# Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators

Authors: Yazheng Dang; Yan Gao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011), based on the very recent work by Dang et al. (2012), in this paper, we propose an inertial iterative algorithm for solving the split common fixed-point problem for quasi-nonexpansive operators in the Hilbert space. We also prove the asymptotical convergence of the algorithm under some suitable conditions. The results improve and develop previously discussed feasibility problems and related algorithms.


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## Body

## 1. Introduction

The convex feasibility problem (CFP), as an important optimization problem [1], is to find a common point in the intersection of finitely many convex sets. It has been applied to many areas, for instance, approximation theory [2], image reconstruction from projections [3, 4], control [5], and so on. When there are only two sets and constraints are imposed on the solutions in the domain of a linear operator as well as in this operator’s ranges, the problem is said to be a split feasibility problem (SFP) which has the following formula: finding a point x satisfying
(1)x∈C,Ax∈Q,
where C is a closed convex subset of a Hilbert space H1, Q is a closed convex subset of a Hilbert space H2, and A:H1→H2 is a bounded linear operator. The SFP was originally introduced in [6], and it has also broad applications in many fields, such as image reconstruction problem, signal processing, and radiation therapy. Many projection methods have also been developed for solving the SFP; see [7–9]. Denote by PC the orthogonal projection onto C; that is, PC(x)=argminy∈C∥x-y∥, over all x∈C. Assuming that the SFP is consistent (i.e., (1) has a solution), it is not hard to see that x∈C solves (1) if and only if it solves the fixed-point equation:
(2)x=PC[(I-γA*(I-PQ)A)(x)],
where 0<γ is any positive constant and A* denotes the adjoint of A.To solve (2), in [10], Byrne introduced the so-called CQ algorithm, which generates a sequence {xk} by
(3)xk+1=PC[(I-γA*(I-PQ)A)(xk)],
where 0<γ<2/ρ(ATA) and ρ(ATA) is the spectral radius of A*A.The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [11]. Our main purpose here is to give an extension of the results developed in [12] to the split common fixed-point problem for quasi-nonexpansive operators, and we will introduce weak symposium convergence result of the algorithm under some suitable conditions. This will be done in the context of general Hilbert spaces.The paper is organized as follows. In Section2, we recall some preliminaries. In Section 3, we present an inertial CQ algorithm and show its convergence.


## 2. Preliminaries

Throughout the rest of the paper,I denotes the identity operator and Fix(T) denotes the set of the fixed points of an operator T, that is, Fix(T):={x∣x=T(x)}.Recall that a mappingT is said to be quasi-nonexpansive (εQ) if
(4)∥Tx-q∥≤∥x-q∥,∀(x,q)∈H×Fix(T).
A mapping T is called nonexpansive (εN) if
(5)∥T(x)-T(y)∥≤∥x-y∥,∀(x,y)∈H×H.
A mapping T is called firmly nonexpansive (εFN) if
(6)∥T(x)-T(y)∥2≤∥x-y∥2-∥(x-y)-(T(x)-T(y))∥2,∀(x,y)∈H×H.
A mapping T is called firmly quasi-nonexpansive (εFQ) if
(7)∥T(x)-q∥2≤∥x-q∥2-∥x-T(x)∥2,∀(x,q)∈H×Fix(H).
It is easily observed that εFN⊂εN⊂εQ and that εFN⊂εFQ⊂εQ. Furthermore, εFN is well known to include resolvents and projection operators, while εFQ contains subgradient projection operators (see, e.g., [13], and the references therein).Recently, Bauschke and Combettes [14] have considered a class of mappings satisfying the condition
(8)〈q-Tx,x-Tx〉≤0,∀(x,q)∈H×Fix(T).
It can easily be seen that the class of mappings satisfying the latter condition coincides with that of firmly quasi-nonexpansive mappings.Usually, the convergence of fixed-point algorithms requires some additional smoothness properties of the mappingT such as demiclosedness.Definition 1.
A mappingT is said to be demiclosed if for any sequence {xk} which weakly converges to y and if the sequence {T(xk)} strongly converges to z, then T(y)=z.
In what follows, only the particular case of demiclosedness at zero will be used, which is the particular case whenz=0.
The following lemmas will be needed in the proof of the convergence of the algorithm.Lemma 2.
LetT be a quasi-nonexpansive mapping.   Set Tα:=(1-α)I+αT. Then, it is immediate that for all (x,q)∈H×Fix(T):(1)
〈x-T(x),x-q〉≥(1/2)∥x-T(x)∥2 and 〈x-T(x),q-T(x)〉≤(1/2)∥x-T(x)∥2;(2)
∥Tα(x)-q∥2≤∥x-q∥2-α(1-α)∥x-T(x)∥;(3)
〈x-Tα(x),x-q〉≥(α/2)∥x-T(x)∥2.Lemma 3 (see [8]).
Assumeφk∈[0,∞) and δk∈[0,∞) satisfy(1)
φk+1-φk≤θk(φk-φk-1)+δk,(2)
∑k=1+∞‍δk<∞,(3)
{θk}⊂[0,θ],  where θ∈[0,1).Then, the sequence  {φk} is convergent with ∑k=1+∞‍[φk+1-φk]+<∞, where [t]+:=max{t,0}  (for any t∈R).


## 3. The Inertial Algorithm and Its Asymptotic Convergence

In what follows, we will focus our attention on the following general two-operator split common fixed-point problem:(9)findx*∈CsuchthatAx*∈Q,
where A:H1→H2 is a bounded linear operator and U:H1→H1 and T:H2→H2 are two quasi-nonexpansive operators with nonempty fixed-point sets Fix(U)=C and Fix(T)=Q, and denote the solution set of the two-operator SCFP by
(10)Γ={y∈C:Ay∈Q}.


### 3.1. The Inertial Algorithm

To solve (9), Moudafi [15] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm:
(11)xk+1=Uαk(xk+γ(A*Tβ-I)A(xk)),k∈N,
where β∈(0,1), αk∈(0,1) are relaxation parameters and γ>0. Inspired by the inertial technique, we introduce the following inertial method and then present its convergence analysis.Algorithm 4.
Initialization: Letx0∈H1 be arbitrary.
Iterative step: Fork∈N, set u=I+γηA*(T-I)A, and let
(12)yk=xk+θk(xk-xk-1)xk+1=(1-αk)u(yk)+αkU(u(yk)),k∈N,where η∈(0,1), αk∈(0,1), and γ∈(0,1/(λη)), with λ being the spectral radius of the operator A*A, θk∈[0,1).


### 3.2. Asymptotic Convergence of the Inertial Algorithm

In this subsection, we establish the asymptotic convergence of Algorithm4.Lemma 5 (Opial [16]).
LetH be a Hilbert space and let {xk} be a sequence in H such that there exists a nonempty set S⊂H satisfying(1)
for everyx*, limk∥xk-x*∥ exists,(2)
any weak cluster point of the sequence{xk} belongs to S. Then, there exists z∈S such that {xk} weakly converges to z.Theorem 6.
Given a bounded linear operatorA:H1→H2, let U:H1→H1 be a quasi-nonexpansive operator with nonempty Fix(U)=C and let T:H2→H2 be a quasi-nonexpansive operator with nonempty Fix(T)=Q. Assume that U-I and T-I are demiclosed at 0. If Γ≠∅, then any sequence {xk} generated by Algorithm 4 weakly converges to a split common fixed point, provided that we choose θk satisfying θk∈[0,θ-k] with θ-k:=min{θ,1/(k∥xk-xk-1∥)2}, θ∈[0,1). γ∈(0,1/(λη)) and αk∈(δ,1-δ) for a small enough δ>0.Proof.
Takingz∈Γ, and using (2) in Lemma 2, we obtain
(13)∥xk+1-z∥2=∥(1-αk)u(yk)+αkU(u(yk))-z∥2≤∥u(yk)-z∥2-αk(1-αk)∥U(u(yk))-u(yk)∥2.
On the other hand, we have
(14)∥u(yk)-z∥2=∥yk+γηA*(T-I)(Ayk)-z∥2=∥yk-z∥2+γ2η2∥A*(T-I)(Ayk)∥2+2γη〈yk-z,A*(T-I)(Ayk)〉≤∥yk-z∥2+λγ2η2∥(T-I)(Ayk)∥2+2γη〈Ayk-Az,(T-I)(Ayk)〉,
that is,
(15)∥u(yk)-z∥2≤∥yk-z∥2+λγ2η2∥(T-I)(Ayk)∥2+2γη〈Ayk-Az,(T-I)(Ayk)〉.
Now, by setting υ:=2γη〈Ayk-Az,(T-I)(Ayk)〉 and using (1) of Lemma 2, we obtain
(16)υ=2γη〈Ayk-Az,(T-I)(Ayk)〉=2γη〈Ayk-Az+(T-I)(Ayk)-(T-I)(Ayk),(T-I)(Ayk)〉=2γη(〈T(Ayk)-Az,(T-I)(Ayk)〉∥(T-I)(Ayk)∥2-∥(T-I)(Ayk)∥2)≤2γη(12∥(T-I)(Ayk)∥2-∥(T-I)(Ayk)∥2)≤-γη∥(T-I)(Ayk)∥2.
Combining the key inequality above with (15) yields
(17)∥xk+1-z∥2≤∥yk-z∥2-γη(1-λγη)∥(T-I)(Ayk)∥2-αk(1-αk)∥U(u(yk))-u(yk)∥2.
Define the auxiliary real sequence φk:=(1/2)∥xk-z∥2. Therefore, from (17), we have
(18)φk+1≤12∥yk-z∥2-12γη(1-λγη)∥(T-I)(Ayk)∥2-12αk(1-αk)∥U(u(yk))-u(yk)∥2.
By deducing, we have
(19)12∥yk-z∥2=12∥xk+θk(xk-xk-1)-z∥2=12∥xk-z∥2+θk〈xk-z,xk-xk-1〉+θk22∥xk-xk-1∥2=φk+θk〈xk-z,xk-xk-1〉+θk22∥xk-xk-1∥2.
It is easy to check that φk=φk-1+〈xk-z,xk-xk-1〉-(1/2)∥xk-xk-1∥2.
Hence,(20)12∥yk-z∥2=φk+θk(φk-φk-1)+θk+θk22∥xk-xk-1∥2.
Putting (20) into (18), we get
(21)φk+1≤φk+θk(φk-φk-1)+θk+θk22∥xk-xk-1∥2-12γη(1-λγη)∥(T-I)(Ayk)∥2-12αk(1-αk)∥U(u(yk))-u(yk)∥2.
Since γ∈(0,1/(λη)), according to θk2≤θk, αk∈(0,1) and (21), we derive
(22)φk+1≤φk+θk(φk-φk-1)+θk∥xk-xk-1∥2.
Evidently,
(23)∑k=1+∞‍θk∥xk-xk-1∥2<∞,
due to θk∥xk-xk-1∥2≤1/k2. Let δk:=θk∥xk-xk-1∥2 in Lemma 3. We deduce that the sequence {∥xk-z∥} is convergent (hence, {xk} is bounded). By (23) and Lemma 3, we obtain ∑k=1+∞‍[∥xk-z∥2-∥xk-1-z∥2]+<∞. By reason of (21), we have
(24)12γη(1-λγη)∥(T-I)(Ayk)∥2≤φk-φk+1+θk(φk-φk-1)+θk∥xk-xk-1∥2,12αk(1-αk)∥U(u(yk))-u(yk)∥2≤φk-φk+1+θk(φk-φk-1)+θk∥xk-xk-1∥2.
Hence,
(25)∑k=1+∞‍12γη(1-λγη)∥(T-I)(Ayk)∥2<∞,∑k=1+∞‍12αk(1-αk)∥U(u(yk))-u(yk)∥2<∞.
By γ∈(0,1/(λη)) and the assumption on αk, we get
(26)∥(T-I)(Ayk)∥2⟶0,(27)∥U(u(yk))-u(yk)∥2⟶0.
Denoting by x* a weak-cluster point {xk}, let {xkσ} be a subsequence of {xk}. Obviously,
(28)w-limσykσ=w-limσxkσ=x*.
Then, from (26) and the demiclosedness of T-I at 0, we obtain
(29)T(Ax*)=Ax*,
it follows that Ax*∈Q.Now, by settinguk=yk+γηA*(T-I)(Ayk), it follows that w-limσukσ=x*. By the demiclosedness of U-I at 0, from (27), we have
(30)U(x*)=x*.
Hence, x*∈C, and therefore x*∈Γ.Since there is no more than one weak-cluster point, the weak convergence of the whole sequence{xk} follows by applying Lemma 5 with S=Γ.Remark 7.
Since the current value of∥xk-xk-1∥ is known before choosing the parameter θk, then θk is well-defined in Theorem 6. In fact, from the process of proof for Theorem 6, we can get the following assert: the convergence result of Theorem 6 always holds provided that we take θk∈[0,θ], θ∈[0,1),forallk≥0, with
(31)∑k=1+∞‍θk∥xk-xk-1∥2<∞.To conclude, we have proposed an algorithm for solving the SCFP in the wide class of quasi-nonexpansive operators and proved its convergence in general Hilbert spaces. Next, we will improve the algorithm to assure the strong convergence in infinite Hilbert spaces.


## 3.1. The Inertial Algorithm

To solve (9), Moudafi [15] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm:
(11)xk+1=Uαk(xk+γ(A*Tβ-I)A(xk)),k∈N,
where β∈(0,1), αk∈(0,1) are relaxation parameters and γ>0. Inspired by the inertial technique, we introduce the following inertial method and then present its convergence analysis.Algorithm 4.
Initialization: Letx0∈H1 be arbitrary.
Iterative step: Fork∈N, set u=I+γηA*(T-I)A, and let
(12)yk=xk+θk(xk-xk-1)xk+1=(1-αk)u(yk)+αkU(u(yk)),k∈N,where η∈(0,1), αk∈(0,1), and γ∈(0,1/(λη)), with λ being the spectral radius of the operator A*A, θk∈[0,1).


## 3.2. Asymptotic Convergence of the Inertial Algorithm

In this subsection, we establish the asymptotic convergence of Algorithm4.Lemma 5 (Opial [16]).
LetH be a Hilbert space and let {xk} be a sequence in H such that there exists a nonempty set S⊂H satisfying(1)
for everyx*, limk∥xk-x*∥ exists,(2)
any weak cluster point of the sequence{xk} belongs to S. Then, there exists z∈S such that {xk} weakly converges to z.Theorem 6.
Given a bounded linear operatorA:H1→H2, let U:H1→H1 be a quasi-nonexpansive operator with nonempty Fix(U)=C and let T:H2→H2 be a quasi-nonexpansive operator with nonempty Fix(T)=Q. Assume that U-I and T-I are demiclosed at 0. If Γ≠∅, then any sequence {xk} generated by Algorithm 4 weakly converges to a split common fixed point, provided that we choose θk satisfying θk∈[0,θ-k] with θ-k:=min{θ,1/(k∥xk-xk-1∥)2}, θ∈[0,1). γ∈(0,1/(λη)) and αk∈(δ,1-δ) for a small enough δ>0.Proof.
Takingz∈Γ, and using (2) in Lemma 2, we obtain
(13)∥xk+1-z∥2=∥(1-αk)u(yk)+αkU(u(yk))-z∥2≤∥u(yk)-z∥2-αk(1-αk)∥U(u(yk))-u(yk)∥2.
On the other hand, we have
(14)∥u(yk)-z∥2=∥yk+γηA*(T-I)(Ayk)-z∥2=∥yk-z∥2+γ2η2∥A*(T-I)(Ayk)∥2+2γη〈yk-z,A*(T-I)(Ayk)〉≤∥yk-z∥2+λγ2η2∥(T-I)(Ayk)∥2+2γη〈Ayk-Az,(T-I)(Ayk)〉,
that is,
(15)∥u(yk)-z∥2≤∥yk-z∥2+λγ2η2∥(T-I)(Ayk)∥2+2γη〈Ayk-Az,(T-I)(Ayk)〉.
Now, by setting υ:=2γη〈Ayk-Az,(T-I)(Ayk)〉 and using (1) of Lemma 2, we obtain
(16)υ=2γη〈Ayk-Az,(T-I)(Ayk)〉=2γη〈Ayk-Az+(T-I)(Ayk)-(T-I)(Ayk),(T-I)(Ayk)〉=2γη(〈T(Ayk)-Az,(T-I)(Ayk)〉∥(T-I)(Ayk)∥2-∥(T-I)(Ayk)∥2)≤2γη(12∥(T-I)(Ayk)∥2-∥(T-I)(Ayk)∥2)≤-γη∥(T-I)(Ayk)∥2.
Combining the key inequality above with (15) yields
(17)∥xk+1-z∥2≤∥yk-z∥2-γη(1-λγη)∥(T-I)(Ayk)∥2-αk(1-αk)∥U(u(yk))-u(yk)∥2.
Define the auxiliary real sequence φk:=(1/2)∥xk-z∥2. Therefore, from (17), we have
(18)φk+1≤12∥yk-z∥2-12γη(1-λγη)∥(T-I)(Ayk)∥2-12αk(1-αk)∥U(u(yk))-u(yk)∥2.
By deducing, we have
(19)12∥yk-z∥2=12∥xk+θk(xk-xk-1)-z∥2=12∥xk-z∥2+θk〈xk-z,xk-xk-1〉+θk22∥xk-xk-1∥2=φk+θk〈xk-z,xk-xk-1〉+θk22∥xk-xk-1∥2.
It is easy to check that φk=φk-1+〈xk-z,xk-xk-1〉-(1/2)∥xk-xk-1∥2.
Hence,(20)12∥yk-z∥2=φk+θk(φk-φk-1)+θk+θk22∥xk-xk-1∥2.
Putting (20) into (18), we get
(21)φk+1≤φk+θk(φk-φk-1)+θk+θk22∥xk-xk-1∥2-12γη(1-λγη)∥(T-I)(Ayk)∥2-12αk(1-αk)∥U(u(yk))-u(yk)∥2.
Since γ∈(0,1/(λη)), according to θk2≤θk, αk∈(0,1) and (21), we derive
(22)φk+1≤φk+θk(φk-φk-1)+θk∥xk-xk-1∥2.
Evidently,
(23)∑k=1+∞‍θk∥xk-xk-1∥2<∞,
due to θk∥xk-xk-1∥2≤1/k2. Let δk:=θk∥xk-xk-1∥2 in Lemma 3. We deduce that the sequence {∥xk-z∥} is convergent (hence, {xk} is bounded). By (23) and Lemma 3, we obtain ∑k=1+∞‍[∥xk-z∥2-∥xk-1-z∥2]+<∞. By reason of (21), we have
(24)12γη(1-λγη)∥(T-I)(Ayk)∥2≤φk-φk+1+θk(φk-φk-1)+θk∥xk-xk-1∥2,12αk(1-αk)∥U(u(yk))-u(yk)∥2≤φk-φk+1+θk(φk-φk-1)+θk∥xk-xk-1∥2.
Hence,
(25)∑k=1+∞‍12γη(1-λγη)∥(T-I)(Ayk)∥2<∞,∑k=1+∞‍12αk(1-αk)∥U(u(yk))-u(yk)∥2<∞.
By γ∈(0,1/(λη)) and the assumption on αk, we get
(26)∥(T-I)(Ayk)∥2⟶0,(27)∥U(u(yk))-u(yk)∥2⟶0.
Denoting by x* a weak-cluster point {xk}, let {xkσ} be a subsequence of {xk}. Obviously,
(28)w-limσykσ=w-limσxkσ=x*.
Then, from (26) and the demiclosedness of T-I at 0, we obtain
(29)T(Ax*)=Ax*,
it follows that Ax*∈Q.Now, by settinguk=yk+γηA*(T-I)(Ayk), it follows that w-limσukσ=x*. By the demiclosedness of U-I at 0, from (27), we have
(30)U(x*)=x*.
Hence, x*∈C, and therefore x*∈Γ.Since there is no more than one weak-cluster point, the weak convergence of the whole sequence{xk} follows by applying Lemma 5 with S=Γ.Remark 7.
Since the current value of∥xk-xk-1∥ is known before choosing the parameter θk, then θk is well-defined in Theorem 6. In fact, from the process of proof for Theorem 6, we can get the following assert: the convergence result of Theorem 6 always holds provided that we take θk∈[0,θ], θ∈[0,1),forallk≥0, with
(31)∑k=1+∞‍θk∥xk-xk-1∥2<∞.To conclude, we have proposed an algorithm for solving the SCFP in the wide class of quasi-nonexpansive operators and proved its convergence in general Hilbert spaces. Next, we will improve the algorithm to assure the strong convergence in infinite Hilbert spaces.



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*Source: 386930-2013-06-25.xml*