)

Then, he proved the weak convergence of the sequence {𝑥_𝑘,𝑦_𝑘} to a solution of the SEP provided that the solution set Γ∶={𝑥∈𝐶,𝑦∈𝑄;𝐴𝑥=𝐵𝑦} is nonempty and some conditions on the sequence of positive parameters (𝛾_𝑘) are satisfied.

The ACQA involves two projections 𝑃_𝐶 and 𝑃_𝑄 and, hence, might be hard to be implemented in the case where one of them fails to have a closed-form expression. So, Moudafi proposed the following relaxed CQ-algorithm (RACQA) in [11]: 𝑥_𝑘+1=𝑃_𝐶𝑘𝑥_𝑘−𝛾𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝑃_𝑄𝑘𝑦_𝑘+𝛽𝐵^∗𝐴𝑥_𝑘+1−𝐵𝑦_𝑘, (3) where 𝐶_𝑘, 𝑄_𝑘 were defined in [11], and then he proved the weak convergence of the sequence {𝑥_𝑘,𝑦_𝑘} to a solution of the SEP.

In [12], Byrne considered and studied the algorithms to solve the approximate split equality problem (ASEP), which can be regarded as containing the consistent case and the inconsistent case of the SEP. There, he proposed a simultaneous iterative algorithm (SSEA) as follows: 𝑥^𝑘+1=𝑃_𝐶𝑥^𝑘−𝛾_𝑘𝐴^𝑇𝐴𝑥^𝑘−𝐵𝑦^𝑘,𝑦^𝑘+1=𝑃_𝑄𝑦^𝑘+𝛾_𝑘𝐵^𝑇𝐴𝑥^𝑘−𝐵𝑦^𝑘, (4) where 𝜖≤𝛾_𝑘≤(2/𝜌(𝐺^𝑇𝐺))−𝜖. Then, he proposed the relaxed SSEA (RSSEA) and the perturbed version of the SSEA (PSSEA) for solving the ASEP, and he proved their convergence. Furthermore, he used these algorithms to solve the approximate split feasibility problem (ASFP), which is a special case of the ASEP. Note that he used the projected Landweber algorithm as a tool in that article.

Note that the algorithms proposed by Moudafi and Byrne have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we use the regularization method to establish a single-step iterative to solve the ASEP in infinite-dimensional Hilbert spaces, and we will prove its strong convergence.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively, and let 𝐾 be a nonempty closed convex subset of 𝐻. Recall that the projection from 𝐻 onto 𝐾, denoted as 𝑃_𝐾, is defined in such a way that, for each 𝑥∈𝐻, 𝑃_𝐾𝑥 is the unique point in 𝐾 with the property 𝑥−𝑃_𝐾𝑥=min𝑥−𝑦∶𝑦∈𝐾. (5)

The following important properties of projections are useful to our study.

Proposition 1. Given that 𝑥∈𝐻 and 𝑧∈𝐾;(a)𝑧=𝑃_𝐾𝑥   if and only if ⟨𝑥−𝑧,𝑦−𝑧⟩≤0, for all 𝑦∈𝐾;(b)⟨𝑃_𝐾𝑢−𝑃_𝐾𝑣,𝑢−𝑣⟩≥‖𝑃_𝐾𝑢−𝑃_𝐾𝑣‖², for all 𝑢,𝑣∈𝐻.

Definition 2. A mapping 𝑇∶𝐻→𝐻 is said to be(a)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻; (6)(b)firmly nonexpansive if 2𝑇−𝐼 is nonexpansive, or equivalently, 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐻; (7)alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (8) where 𝑆∶𝐻→𝐻 is nonexpansive. It is well known that projections are (firmly) nonexpansive.

Definition 3. Let 𝑇 be a nonlinear operator whose domain is 𝐷(𝑇)⊆𝐻 and whose range is 𝑅(𝑇)⊆𝐻.(a)𝑇 is said to be monotone if 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐷(𝑇). (9)
(b) Given a number 𝛽>0, 𝑇 is said to be 𝛽-strongly monotone if 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (10)
(c) Given a number 𝐿>0,  𝑇 is said to be 𝐿-Lipschitz if 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐷(𝑇). (11)

Lemma 4 (see [13]). Assume that 𝑎_𝑛 is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0, (12) where 𝛾_𝑛, 𝛿_𝑛 are sequences of real numbers such that(i)𝛾_𝑛⊂(0,1) and ∑^∞_𝑛=0𝛾_𝑛=∞;(ii)either limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=1𝛾_𝑛|𝛿_𝑛|<∞.
Then, lim_𝑛→∞𝑎_𝑛=0.

Next, we will state and prove our main result in this paper.

3. Regularization Method for the ASEP

Let 𝑆=𝐶×𝑄. Define ,𝑥𝑦.𝐺=𝐴−𝐵𝜔= (13)

The ASEP can now be reformulated as finding 𝜔∈𝑆 with minimizing the function ‖𝐺𝜔‖ over 𝜔∈𝑆. Therefore, solving the ASEP (1) is equivalent to solving the following minimization problem (14).

The minimization problem min_𝜔∈𝑆𝑓(𝜔)=12‖𝐺𝜔‖² (14) is generally ill-posed. We consider the Tikhonov regularization (for more details about Tikhonov approximation, please see [8, 14] and the references therein) min_𝜔∈𝑆𝑓_𝜀(𝜔)=12‖𝐺𝜔‖²+12𝜀‖𝜔‖², (15) where 𝜀>0 is the regularization parameter. The regularization minimization (15) has a unique solution which is denoted by 𝜔_𝜀. Assume that the minimization (14) is consistent, and let 𝜔_min be its minimum-norm solution; namely, 𝜔_min∈Γ (Γ is the solution set of the minimization (14)) has the property 𝜔_min=min{‖𝜔‖∶𝜔∈Γ}. (16)

The following result is easily proved.

Proposition 5. If the minimization (14) is consistent, then the strong lim_𝜀→0𝜔_𝜀 exists and is the minimum-norm solution of the minimization (14).

Proof. For any 𝜔∈Γ, we have 𝑓(𝜔)+𝜀2𝜔_𝜀²≤𝑓𝜔_𝜀+𝜀2𝜔_𝜀²=𝑓_𝜀𝜔_𝜀≤𝑓_𝜀(𝜔)=𝑓(𝜔)+𝜀2‖𝜔‖². (17)
It follows that, for all 𝜀>0 and 𝜔∈Γ, 𝜔_𝜀≤‖𝜔‖. (18)
Therefore, 𝜔_𝜀 is bounded. Assume that 𝜀_𝑗→0 is such that 𝜔_𝜀𝑗⇀𝜔^∗. Then, the weak lower semicontinuity of 𝑓 implies that, for any 𝜔∈𝑆, 𝑓𝜔^∗≤liminf_𝑗→∞𝑓𝜔_𝜀𝑗≤liminf_𝑗→∞𝑓_𝜀𝑗𝜔_𝜀𝑗≤liminf_𝑗→∞𝑓_𝜀𝑗(𝜔)=liminf_𝑗→∞𝑓(𝜔)+𝜀_𝑗2‖𝜔‖²=𝑓(𝜔). (19)
This means that 𝜔^∗∈Γ. Since the norm is weak lower semicontinuous, we get from (18) that ‖𝜔^∗‖≤‖𝜔‖ for all 𝜔∈Γ; hence, 𝜔^∗=𝜔_min. This is sufficient to ensure that 𝜔_𝜀⇀𝜔_min. To obtain the strong convergence, noting that (18) holds for 𝜔_min, we compute 𝜔_𝜀−𝜔_min²=𝜔_𝜀²−2𝜔_𝜀,𝜔_min+𝜔_min²≤2𝜔_min²−𝜔_𝜀,𝜔_min. (20)
Since 𝜔_𝜀⇀𝜔_min, we get 𝜔_𝜀→𝜔_min in norm. So, we complete the proof.

Next we will state that 𝜔_min can be obtained by two steps. First, observing that the gradient ∇𝑓_𝜀=∇𝑓+𝜀𝐼=𝐺^𝑇𝐺+𝜀𝐼 (21) is (𝜀+‖𝐺‖²)-Lipschitz and 𝜀-strongly monotone, the mapping 𝑃_𝑆(𝐼−𝛾∇𝑓_𝜀) is a contraction with coefficient 1−𝛾2𝜀−𝛾‖𝐺‖²+𝜀²≤11−𝜀𝛾≤1−2𝜀𝛾, (22) where 𝜀0<𝛾≤‖𝐺‖²+𝜀². (23)

Indeed, observe that 𝑃_𝑆𝐼−𝛾∇𝑓_𝜀(𝑥)−𝑃_𝑆𝐼−𝛾∇𝑓_𝜀𝑦²≤𝐼−𝛾∇𝑓_𝜀(𝑥)−𝐼−𝛾∇𝑓_𝜀𝑦²=𝑥−𝑦²−2𝛾∇𝑓_𝜀(𝑥)−∇𝑓_𝜀𝑦,𝑥−𝑦+𝛾²∇𝑓_𝜀(𝑥)−∇𝑓_𝜀𝑦²≤1−2𝛾𝜀+𝛾²𝜀+‖𝐺‖²²𝑥−𝑦²=1−𝛾2𝜀−𝛾𝜀+‖𝐺‖²²𝑥−𝑦²≤1−𝜀𝛾𝑥−𝑦². (24)

Note that 𝜔_𝜀 is a fixed point of the mapping 𝑃_𝑆(𝐼−𝛾∇𝑓_𝜀) for any 𝛾>0 satisfying (23) and can be obtained through the limit as 𝑛→∞ of the sequence of Picard iterates as follows: 𝜔^𝜀_𝑛+1=𝑃_𝑆𝐼−𝛾∇𝑓_𝜀𝜔^𝜀_𝑛. (25)

Secondly, letting 𝜀→0 yields 𝜔_𝜀→𝜔_min in norm. It is interesting to know whether these two steps can be combined to get 𝜔_min in a single step. The following result shows that for suitable choices of 𝛾 and 𝜀, the minimum-norm solution 𝜔_min can be obtained by a single step, motivated by Xu [8].

Theorem 6. Assume that the minimization problem (14) is consistent. Define a sequence 𝜔_𝑛 by the iterative algorithm 𝜔_𝑛+1=𝑃_𝑆𝐼−𝛾_𝑛∇𝑓_𝜀𝑛𝜔_𝑛=𝑃_𝑆1−𝜀_𝑛𝛾_𝑛𝜔_𝑛−𝛾_𝑛𝐺^𝑇𝐺𝜔_𝑛, (26) where 𝜀_𝑛 and 𝛾_𝑛 satisfy the following conditions:(i)0<𝛾_𝑛≤𝜀_𝑛/(‖𝐺‖²+𝜀_𝑛)² for all (large enough) 𝑛;(ii)𝜀_𝑛→0 and 𝛾_𝑛→0;(iii)∑^∞_𝑛=1𝜀_𝑛𝛾_𝑛=∞;(iv)(|𝛾_𝑛+1−𝛾_𝑛|+𝛾_𝑛|𝜀_𝑛+1−𝜀_𝑛|)/(𝜀_𝑛+1𝛾_𝑛+1)²→0.
Then, 𝜔_𝑛 converges in norm to the minimum-norm solution of the minimization problem (14).

Proof. Note that for any 𝛾 satisfying (23), 𝜔_𝜀 is a fixed point of the mapping 𝑃_𝑆(𝐼−𝛾∇𝑓_𝜀). For each 𝑛, let 𝑧_𝑛 be the unique fixed point of the contraction 𝑇_𝑛∶=𝑃_𝑆𝐼−𝛾_𝑛∇𝑓_𝜀𝑛. (27)
Then, 𝑧_𝑛=𝜔_𝜀𝑛, and so 𝑧_𝑛⟶𝜔_mininnorm. (28)
Thus, to prove the theorem, it suffices to prove that 𝜔_𝑛+1−𝑧_𝑛⟶0. (29)
Noting that 𝑇_𝑛 has contraction coefficient of (1−(1/2)𝜀_𝑛𝛾_𝑛), we have 𝜔_𝑛+1−𝑧_𝑛=𝑇_𝑛𝜔_𝑛−𝑇_𝑛𝑧_𝑛≤11−2𝜀_𝑛𝛾_𝑛𝜔_𝑛−𝑧_𝑛≤11−2𝜀_𝑛𝛾_𝑛𝜔_𝑛−𝑧_𝑛−1+𝑧_𝑛−𝑧_𝑛−1. (30)
We now estimate 𝑧_𝑛−𝑧_𝑛−1=𝑇_𝑛𝑧_𝑛−𝑇_𝑛−1𝑧_𝑛−1≤𝑇_𝑛𝑧_𝑛−𝑇_𝑛𝑧_𝑛−1+𝑇_𝑛𝑧_𝑛−1−𝑇_𝑛−1𝑧_𝑛−1≤11−2𝜀_𝑛𝛾_𝑛𝑧_𝑛−𝑧_𝑛−1+𝑇_𝑛𝑧_𝑛−1−𝑇_𝑛−1𝑧_𝑛−1. (31)
This implies that 𝑧_𝑛−𝑧_𝑛−1≤2𝜀_𝑛𝛾_𝑛𝑇_𝑛𝑧_𝑛−1−𝑇_𝑛−1𝑧_𝑛−1. (32)
However, since 𝑧_𝑛 is bounded, we have, for an appropriate constant 𝑀>0, 𝑇_𝑛𝑧_𝑛−1−𝑇_𝑛−1𝑧_𝑛−1=𝑃_𝑆𝐼−𝛾_𝑛∇𝑓_𝜀𝑛𝑧_𝑛−1−𝑃_𝑆𝐼−𝛾_𝑛−1∇𝑓_𝜀𝑛−1𝑧_𝑛−1≤𝐼−𝛾_𝑛∇𝑓_𝜀𝑛𝑧_𝑛−1−𝐼−𝛾_𝑛−1∇𝑓_𝜀𝑛−1𝑧_𝑛−1=𝛾_𝑛∇𝑓_𝜀𝑛𝑧_𝑛−1−𝛾_𝑛−1∇𝑓_𝜀𝑛−1𝑧_𝑛−1=𝛾_𝑛−𝛾_𝑛−1∇𝑓_𝜀𝑛𝑧_𝑛−1+𝛾_𝑛−1∇𝑓_𝜀𝑛𝑧_𝑛−1−∇𝑓_𝜀𝑛−1𝑧_𝑛−1≤𝛾_𝑛−𝛾_𝑛−1∇𝑓𝑧_𝑛−1+𝜀_𝑛𝑧_𝑛−1+𝛾_𝑛−1𝜀_𝑛−𝜀_𝑛−1𝑧_𝑛−1≤𝛾_𝑛−𝛾_𝑛−1+𝛾_𝑛−1𝜀_𝑛−𝜀_𝑛−1𝑀. (33)
Combining (30), (32), and (33), we obtain 𝜔_𝑛+1−𝑧_𝑛≤11−2𝜀_𝑛𝛾_𝑛𝜔_𝑛−𝑧_𝑛−1+12𝜀_𝑛𝛾_𝑛𝛽_𝑛, (34) where 𝛽_𝑛=4𝑀𝛾_𝑛−𝛾_𝑛−1+𝛾_𝑛−1𝜀_𝑛−𝜀_𝑛−1𝜀_𝑛𝛾_𝑛²⟶0. (35)
Now applying Lemma 4 to (34) and using the conditions (ii)–(iv), we conclude that ‖𝜔_𝑛+1−𝑧_𝑛‖→0; therefore, 𝜔_𝑛→𝜔_min in norm.

Remark 7. Note that 𝜀_𝑛=𝑛^−𝛿 and 𝛾_𝑛=𝑛^−𝜎 with 0<𝛿<𝜎<1 and 𝜎+2𝛿<1  satisfy the conditions (i)–(iv).

Remark 8. We can express the algorithm (26) in terms of 𝑥 and 𝑦, and we get 𝑥_𝑛+1=𝑃_𝐶1−𝜀_𝑛𝛾_𝑛𝑥_𝑛−𝛾_𝑛𝐴^𝑇𝐴𝑥_𝑛−𝐵𝑦_𝑛,𝑦_𝑛+1=𝑃_𝑄1−𝜀_𝑛𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝐵^𝑇𝐴𝑥_𝑛−𝐵𝑦_𝑛. (36)
And we can obtain that the whole sequence (𝑥_𝑛,𝑦_𝑛) generated by the algorithm (36) strongly converges to the minimum-norm solution of the ASEP (1) provided that the ASEP (1) is consistent and 𝜀_𝑛 and 𝛾_𝑛 satisfy the conditions (i)–(iv).

Remark 9. Now, we apply the algorithm (36) to solve the ASFP. Let 𝐵=𝐼; the iteration in (36) becomes 𝑥_𝑛+1=𝑃_𝐶1−𝜀_𝑛𝛾_𝑛𝑥_𝑛−𝛾_𝑛𝐴^𝑇𝐴𝑥_𝑛−𝑦_𝑛,𝑦_𝑛+1=𝑃_𝑄1−𝜀_𝑛𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝐴𝑥_𝑛−𝑦_𝑛. (37)
This algorithm is different from the algorithms that have been proposed to solve the ASFP, but it does solve the ASFP.

In this paper, we considered the ASEP in infinite-dimensional Hilbert spaces, which has broad applicability in modeling significant real-world problems. Then, we used the regularization method to propose a single-step iterative and showed that the sequence generated by such an algorithm strongly converges to the minimum-norm solution of the ASEP (1). We also gave an algorithm for solving the ASFP in Remark 9.

Acknowledgments

The authors wish to thank the referees for their helpful comments and suggestions. This research was supported by NSFC Grants no. 11071279.