# A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces

Authors: Ming Tian; Li-Hua Huang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The constrained convex minimization problem is to find a
pointx∗ with the property that x∗∈C, and h(x∗)=min  h(x), ∀x∈C, where C is a nonempty, closed, and convex subset of a real Hilbert space H, h(x) is a real-valued convex function, and h(x) is not Fréchet differentiable, but lower semicontinuous. In this paper, we discuss an iterative algorithm which is different from traditional gradient-projection algorithms. We firstly construct a bifunction F1(x,y) defined as F1(x,y)=h(y)−h(x). And we ensure the equilibrium problem for F1(x,y) equivalent to the above optimization problem. Then we use iterative methods for equilibrium problems to study the above optimization problem. Based on Jung’s method (2011), we propose a general approximation method and prove the strong convergence of our algorithm to a solution of the above optimization problem. In addition, we apply the proposed iterative algorithm for finding a solution of the split feasibility problem and establish the strong convergence theorem. The results of this paper extend and improve some existing results.


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## Body

## 1. Introduction

LetH be a real Hilbert space with the inner product 〈·,·〉 and the induced norm ∥·∥. Let C be a nonempty, closed, and convex subset of H. Recall that a mapping T is nonexpansive (see [1]) if
(1)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
We denote by F(T)={x∈C:Tx=x} the set of fixed points of T. Recall that a self-mapping f on H is a contraction if there exists a constant α∈(0,1) such that
(2)∥f(x)-f(y)∥≤α∥x-y∥,∀x,y∈H.Consider the following constrained convex minimization problem:(3)min⁡x∈C⁡h(x),
where h:C→R is a real-valued convex function. Assume that the constrained convex minimization problem (3) is solvable, and let S denote the set of solutions of (3). Assume that h(x) is lower semicontinuous. Let F1 be a bifunction from C×C to ℝ defined by F1(x,y)=h(y)-h(x). It is easy to see that S=EP⁡(F1), where EP⁡(F1) denotes the set of solutions of equilibrium problem to find x∈C such that
(4)F1(x,y)≥0,∀y∈C.
Hence, the optimization problem (3) is equivalent to the equilibrium problem (4).For solving the convex optimization problem (3), Su and Li [2] introduced the following iterative scheme in a Hilbert space: x1∈H and
(5)h(y)-h(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+γnun+βnxn,
for all n∈ℕ, where un=Trnxn, {αn},{βn},{γn}⊂(0,1), αn+βn+γn=1, and {rn}⊂(0,∞). Under appropriate conditions, they proved that the sequence {xn} generated by (5) converges strongly to a point z=PSf(z) of optimization problem (3).In 2010, combining Yamada’s method [3] and Marino and Xu’s method [4], Tian [5] proposed a general iterative algorithm:
(6)xn+1=αnγf(xn)+(I-αnμF)T(xn),
where f:H→H is a contraction with coefficient 0<α<1 and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0, η>0. Let 0<μ<2η/k2, 0<γ<μ(η-(μk2/2))/α=τ/α. It is proved that the sequence {xn}n=0∞ generated by (6) converges strongly to a fixed point x~∈Fix⁡(T), which solves the variational inequality
(7)〈(γf-μF)x~,x-x~〉≤0,∀x∈Fix⁡(T).In 2011, Jung [6] introduced the following general iterative scheme for k-strictly pseudocontractive mapping T:
(8)xn+1=αnγf(xn)+((1-βn)I-αnμF)PCS′(xn)+βnxn,
where S′:C→H is a mapping defined by S′x=kx+(1-k)Tx and PC is the metric projection of H onto C. Under appropriate conditions, he established the strong convergence of the sequence {xn} generated by (8) to a fixed point of T, which is a solution of the variational inequality (7).In this paper, motivated and inspired by the above results, we introduce a general iterative method:x1∈H and
(9)h(y)-h(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+((1-βn)I-αnμF)un+βnxn,n∈ℕ,
for solving the optimization problem (3), where un=Trnxn. Under appropriate conditions, it is proved that the sequence {xn} generated by (9) converges strongly to a point z∈S which solves the variational inequality
(10)〈(μF-γf)z,z-x〉≤0,∀x∈S.
Furthermore, by using the above result we studied the split feasibility problem and obtained the iterative algorithm for solving the split feasibility problem.


## 2. Preliminaries

In this section we introduce some useful definitions and lemmas which will be used in the proofs for the main results in the next section.Monotone operators are very useful in the convergence analysis.Definition 1 (see [7] for comprehensive theory of monotone operators).
LetA:H→H be an operator.(i)
Ais monotone, if and only if
(11)〈x-y,Ax-Ay〉≥0,∀x,y∈H.(ii)
A is said to be η-strongly monotone, if there exists a positive constant η such that
(12)〈x-y,Ax-Ay〉≥η∥x-y∥2,∀x,y∈H.(iii)
A is said to be υ-inverse strongly monotone (υ-ism), if there exists a positive constant υ such that
(13)〈x-y,Ax-Ay〉≥υ∥Ax-Ay∥2,∀x,y∈H.
It is known that inverse strongly monotone operators have been studied widely (see [7–9]) and applied to solve practical problems in various fields, for instance, in traffic assignment problems (see [10, 11]).Lemma 2 (see [5]).
LetH be a Hilbert space, f:H→H a contraction with a coefficient 0<ρ<1, and F:H→H a k-Lipschitz continuous and η-strongly monotone operator with constants k>0, η>0. Then, for 0<γ<μη/ρ,
(14)〈x-y,(μF-γf)x-(μF-γf)y〉≥(μη-γρ)∥x-y∥2,〈x-y,(μF-γf)i-(μF-γf)i〉≥(μ-γρ)∀x,y∈H.
That is, μF-γf is strongly monotone with a coefficient μη-γρ.Recall the metric (nearest point) projectionPC from a real Hilbert space H to a closed and convex subset C of H which is defined as follows: given x∈H, PCx is the unique point in C with the property
(15)∥x-PCx∥=inf⁡{∥x-y∥:y∈C}.PC is characterized as follows.Lemma 3.
LetC be a closed and convex subset of a real Hilbert space H. Given x∈H and y∈C, then y=PCx if and only if the following inequality holds:
(16)〈x-y,y-z〉≥0,∀z∈C.For solving the equilibrium problem for a bifunctionF1:C×C→ℝ, let us assume that F1 satisfies the following conditions:(A1)
F1(x,x)=0 for all x∈C;(A2)
F1 is monotone; that is, F1(x,y)+F1(y,x)≤0 for all x,y∈C;(A3)
for eachx,y,z∈C, limsup⁡t→0+F1(tz+(1-t)x,y)≤F1(x,y);(A4)
for eachx∈C, y↦F1(x,y) is convex and lower semicontinuous.Lemma 4 (see [12]).
LetC be a nonempty, closed, and convex subset of H and let F1 be a bifunction of C×C into ℝ satisfying(A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(17)F1(z,y)+1r〈y-z,z-x〉≥0∀y∈C.Lemma 5 (see [13]).
Assume thatF1:C×C→ℝ satisfies(A1)–(A4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(18)Tr(x)={z∈C:F1(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}.
Then, the following hold. (1)
Tr is single-valued;(2)
Tr is firmly nonexpansive; that is, ∥Trx-Try∥2≤〈Trx-Try,x-y〉 for any x,y∈H;(3)
F(Tr)=EP(F1);(4)
EP(F1) is closed and convex.Lemma 6 (see [14]).
Assume that{an}n=0∞ is a sequence of nonnegative real numbers such that
(19)an+1≤(1-γn)an+γnδn,n≥0,
where {γn}n=0∞ is a sequence in (0,1) and {δn}n=0∞ is a sequence in ℝ such that(i)
∑n=0∞‍γn=∞;(ii)
eitherlimsup⁡n→∞δn≤0 or ∑n=0∞‍γn|δn|<∞.Then lim⁡n→∞an=0.We adopt the following notation:(i)
xn→x means that xn→x strongly;(ii)
xn⇀x means that xn→x weakly.


## 3. Main Results

Recall that throughout this paper we always useS to denote the solution set of the constrained convex minimization problem (3).LetH be a real Hilbert space and let C be a nonempty, closed, and convex subset of H. Let h:C→ℝ be a real-valued convex function. Assume that h(x) is lower semicontinuous. Let f:H→H be a contraction with a coefficient 0<ρ<1, and let F:C→H be a k-Lipschitzian and η-strongly monotone operator with constants k>0, η>0. Suppose that 0<μ<2η/k2, 0<γ<τ/ρ, where τ=μη-(μ2k2/2).Next, we study the following iterative method. For a given arbitrary initial guessx1∈H, the sequence {xn}n=0∞ is generated by the following recursive formula:
(20)h(y)-h(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+((1-βn)I-αnμF)un+βnxn,n∈ℕ,
where un=Trnxn and Trn is a mapping defined as in Lemma 5.In order to prove the convergence, we also need the following proposition.Proposition 7.
LetH be a real Hilbert space and let F:H→H be a k-Lipschitzian and η-strongly monotone operator with constants k>0, η>0. Suppose that 0<μ<2η/k2, and {αn}, {βn}⊂(0,1) satisfy 0<αn<min⁡{(1-βn)/2τ,1-βn}. Then, the mapping G(x)=((1-βn)I-αnμF)(x) is a contraction with a contractive constant 1-βn-αnτ, where τ=μη-(μ2k2/2).Proof.
Takingx,y∈H, we have
(21)∥G(x)-G(y)∥2=∥((1-βn)I-αnμF)(x)-((1-βn)I-αnμF)(y)∥2=(1-βn)2∥x-y∥2-2αnμ(1-βn)〈x-y,F(x)-F(y)〉+αn2μ2∥F(x)-F(y)∥2≤[(1-βn)2-2αnμ(1-βn)η+αn2μ2k2]∥x-y∥2≤(1-βn)2[1-2αnμη1-βn+αnμ2k21-βn]∥x-y∥2.
Thus,
(22)∥G(x)-G(y)∥≤(1-βn)1-αn(2μη1-βn-μ2k21-βn)∥x-y∥≤(1-βn)[1-αn(μη1-βn-μ2k22(1-βn))]∥x-y∥=(1-βn-αnτ)∥x-y∥,
where τ=μη-(μ2k2/2).Theorem 8.
LetH be a real Hilbert space and let C be a nonempty, closed, and convex subset of H. Let h:C→ℝ be a real-valued convex function. Assume that h(x) is lower semicontinuous. Let f:H→H be a contraction with a coefficient 0<ρ<1 and let F:C→H be a k-Lipschitzian and η-strongly monotone operator with constants k>0, η>0. Suppose that 0<μ<2η/k2, 0<γ<τ/ρ, where τ=μη-(μ2k2/2). Suppose that the optimization problem (3) is consistent and let S denote its solution set. Let {xn}n=0∞ be generated by x1∈H and
(23)h(y)-h(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+((1-βn)I-αnμF)un+βnxn,n∈ℕ,
where un=Trnxn and Trn is a mapping defined as in Lemma 5. {αn}, {βn}⊂(0,1), {rn}⊂(0,∞) satisfy the following conditions:(C1)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(C2)
∑n=1∞‍|αn+1-αn|<∞;(C3)
limsup⁡n→∞βn<1;(C4)
∑n=1∞‍|βn+1-βn|<∞;(C5)
liminf⁡n→∞rn>0 and ∑n=1∞‍|rn+1-rn|<∞.Then the sequence {xn} generated by (23) converges strongly to a point z∈S, which solves the variational inequality
(24)〈(μF-γf)z,z-x〉≤0,∀x∈S.Proof.
LetF1 be a bifunction from C×C to ℝ defined by F1(x,y)=h(y)-h(x). We consider the following equilibrium problem, that is, to find x∈C, such that
(25)F1(x,y)≥0,∀y∈C.
It is obvious that EP⁡(F1)=S, where EP⁡(F1) denotes the set of solutions of equilibrium problem (25). In addition, it is easy to see that F1(x,y) satisfies the conditions(A1)–(A4) in Section 2. Then, the iterative method (23) is equivalent to x1∈H and
(26)F1(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+((1-βn)I-αnμF)un+βnxn,n∈ℕ,
where un=Trnxn.
It is easy to see the uniqueness of a solution of variational inequality (24). By Lemma 2, μF-γf is strongly monotone, so variational inequality (24) has only one solution. Below, we use z∈S to denote the unique solution of (24). Since z∈S solves variational inequality (24), then (24) can be written as
(27)〈(I-μF+γf)z-z,z-x〉≥0,∀x∈S.
So in terms of Lemma 3, it is equivalent to the following fixed-point equation
(28)z=PS(I-μF+γf)z=PEP⁡(F1)(I-μF+γf)z.
Now, we show that {xn} is bounded. Indeed, picking p∈S=EP⁡(F1), since un=Trnxn, by Lemma 5, we know that
(29)∥un-p∥=∥Trnxn-Trnp∥≤∥xn-p∥,∀n≥1.
From Propositions 3.1 and (29), we derive that
(30)∥xn+1-p∥=∥αnγf(xn)+((1-βn)I-αnμF)un+βnxn-p∥≤αn∥γf(xn)-μF(p)∥+∥((1-βn)I-αnμF)un-((1-βn)I-αnμF)p∥+βn∥xn-p∥≤αnγ∥f(xn)-f(p)∥+αn∥γf(p)-μF(p)∥+(1-βn-αnτ)∥un-p∥+βn∥xn-p∥≤αnγρ∥xn-p∥+αn∥γf(p)-μF(p)∥+(1-βn-αnτ)∥un-p∥+βn∥xn-p∥≤(1-αn(τ-γρ))∥xn-p∥+αn∥γf(p)-μF(p)∥=(1-αn(τ-γρ))∥xn-p∥+αn(τ-γρ)∥γf(p)-μF(p)∥τ-γρ≤max⁡{∥xn-p∥,∥γf(p)-μF(p)∥τ-γρ}.
By induction, we have
(31)∥xn-p∥≤max⁡{∥x0-p∥,∥γf(p)-μF(p)∥τ-γρ}.
Hence {xn} is bounded. From (29), we also derive that {un} is bounded.
Next, we show that∥xn+1-xn∥→0. Consider
(32)∥xn+1-xn∥=∥[αnγf(xn)+((1-βn)I-αnμF)un+βnxn]-[αn-1γf(xn-1)+((1-βn-1)I-αn-1μF)un-1+βn-1xn-1]∥≤αnγ∥f(xn)-f(xn-1)∥+|αn-αn-1|γ∥f(xn-1)∥+∥((1-βn)I-αnμF)un-((1-βn)I-αnμF)un-1∥+∥(βn-1-βn)un-1+(αn-1-αn)μF(un-1)∥+|βn-βn-1|∥xn-1∥+βn∥xn-xn-1∥≤αnγρ∥xn-xn-1∥+|αn-αn-1|γ∥f(xn-1)∥+(1-βn-αnτ)∥un-un-1∥+|βn-βn-1|∥un-1∥+|αn-αn-1|∥μF(un-1)∥+|βn-βn-1|∥xn-1∥+βn∥xn-xn-1∥=(αnγρ+βn)∥xn-xn-1∥+|αn-αn-1|(γ∥f(xn-1)∥+∥μF(un-1)∥)+(1-βn-αnτ)∥un-un-1∥+|βn-βn-1|(∥un-1∥+∥xn-1∥).
From un+1=Trn+1xn+1 and un=Trnxn, we note that
(33)F1(un+1,y)+1rn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C,(34)F1(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C.
Putting y=un in (33) and y=un+1 in (34), we have
(35)F1(un+1,un)+1rn+1〈un-un+1,un+1-xn+1〉≥0,F1(un,un+1)+1rn〈un+1-un,un-xn〉≥0.
So, from (A2), we have
(36)〈un+1-un,un-xnrn-un+1-xn+1rn+1〉≥0,
and hence
(37)〈un+1-un,un-un+1+un+1-xn-rnrn+1(un+1-xn+1)〉≥0.
Since liminf⁡n→∞rn>0, without loss of generality, let us assume that there exists a real number a such that rn>a>0 for all n∈ℕ. Thus, we have
(38)∥un+1-un∥2≤〈un+1-un,xn+1-xn+(1-rnrn+1)(un+1-xn+1)〉≤∥un+1-un∥{∥xn+1-xn∥+|1-rnrn+1|∥un+1-xn+1∥};
thus,
(39)∥un+1-un∥≤∥xn+1-xn∥+1a|rn+1-rn|M1,
where M1=sup⁡{∥un-xn∥:n∈ℕ}.
From (32) and (39), we obtain
(40)∥xn+1-xn∥≤(αnγρ+βn)∥xn-xn-1∥+|αn-αn-1|(γ∥f(xn-1)∥+∥μF(un-1)∥)+(1-βn-αnτ)(∥xn-xn-1∥+1a|rn-rn-1|M1)+|βn-βn-1|(∥un-1∥+∥xn-1∥)≤(1-αnτ+αnγρ)∥xn-xn-1∥+|αn-αn-1|(γ∥f(xn-1)∥+∥μF(un-1)∥)+M1a|rn-rn-1|+|βn-βn-1|(∥un-1∥+∥xn-1∥)≤(1-αn(τ-γρ))∥xn-xn-1∥+M2(|αn-αn-1|+|rn-rn-1|+|βn-βn-1|),
where M2≥max⁡{γ∥f(xn-1)∥+∥μF(un-1)∥,M1/a,∥un-1∥+∥xn-1∥}, for all n∈ℕ. Hence, by Lemma 6, we have
(41)lim⁡n→∞∥xn+1-xn∥=0.
Then, from (39), (41), and |rn+1-rn|→0, we have
(42)lim⁡n→∞∥un+1-un∥=0.
Next, we show that ∥xn-un∥→0.
Indeed, for anyp∈EP⁡(F1), by Lemma 5, we have
(43)∥un-p∥2=∥Trnxn-Trnp∥2≤〈xn-p,un-p〉=12(∥xn-p∥2+∥un-p∥2-∥un-xn∥2).
This implies that
(44)∥un-p∥2≤∥xn-p∥2-∥un-xn∥2.
Then from (29) and (44), we derive that
(45)∥xn+1-p∥2=∥αnγf(xn)+((1-βn)I-αnμF)un+βnxn-p∥2≤(αn∥γf(xn)-μF(p)∥+βn∥xn-p∥+∥((1-βn)I-αnμF)un-((1-βn)I-αnμF)p∥)2≤(αnγ∥f(xn)-f(p)∥+αn∥γf(p)-μF(p)∥+(1-βn-αnτ)∥un-p∥+βn∥xn-p∥)2≤((αnγρ+βn)∥xn-p∥+(1-βn-αnτ)∥un-p∥+αn∥γf(p)-μF(p)∥(αnγρ+βn)∥xn-p∥)2≤((αnτ+βn)∥xn-p∥+(1-βn-αnτ)∥un-p∥+αn∥γf(p)-μF(p)∥(αnτ+βn)∥xn-p∥)2≤(αnτ+βn)∥xn-p∥2+(1-βn-αnτ)∥un-p∥2+αn2∥γf(p)-μF(p)∥2+2[(αnτ+βn)∥xn-p∥+(1-βn-αnτ)∥un-p∥]×αn∥γf(p)-μF(p)∥≤(αnτ+βn)∥xn-p∥2+(1-βn-αnτ)×(∥xn-p∥2-∥un-xn∥2)+αn2∥γf(p)-μF(p)∥2+2∥xn-p∥αn∥γf(p)-μF(p)∥=∥xn-p∥2-(1-βn-αnτ)∥un-xn∥2+αn2∥γf(p)-μF(p)∥2+2∥xn-p∥αn∥γf(p)-μF(p)∥≤(∥xn-xn+1∥+∥xn+1-p∥)2-(1-βn-αnτ)∥un-xn∥2+αn2∥γf(p)-μF(p)∥2+2∥xn-p∥αn∥γf(p)-μF(p)∥=∥xn-xn+1∥2+∥xn+1-p∥2+2∥xn-xn+1∥∥xn+1-p∥-(1-βn-αnτ)∥un-xn∥2+αn2∥γf(p)-μF(p)∥2+2∥xn-p∥αn∥γf(p)-μF(p)∥.
Thus,
(46)(1-βn-αnτ)∥un-xn∥2≤∥xn-xn+1∥2+2∥xn-xn+1∥∥xn+1-p∥+αn2∥γf(p)-μF(p)∥2+2∥xn-p∥αn∥γf(p)-μF(p)∥.
From (41) and condition (C1), we obtain that
(47)(1-βn-αnτ)∥un-xn∥2⟶0,n⟶∞.
From conditions (C1) and (C3), we get that
(48)lim⁡n→∞∥un-xn∥=0.
Next, we show that
(49)limsup⁡n→∞〈(γf-μF)z,xn-z〉≤0,
where z∈S which solves the variational inequality
(50)〈(μF-γf)z,z-x〉≤0,∀x∈S.
Since {xn} is bounded, without loss of generality, we may assume that xni⇀z~ such that
(51)limsup⁡n→∞〈(γf-μF)z,xn-z〉=lim⁡i→∞〈(γf-μF)z,xni-z〉.
From ∥un-xn∥→0 and xni⇀z~, we obtain that uni⇀z~.
Next, we show thatz~∈S=EP⁡(F1).
Indeed, fromun=Trnxn, for any y∈C, we obtain
(52)F1(un,y)+1rn〈y-un,un-xn〉≥0.
From (A2), we have
(53)1rn〈y-un,un-xn〉≥F1(y,un)
and hence
(54)〈y-uni,uni-xnirni〉≥F1(y,uni).
Since (uni-xni)/rni→0 and uni⇀z~, it follows from (A4) that F1(y,z~)≤0, for any y∈C. Let yt=ty+(1-t)z~, for all t∈(0,1], y∈C. Since {uni}⊂C, uni⇀z~, and C is closed and convex set, then we get z~∈C. Since y∈C and z~∈C, then we have yt∈C. Hence we have F1(yt,z~)≤0.
Thus, from (A1) and (A4), we have(55)0=F1(yt,yt)=F1(yt,ty+(1-t)z~)≤tF1(yt,y)+(1-t)F1(yt,z~)≤tF1(yt,y),
and hence F1(yt,y)≥0. From (A3), we have F1(z~,y)≥0, for any y∈C. Hence z~∈EP⁡(F1)=S.
Therefore,(56)limsup⁡n→∞〈(γf-μF)z,xn-z〉=lim⁡i→∞〈(γf-μF)z,xni-z〉=〈(γf-μF)z,z~-z〉≤0.
Finally, we show that xn→z.
As a matter of fact,(57)xn+1-z=αnγf(xn)+((1-βn)I-αnμF)un+βnxn-z=αnγf(xn)-αnμF(z)+βnxn-βn(z)+((1-βn)I-αnμF)un-((1-βn)I-αnμF)z=αnγ(f(xn)-f(z))+αn(γf(z)-μF(z))+βn(xn-z)+((1-βn)I-αnμF)un-((1-βn)I-αnμF)z.
So, from (29), we derive
(58)∥xn+1-z∥2=αnγ〈f(xn)-f(z),xn+1-z〉+〈((1-βn)I-αnμF)un-((1-βn)I-αnμF)z,xn+1-z〉+αn〈γf(z)-μF(z),xn+1-z〉+βn〈xn-z,xn+1-z〉≤αnγρ∥xn-z∥·∥xn+1-z∥+(1-βn-αnτ)∥un-z∥·∥xn+1-z∥+αn〈γf(z)-μF(z),xn+1-z〉+βn∥xn-z∥·∥xn+1-z∥≤(1-αn(τ-γρ))∥xn-z∥·∥xn+1-z∥+αn〈γf(z)-μF(z),xn+1-z〉≤1-αn(τ-γρ)2(∥xn-z∥2+∥xn+1-z∥2)+αn〈γf(z)-μF(z),xn+1-z〉.
It follows that
(59)∥xn+1-z∥2≤1-αn(τ-γρ)1+αn(τ-γρ)∥xn-z∥2+2αn1+αn(τ-γρ)〈γf(z)-μF(z),xn+1-z〉≤(1-αn(τ-γρ))∥xn-z∥2+2αn1+αn(τ-γρ)〈γf(z)-μF(z),xn+1-z〉.
Hence,
(60)∥xn+1-z∥2≤(1-γn)∥xn-z∥2+γnδn,
where γn=αn(τ-γρ), γnδn=(2αn/(1+αn(τ-γρ)))〈γf(z)-μF(z),xn+1-z〉.
Sinceαn→0 and ∑n=0∞‍αn=∞, we get γn→0 and ∑n=0∞‍γn=∞. From (49), we get
(61)limsup⁡n→∞δn=limsup⁡n→∞2(1+αn(τ-γρ))(τ-γρ)ghghgh×〈γf(z)-μF(z),xn+1-z〉≤0.
Now, applying Lemma 6 to (60) concludes that xn→z as n→∞.Remark 9.
Our proposed algorithm (23) is more general than Su and Li’s algorithm (5), because of introducing nonlinear operator F.
Ifγ=1, μ=1, and F=I, we obtain the following Corollary.Corollary 10.
LetH be a real Hilbert space and let C be a nonempty, closed, and convex subset of H. Let h:C→ℝ be a real-valued convex function. Assume that h(x) is lower semicontinuous. Let f:H→H be a contraction with a coefficient 0<ρ<1. Suppose that the optimization problem (3) is consistent and let S denote its solution set. Let {xn}n=0∞ be generated by x1∈H and
(62)h(y)-h(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-βn-αn)un+βnxn,n∈ℕ,
where un=Trnxn. Let {αn}, {βn}⊂(0,1), {rn}⊂(0,∞) satisfy the following conditions:(C1)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(C2)
∑n=1∞‍|αn+1-αn|<∞;(C3)
limsup⁡n→∞βn<1 and ∑n=1∞‍|βn+1-βn|<∞;(C4)
liminf⁡n→∞rn>0 and ∑n=1∞‍|rn+1-rn|<∞.Then the sequence {xn} generated by (62) converges strongly to a point z∈S, which solves the variational inequality
(63)〈(I-f)z,z-x〉≤0,∀x∈S.Remark 11.
We would like to point out that the conditions of algorithm (62) are different from those of algorithm (4.3) in [2]. The conditions of the sequence {αn} and the sequence {rn} of parameters in (4.3) are the same as the ones of algorithm (62), but the conditions of the sequence {βn} of parameters in (4.3) are βn=o(αn) and ∑n=1∞‍|βn+1-βn|<∞. It is obvious that our conditions are much weaker.


## 4. Application

In this section, we give an application of Theorem8 to the split feasibility problem (say SFP, for short) which was introduced by Censor and Elfving [15]. Since its inception in 1994, the split feasibility problem (SFP) has received much attention (see [16–18]) due to its applications to signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy.The SFP can mathematically be formulated as the problem of finding a pointx with the property
(64)x∈C,Bx∈Q,
where C and Q are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively. B:H1→H2 is a bounded linear operator.It is clear thatx* is a solution to the split feasibility problem (64) if and only if x*∈C and Bx*-PQBx*=0. We define the proximity function h by
(65)h(x)=12∥Bx-PQBx∥2
and consider the following optimization problem:
(66)min⁡x∈Ch(x)=min⁡x∈C12∥Bx-PQBx∥2.
Then, x* solves the split feasibility problem (64) if and only if x* solves the minimization problem (66) with the minimization being equal to 0. Byrne [19] introduced the so-called CQ algorithm to solve the SFP:
(67)xn+1=PC(I-γB*(I-PQ)B)xn,n≥0,
where 0<γ<2/∥B∥2. He obtained that the sequence {xn} generated by (67) converges weakly to a solution of the SFP.In order to obtain strong convergence iterative sequence to solve the SFP (64), we propose the following iterative algorithm by x1∈H1 and
(68)12∥By-PQBy∥2-12∥Bun-PQBun∥2+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+((1-βn)I-αnμF)un+βnxn,ghhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhghn∈ℕ,
where un=Trnxn and Trn:H1→C is a mapping defined as in Lemma 5, where f:H1→H1 is a contraction with a coefficient 0<ρ<1 and F:C→H1 is a k-Lipschitzian and η-strongly monotone operator with constants k>0, η>0. Suppose that 0<μ<2η/k2, 0<γ<τ/ρ, where τ=μη-(μ2k2/2). We can show that the sequence {xn} generated by (68) converges strongly to a solution of the SFP (64), if the sequence {αn}, {βn}⊂(0,1) and the sequence {rn}⊂(0,∞) of parameters satisfy appropriate conditions.Applying Theorem8, we obtain the following result.Theorem 12.
Assume that the split feasibility problem (64) is consistent. Let the sequence {xn} be generated by (68), where the sequence {αn}, {βn}⊂(0,1) and the sequence {rn}⊂(0,∞) satisfy the conditions(C1)–(C5). Then the sequence {xn} converges strongly to a solution of the split feasibility problem (64).Proof.
We define the proximity functionh by
(69)h(x)=12∥Bx-PQBx∥2
and consider the following optimization problem:
(70)min⁡x∈Ch(x)=min⁡x∈C12∥Bx-PQBx∥2,
where h:C→ℝ is a real-valued convex function and h(x) is lower semicontinuous. Then, x* solves the split feasibility problem (64) if and only if x* solves the minimization problem (66) with the minimization being equal to 0.
Consequently, the iterative scheme (68) is equivalent to
(71)x1∈H1,h(y)-h(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+((1-βn)I-αnμF)un+βnxn,n∈ℕ,
where un=Trnxn and Trn:H1→C is a mapping defined as in Lemma 5. Due to Theorem 8, we have the conclusion immediately.



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*Source: 156073-2014-04-17.xml*