# Regularization Method for the Approximate Split

Authors: Rudong Chen; Junlei Li; Yijie Ren
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. LetH1, 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
fx,y=(1/2)Ax-By22 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).


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## Body

## 1. Introduction

LetC⊆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 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. LetH1, 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
(1)tominimizethefunctionf(x,y)=12∥Ax-By∥22
over x∈C and y∈Q.Very recently, for solving the SEP, Moudafi introduced the following alternating CQ-algorithms (ACQA) in [10]:
(2)xk+1=PC(xk-γkA*(Axk-Byk)),yk+1=PQ(yk+γkB*(Axk+1-Byk)).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 projectionsPC 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]:
(3)xk+1=PCk(xk-γA*(Axk-Byk)),yk+1=PQk(yk+βB*(Axk+1-Byk)),
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:
(4)xk+1=PC(xk-γkAT(Axk-Byk)),yk+1=PQ(yk+γkBT(Axk-Byk)),
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

LetH 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
(5)∥x-PKx∥=min{∥x-y∥:y∈K}.The following important properties of projections are useful to our study.Proposition 1.
Given thatx∈H and z∈K;(a)
z=PKx   if and only if〈x-z,y-z〉≤0, for ally∈K;(b)
〈PKu-PKv,u-v〉≥∥PKu-PKv∥2, for allu,v∈H.Definition 2.
A mappingT:H→H is said to be(a)
nonexpansive if(6)∥Tx-Ty∥≤∥x-y∥,∀x,y∈H;(b)
firmly nonexpansive if2T-I is nonexpansive, or equivalently,
(7)〈Tx-Ty,x-y〉≥∥Tx-Ty∥2,∀x,y∈H;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(8)T=(12)(I+S),
where S:H→H is nonexpansive. It is well known that projections are (firmly) nonexpansive.Definition 3.
LetT 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 (9)〈Tx-Ty,x-y〉≥0,∀x,y∈D(T).
(b)
Given a numberβ>0, T is said to be β-strongly monotone if
(10)〈Tx-Ty,x-y〉≥β∥x-y∥2,∀x,y∈D(T).
(c)
Given a numberL>0,  T is said to be L-Lipschitz if
(11)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈D(T).Lemma 4 (see [13]).
Assume thatan is a sequence of nonnegative real numbers such that
(12)an+1≤(1-γn)an+γnδn,n≥0,
where γn, δn are sequences of real numbers such that(i)
γn⊂(0,1) and ∑n=0∞‍γn=∞;(ii)
eitherlimsupn→∞δn≤0 or ∑n=1∞‍γn|δn|<∞.
Then,limn→∞an=0.Next, we will state and prove our main result in this paper.


## 3. Regularization Method for the ASEP

LetS=C×Q. Define
(13)G=[A-B],ω=[xy].The ASEP can now be reformulated as findingω∈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(14)minω∈Sf(ω)=12∥Gω∥2
is generally ill-posed. We consider the Tikhonov regularization (for more details about Tikhonov approximation, please see [8, 14] and the references therein)
(15)minω∈Sfε(ω)=12∥Gω∥2+12ε∥ω∥2,
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
(16)∥ωmin∥=min{∥ω-∥:ω-∈Γ}.The following result is easily proved.Proposition 5.
If the minimization (14) is consistent, then the strong limε→0ωε exists and is the minimum-norm solution of the minimization (14).Proof.
For anyω-∈Γ, we have
(17)f(ω-)+ε2∥ωε∥2≤f(ωε)+ε2∥ωε∥2=fε(ωε)≤fε(ω-)=f(ω-)+ε2∥ω-∥2.
It follows that, for allε>0 and ω-∈Γ,
(18)∥ωε∥≤∥ω-∥.
Therefore,ωε is bounded. Assume that εj→0 is such that ωεj⇀ω*. Then, the weak lower semicontinuity of f implies that, for any ω∈S,
(19)f(ω*)≤liminfj→∞f(ωεj)≤liminfj→∞fεj(ωεj)≤liminfj→∞fεj(ω)=liminfj→∞[f(ω)+εj2∥ω∥2]=f(ω).
This means thatω*∈Γ. Since the norm is weak lower semicontinuous, we get from (18) that ∥ω*∥≤∥ω-∥ for all ω-∈Γ; hence, ω*=ωmin. This is sufficient to ensure that ωε⇀ωmin. To obtain the strong convergence, noting that (18) holds for ωmin, we compute
(20)∥ωε-ωmin∥2=∥ωε∥2-2〈ωε,ωmin〉+∥ωmin∥2≤2(∥ωmin∥2-〈ωε,ωmin〉).
Sinceωε⇀ωmin, we get ωε→ωmin in norm. So, we complete the proof.Next we will state thatωmin can be obtained by two steps. First, observing that the gradient
(21)∇fε=∇f+εI=GTG+εI
is (ε+∥G∥2)-Lipschitz and ε-strongly monotone, the mapping PS(I-γ∇fε) is a contraction with coefficient
(22)1-γ(2ε-γ(∥G∥2+ε)2)(≤1-εγ≤1-12εγ),
where
(23)0<γ≤ε(∥G∥2+ε)2.Indeed, observe that(24)∥PS(I-γ∇fε)(x)-PS(I-γ∇fε)(y)∥2≤∥(I-γ∇fε)(x)-(I-γ∇fε)(y)∥2=∥x-y∥2-2γ〈∇fε(x)-∇fε(y),x-y〉+γ2∥∇fε(x)-∇fε(y)∥2≤(1-2γε+γ2(ε+∥G∥2)2)∥x-y∥2=[1-γ(2ε-γ(ε+∥G∥2)2)]∥x-y∥2≤(1-εγ)∥x-y∥2.Note thatωε 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:
(25)ωn+1ε=PS(I-γ∇fε)ωnε.Secondly, lettingε→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
(26)ωn+1=PS(I-γn∇fεn)ωn=PS((1-εnγn)ωn-γnGTGωn),
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)
∑n=1∞‍εnγn=∞;(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 anyγ 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
(27)Tn:=PS(I-γn∇fεn).
Then,zn=ωεn, and so
(28)zn⟶ωmininnorm.
Thus, to prove the theorem, it suffices to prove that(29)∥ωn+1-zn∥⟶0.
Noting thatTn has contraction coefficient of (1-(1/2)εnγn), we have
(30)∥ωn+1-zn∥=∥Tnωn-Tnzn∥≤(1-12εnγn)∥ωn-zn∥≤(1-12εnγn)∥ωn-zn-1∥+∥zn-zn-1∥.
We now estimate(31)∥zn-zn-1∥=∥Tnzn-Tn-1zn-1∥≤∥Tnzn-Tnzn-1∥+∥Tnzn-1-Tn-1zn-1∥≤(1-12εnγn)∥zn-zn-1∥+∥Tnzn-1-Tn-1zn-1∥.
This implies that(32)∥zn-zn-1∥≤2εnγn∥Tnzn-1-Tn-1zn-1∥.
However, sincezn is bounded, we have, for an appropriate constant M>0,
(33)∥Tnzn-1-Tn-1zn-1∥=∥PS(I-γn∇fεn)zn-1-PS(I-γn-1∇fεn-1)zn-1∥≤∥(I-γn∇fεn)zn-1-(I-γn-1∇fεn-1)zn-1∥=∥γn∇fεn(zn-1)-γn-1∇fεn-1(zn-1)∥=∥(γn-γn-1)∇fεn(zn-1)+γn-1(∇fεn(zn-1)-∇fεn-1(zn-1))∥≤|γn-γn-1|∥∇f(zn-1)+εnzn-1∥+γn-1|εn-εn-1|∥zn-1∥≤(|γn-γn-1|+γn-1|εn-εn-1|)M.
Combining (30), (32), and (33), we obtain
(34)∥ωn+1-zn∥≤(1-12εnγn)∥ωn-zn-1∥+(12εnγn)βn,
where
(35)βn=4M(|γn-γn-1|+γn-1|εn-εn-1|)(εnγn)2⟶0.
Now applying Lemma4 to (34) and using the conditions (ii)–(iv), we conclude that ∥ωn+1-zn∥→0; therefore, ωn→ωmin in norm.Remark 7.
Note thatεn=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
(36)xn+1=PC((1-εnγn)xn-γnAT(Axn-Byn)),yn+1=PQ((1-εnγn)yn+γnBT(Axn-Byn)).
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
(37)xn+1=PC((1-εnγn)xn-γnAT(Axn-yn)),yn+1=PQ((1-εnγn)yn+γn(Axn-yn)).
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.



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