Abstract and Applied Analysis
Volume 2013, Issue 1 813635
Research Article
Open Access

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

Rudong Chen, 

Corresponding Author

Rudong Chen

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China tjpu.edu.cn

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Junlei Li, 

Junlei Li

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China tjpu.edu.cn

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Yijie Ren, 

Yijie Ren

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China tjpu.edu.cn

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First published: 07 May 2013
https://doi.org/10.1155/2013/813635
Citations: 2
Academic Editor: Yisheng Song

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 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⊆RN and Q⊆RM 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 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
()
over x ∈ C and y ∈ Q.
Very recently, for solving the SEP, Moudafi introduced the following alternating CQ-algorithms (ACQA) in [10]:
()

Then, he proved the weak convergence of the sequence {xk, yk} 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 (γk) are satisfied.

The ACQA involves two projections PC and PQ 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]:
()
where Ck, Qk were defined in [11], and then he proved the weak convergence of the sequence {xk, yk} 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:
()
where ϵ ≤ γk ≤ (2/ρ(GTG)) − 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 PK, is defined in such a way that, for each x ∈ H, PKx is the unique point in K with the property
()

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

Proposition 1. Given that x ∈ H and z ∈ K;

  • (a)

    z = PKx   if and only if 〈x − z, y − z〉≤0, for all y ∈ K;

  • (b)

    , for all u, v ∈ H.

Definition 2. A mapping T : H → H is said to be

  • (a)

    nonexpansive if

    ()

  • (b)

    firmly nonexpansive if 2T − I is nonexpansive, or equivalently,

    ()

alternatively, T is firmly nonexpansive if and only if T can be expressed as
()
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

    ()

  • (b)

    Given a number β > 0, T is said to be β-strongly monotone if

    ()

  • (c)

    Given a number L > 0,  T is said to be L-Lipschitz if

    ()

Lemma 4 (see [13].)Assume that an is a sequence of nonnegative real numbers such that

()
where γn, δn are sequences of real numbers such that
  • (i)

    γn ⊂ (0,1) and ;

  • (ii)

    either limsup n→∞δn ≤ 0 or .

Then, lim n→∞an = 0.

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

3. Regularization Method for the ASEP

Let S = C × Q. Define
()

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
()
is generally ill-posed. We consider the Tikhonov regularization (for more details about Tikhonov approximation, please see [8, 14] and the references therein)
()
where ε > 0 is the regularization parameter. The regularization minimization (15) has a unique solution which is denoted by ωε. Assume that the minimization (14) is consistent, and let ωmin  be its minimum-norm solution; namely, ωmin  ∈ Γ (Γ is the solution set of the minimization (14)) has the property
()

The following result is easily proved.

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

Proof. For any , we have

()

It follows that, for all ε > 0 and ,

()

Therefore, ωε is bounded. Assume that εj → 0 is such that . Then, the weak lower semicontinuity of f implies that, for any ω ∈ S,

()

This means that ω* ∈ Γ. Since the norm is weak lower semicontinuous, we get from (18) that for all ; hence, ω* = ωmin . This is sufficient to ensure that ωε⇀ωmin . To obtain the strong convergence, noting that (18) holds for ωmin , we compute

()

Since ωε⇀ωmin , we get ωε → ωmin  in norm. So, we complete the proof.

Next we will state that ωmin  can be obtained by two steps. First, observing that the gradient
()
is (ε + ∥G∥2)-Lipschitz and ε-strongly monotone, the mapping PS(I − γ∇fε) is a contraction with coefficient
()
where
()
Indeed, observe that
()
Note that ωε is a fixed point of the mapping PS(I − γ∇fε) for any γ > 0 satisfying (23) and can be obtained through the limit as n → ∞ of the sequence of Picard iterates as follows:
()

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

Theorem 6. Assume that the minimization problem (14) is consistent. Define a sequence ωn by the iterative algorithm

()
where εn and γn satisfy the following conditions:
  • (i)

    0 < γn ≤ εn/(∥G∥2 + εn) 2 for all (large enough) n;

  • (ii)

    εn → 0 and γn → 0;

  • (iii)

    ;

  • (iv)

    (|γn+1 − γn | + γn | εn+1 − εn|)/(εn+1γn+1) 2 → 0.

Then, ωn converges in norm to the minimum-norm solution of the minimization problem (14).

Proof. Note that for any γ satisfying (23), ωε is a fixed point of the mapping PS(I − γ∇fε). For each n, let zn be the unique fixed point of the contraction

()

Then, , and so

()

Thus, to prove the theorem, it suffices to prove that

()

Noting that Tn has contraction coefficient of (1 − (1/2)εnγn), we have

()

We now estimate

()

This implies that

()

However, since zn is bounded, we have, for an appropriate constant M > 0,

()

Combining (30), (32), and (33), we obtain

()
where
()

Now applying Lemma 4 to (34) and using the conditions (ii)–(iv), we conclude that ∥ωn+1 − zn∥ → 0; therefore, ωn → ωmin  in norm.

Remark 7. Note that εn = n−δ and γn = 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

()

And we can obtain that the whole sequence (xn, yn) generated by the algorithm (36) strongly converges to the minimum-norm solution of the ASEP (1) provided that the ASEP (1) is consistent and εn and γn 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

()

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.

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