# A Regularized Algorithm for the Proximal Split Feasibility Problem

Authors: Zhangsong Yao; Sun Young Cho; Shin Min Kang; Li-Jun Zhu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The proximal split feasibility problem has been studied. A regularized method has been presented for solving the proximal split feasibility problem. Strong convergence theorem is given.


---

## Body

## 1. Introduction

Throughout, we assume thatH
1 and H
2 are two real Hilbert spaces, f
:
H
1
→
R
∪
{
+
∞
} and g
:
H
2
→
R
∪
{
+
∞
} are two proper, lower semicontinuous convex functions, and A
:
H
1
→
H
2 is a bounded linear operator.In the present paper, we are devoted to solving the following minimization problem:(1)
min
⁡
x
†
∈
H
1
{
f
(
x
†
)
+
g
λ
(
A
x
†
)
}
,
where g
λ stands for the Moreau-Yosida approximation of the function g of parameter λ; that is,
(2)
g
λ
(
u
)
=
min
⁡
v
∈
H
2
{
g
(
v
)
+
1
2
λ
∥
u
-
v
∥
2
}
.Problem (1) includes the split feasibility problem as a special case. In fact, we choose f and g as the indicator functions of two nonempty closed convex sets C
⊂
H
1 and Q
∈
H
2; that is,
(3)
f
(
x
†
)
=
δ
C
(
x
†
)
=
{
0
,
if
x
†
∈
C
,
+
∞
,
otherwise
,
g
(
x
†
)
=
δ
Q
(
x
†
)
=
{
0
,
if
x
†
∈
Q
,
+
∞
,
otherwise
.
Then, problem (1) reduces to
(4)
min
⁡
x
†
∈
H
1
{
δ
C
(
x
†
)
+
(
δ
Q
)
λ
(
A
x
†
)
}
,
which equals
(5)
min
⁡
x
†
∈
C
{
1
2
λ
∥
(
I
-
pro
j
Q
)
(
A
x
†
)
∥
2
}
.
Now we know that solving (5) is exactly to solve the following split feasibility problem of finding x
‡ such that
(6)
x
‡
∈
C
,
A
x
‡
∈
Q
,
provided C
∩
A
-
1
(
Q
)
≠
∅.The split feasibility problem in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Recently, the split feasibility problem (6) has been studied extensively by many authors; see, for instance, [2–8].In order to solve (6), one of the key ideas is to use fixed point technique according to x
† which solves (6) if and only if
(7)
x
†
=
pro
j
C
(
I
-
γ
A
*
(
I
-
pro
j
Q
)
A
)
x
†
.
Next, we will use this idea to solve (1). First, by the differentiability of the Yosida approximation g
λ, we have
(8)
∂
(
f
(
x
†
)
+
g
λ
(
A
x
†
)
)
=
∂
f
(
x
†
)
+
A
*
∇
g
λ
(
A
x
†
)
=
∂
f
(
x
†
)
+
A
*
(
I
-
pro
x
λ
g
λ
)
(
A
x
†
)
,
where ∂
f
(
x
†
) denotes the subdifferential of f at x
† and pro
x
λ
g
(
x
†
) is the proximal mapping of g. That is,
(9)
∂
f
(
x
†
)
=
{
x
*
∈
H
1
:
f
(
x
‡
)
≥
f
(
x
†
)
+
〈
x
*
,
x
‡
-
x
†
〉
,
h
h
h
∀
x
‡
∈
H
1
}
,
pro
x
λ
g
(
x
†
)
=
arg
min
⁡
x
‡
∈
H
2
{
g
(
x
‡
)
+
1
2
λ
∥
x
‡
-
x
†
∥
2
}
.
Note that the optimality condition of (8) is as follows:
(10)
0
∈
∂
f
(
x
†
)
+
A
*
(
I
-
pro
x
λ
g
λ
)
(
A
x
†
)
,
which can be rewritten as
(11)
0
∈
μ
λ
∂
f
(
x
†
)
+
μ
A
*
(
I
-
pro
x
λ
g
)
(
A
x
†
)
,
which is equivalent to the fixed point equation
(12)
x
†
=
pro
x
μ
λ
f
(
x
†
-
μ
A
*
(
I
-
pro
x
λ
g
)
)
(
A
x
†
)
.
If arg
min
⁡
⁡
f
∩
A
-
1
(
arg
min
⁡
⁡
g
)
≠
∅, then (1) is reduced to the following proximal split feasibility problem of finding x
† such that
(13)
x
†
∈
arg
min
⁡
⁡
f
,
A
x
†
∈
arg
min
⁡
⁡
g
,
where
(14)
arg
min
⁡
⁡
f
=
{
x
*
∈
H
1
:
f
(
x
*
)
≤
f
(
x
†
)
,
∀
x
†
∈
H
1
}
,
arg
min
⁡
⁡
g
=
{
x
†
∈
H
2
:
g
(
x
†
)
≤
g
(
x
)
,
∀
x
∈
H
2
}
.
In the sequel, we will use Γ to denote the solution set of (13).Recently, in order to solve (13), Moudafi and Thakur [9] presented the following split proximal algorithm with a way of selecting the stepsizes such that its implementation does not need any prior information about the operator norm.Split Proximal AlgorithmStep 1 (initialization).
(15)
x
0
∈
H
1
.Step 2.
Assume thatx
n has been constructed and θ
(
x
n
)
≠
∅. Then compute x
n
+
1 via the manner
(16)
x
n
+
1
=
pro
x
μ
n
λ
f
[
x
n
-
μ
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
]
,
∀
n
≥
0
,
where the stepsize μ
n
=
ρ
n
(
(
h
(
x
n
)
+
l
(
x
n
)
)
/
θ
2
(
x
n
)
) in which 0
<
ρ
n
<
4
,
h
(
x
n
)
=
(
1
/
2
)
∥
(
I
-
pro
x
λ
g
)
A
x
n
∥
2
,
l
(
x
n
)
=
(
1
/
2
)
∥
(
I
-
pro
x
μ
n
λ
f
)
x
n
∥
2  and θ
(
x
n
)
=
∥
∇
h
(
x
n
)
∥
2
+
∥
∇
l
(
x
n
)
∥
2.Ifθ
(
x
n
)
=
0, then x
n
+
1
=
x
n is a solution of (13) and the iterative process stops; otherwise, we set n
:
=
n
+
1 and go to (16).Consequently, they demonstrated the following weak convergence of the above split proximal algorithm.Theorem 1.
Suppose thatΓ
≠
∅. Assume that the parameters satisfy the condition:
(17)
ϵ
≤
ρ
n
≤
4
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
-
ϵ
f
o
r
s
o
m
e
ϵ
>
0
s
m
a
l
l
e
n
o
u
g
h
.
Then the sequence x
n weakly converges to a solution of (13).Note that the proximal mapping ofg is firmly nonexpansive, namely,
(18)
〈
pro
x
λ
g
x
-
pro
x
λ
g
y
,
x
-
y
〉
≥
∥
pro
x
λ
g
x
-
pro
x
λ
g
y
∥
2
,
∀
x
,
y
∈
H
2
,
and it is also the case for complement I
-
pro
x
λ
g. Thus, A
*
(
I
-
pro
x
λ
g
)
A is cocoercive with coefficient 1
/
∥
A
∥
2 (recall that a mapping B
:
H
1
→
H
1 is said to becocoercive if 〈
B
x
-
B
y
,
x
-
y
〉
≥
α
∥
B
x
-
B
y
∥
2 for all x
,
y
∈
H
1 and some α
>
0). If μ
∈
(
0
,
1
/
∥
A
∥
2
), then I
-
μ
A
*
(
I
-
pro
x
λ
g
)
A is nonexpansive. Hence, we need to regularize (16) such that the strong convergence is obtained. This is the main purpose of this paper. In the next section, we will collect some useful lemmas and in the last section we will present our algorithm and prove its strong convergence.


## 2. Lemmas

Lemma 2 (see [10]).
Let{
a
n
}
n
∈
N be a sequence of nonnegative real numbers satisfying the following relation:
(19)
a
n
+
1
≤
(
1
-
α
n
)
a
n
+
α
n
σ
n
+
δ
n
,
n
≥
0
,
where(i)
{
α
n
}
n
∈
N
⊂
[
0,1
] and ∑
n
=
1
∞
‍
α
n
=
∞;
(ii)
limsup
⁡
n
→
∞
σ
n
≤
0;
(iii)
∑
n
=
1
∞
‍
δ
n
<
∞.Then, lim
⁡
n
→
∞
a
n
=
0.Lemma 3 (see [11]).
Let{
γ
n
}
n
∈
N be a sequence of real numbers such that there exists a subsequence {
γ
n
i
}
i
∈
N of {
γ
n
}
n
∈
N such that γ
n
i
<
γ
n
i
+
1 for all i
∈
N. Then, there exists a nondecreasing sequence {
m
k
}
k
∈
N of N such that lim
⁡
k
→
∞
m
k
=
∞ and the following properties are satisfied by all (sufficiently large) numbers k
∈
N:
(20)
γ
m
k
≤
γ
m
k
+
1
,
γ
k
≤
γ
m
k
+
1
.
In fact, m
k is the largest number n in the set {
1
,
…
,
k
} such that the condition γ
n
<
γ
n
+
1 holds.


## 3. Main results

LetH
1 and H
2 be two real Hilbert spaces. Let f
:
H
1
→
R
∪
{
+
∞
} and g
:
H
2
→
R
∪
{
+
∞
} be two proper, lower semicontinuous convex functions and A
:
H
1
→
H
2 a bounded linear operator.Now, we firstly introduce our algorithm.Algorithm  4Step 1 (initialization).
(21)
x
0
∈
H
1
.Step 2.
Assume thatx
n has been constructed. Set h
(
x
n
)
=
(
1
/
2
)
∥
(
I
-
pro
x
λ
g
)
A
x
n
∥
2
,
l
(
x
n
)
=
(
1
/
2
)
∥
(
I
-
pro
x
μ
n
λ
f
)
x
n
∥
2 and θ
(
x
n
)
=
∥
∇
h
(
x
n
)
∥
2
+
∥
∇
l
(
x
n
)
∥
2 for all n
∈
N.
Ifθ
(
x
n
)
≠
∅, then compute x
n
+
1 via the manner
(22)
x
n
+
1
=
pro
x
μ
n
λ
f
[
α
n
u
+
(
1
-
α
n
)
x
n
-
μ
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
]
,
HHHHHHHHHHHHHHHHHHHHHHHHLH
∀
n
≥
0
,
where u
∈
H
1 is a fixed point and {
α
n
}
n
∈
N
⊂
[
0,1
] is a real number sequence and μ
n is the stepsize satisfying μ
n
=
ρ
n
(
(
h
(
x
n
)
+
l
(
x
n
)
)
/
θ
2
(
x
n
)
) with 0
<
ρ
n
<
4.Ifθ
(
x
n
)
=
0, then x
n
+
1
=
x
n is a solution of (13) and the iterative process stops; otherwise, we set n
:
=
n
+
1 and go to (22).Theorem 5.
Suppose thatΓ
≠
∅. Assume that the parameters {
α
n
} and {
ρ
n
} satisfy the conditions:(C1)
lim
⁡
n
→
∞
α
n
=
0;
(C2)
∑
n
=
0
∞
‍
α
n
=
∞;
(C3)
ϵ
≤
ρ
n
≤
(
4
h
(
x
n
)
/
(
h
(
x
n
)
+
l
(
x
n
)
)
)
-
ϵ for some ϵ
>
0 small enough. Then the sequence x
n converges strongly to p
r
o
j
Γ
(
u
).Proof.
Letx
*
∈
Γ. Since minimizers of any function are exactly fixed points of its proximal mappings, we have x
*
=
pro
x
μ
n
λ
f
x
* and A
x
*
=
pro
x
λ
g
A
x
*. By (22) and the nonexpansivity of pro
x
μ
n
λ
f, we derive
(23)
∥
x
n
+
1
-
x
*
∥
2
=
∥
pro
x
μ
n
λ
f
[
α
n
u
+
(
1
-
α
n
)
x
n
-
μ
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
]
h
h
h
-
pro
x
μ
n
λ
f
x
*
∥
2
≤
∥
α
n
u
+
(
1
-
α
n
)
x
n
-
μ
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
x
*
∥
2
=
∥
μ
n
1
-
α
n
α
n
(
u
-
x
*
)
+
(
1
-
α
n
)
hhh
×
[
x
n
-
μ
n
1
-
α
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
x
*
]
∥
2
≤
α
n
∥
u
-
x
*
∥
2
+
(
1
-
α
n
)
×
∥
x
n
-
μ
n
1
-
α
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
x
*
∥
2
.
Since pro
x
λ
g is firmly nonexpansive, we deduce that I
-
pro
x
λ
g is also firmly nonexpansive. Hence, we have
(24)
〈
A
*
(
I
-
pro
x
λ
g
)
A
x
n
,
x
n
-
x
*
〉
=
〈
(
I
-
pro
x
λ
g
)
A
x
n
,
A
x
n
-
A
x
*
〉
=
〈
(
I
-
pro
x
λ
g
)
A
x
n
-
(
I
-
pro
x
λ
g
)
A
x
*
,
A
x
n
-
A
x
*
〉
≥
∥
(
I
-
pro
x
λ
g
)
A
x
n
∥
2
=
2
h
(
x
n
)
.
Note that ∇
h
(
x
n
)
=
A
*
(
I
-
prox
λ
g
)
A
x
n and ∇
l
(
x
n
)
=
(
I
-
prox
μ
n
λ
f
)
x
n. From (24), we obtain
(25)
∥
x
n
-
μ
n
1
-
α
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
x
*
∥
2
=
∥
x
n
-
x
*
∥
2
+
μ
n
2
(
1
-
α
n
)
2
∥
A
*
(
I
-
pro
x
λ
g
)
A
x
n
∥
2
-
2
μ
n
1
-
α
n
〈
A
*
(
I
-
pro
x
λ
g
)
A
x
n
,
x
n
-
x
*
〉
=
∥
x
n
-
x
*
∥
2
+
μ
n
2
(
1
-
α
n
)
2
∥
∇
h
(
x
n
)
∥
2
-
2
μ
n
1
-
α
n
〈
∇
h
(
x
n
)
,
x
n
-
x
*
〉
≤
∥
x
n
-
x
*
∥
2
+
μ
n
2
(
1
-
α
n
)
2
∥
∇
h
(
x
n
)
∥
2
-
4
μ
n
h
(
x
n
)
1
-
α
n
=
∥
x
n
-
x
*
∥
2
+
ρ
n
2
(
h
(
x
n
)
+
l
(
x
n
)
)
2
(
1
-
α
n
)
2
θ
4
(
x
n
)
∥
∇
h
(
x
n
)
∥
2
-
4
ρ
n
h
(
x
n
)
+
l
(
x
n
)
(
1
-
α
n
)
θ
2
(
x
n
)
h
(
x
n
)
≤
∥
x
n
-
x
*
∥
2
+
ρ
n
2
(
h
(
x
n
)
+
l
(
x
n
)
)
2
(
1
-
α
n
)
2
θ
2
(
x
n
)
-
4
ρ
n
(
h
(
x
n
)
+
l
(
x
n
)
)
2
(
1
-
α
n
)
θ
2
(
x
n
)
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
=
∥
x
n
-
x
*
∥
2
-
ρ
n
(
4
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
-
ρ
n
1
-
α
n
)
×
(
h
(
x
n
)
+
l
(
x
n
)
)
2
(
1
-
α
n
)
θ
2
(
x
n
)
.
By condition (C3), without loss of generality, we can assume that (
4
h
(
x
n
)
/
(
h
(
x
n
)
+
l
(
x
n
)
)
)
-
(
ρ
n
/
(
1
-
α
n
)
)
≥
0 for all n
≥
0. Thus, from (23) and (25), we obtain
(26)
∥
x
n
+
1
-
x
*
∥
2
≤
α
n
∥
u
-
x
*
∥
2
+
(
1
-
α
n
)
×
[
(
h
(
x
n
)
+
l
(
x
n
)
)
2
(
1
-
α
n
)
θ
2
(
x
n
)
∥
x
n
-
x
*
∥
2
hhhlh
-
ρ
n
(
4
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
-
ρ
n
1
-
α
n
)
(
h
(
x
n
)
+
l
(
x
n
)
)
2
(
1
-
α
n
)
θ
2
(
x
n
)
]
=
α
n
∥
u
-
x
*
∥
2
+
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
-
ρ
n
(
4
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
-
ρ
n
1
-
α
n
)
(
h
(
x
n
)
+
l
(
x
n
)
)
2
θ
2
(
x
n
)
≤
α
n
∥
u
-
x
*
∥
2
+
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
≤
max
⁡
{
∥
u
-
x
*
∥
2
,
∥
x
n
-
x
*
∥
2
}
.
Hence, {
x
n
} is bounded.
Letz
=
P
Γ
u. From (26), we deduce
(27)
0
≤
ρ
n
(
4
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
-
ρ
n
1
-
α
n
)
(
h
(
x
n
)
+
l
(
x
n
)
)
2
θ
2
(
x
n
)
≤
α
n
∥
u
-
z
∥
2
+
(
1
-
α
n
)
∥
x
n
-
z
∥
2
-
∥
x
n
+
1
-
z
∥
2
.
We consider the following two cases.
Case  1.  One has ∥
x
n
+
1
-
z
∥
≤
∥
x
n
-
z
∥ for every n
≥
n
0 large enough.
In this case,lim
⁡
n
→
∞
∥
x
n
-
z
∥ exists as finite and hence
(28)
lim
⁡
n
→
∞
(
∥
x
n
+
1
-
z
∥
-
∥
x
n
-
z
∥
)
=
0
.
This together with (27) implies that
(29)
ρ
n
(
4
h
(
x
n
)
h
(
x
n
)
+
l
(
x
n
)
-
ρ
n
1
-
α
n
)
(
h
(
x
n
)
+
l
(
x
n
)
)
2
θ
2
(
x
n
)
⟶
0
.
Since liminf
⁡
n
→
∞
ρ
n
(
(
4
h
(
x
n
)
/
(
h
(
x
n
)
+
l
(
x
n
)
)
)
-
(
ρ
n
/
(
1
-
α
n
)
)
)
≥
2
ϵ (by condition (C3)), we get
(30)
(
h
(
x
n
)
+
l
(
x
n
)
)
2
θ
2
(
x
n
)
⟶
0
.
Noting that θ
2
(
x
n
)
=
∥
∇
h
(
x
n
)
∥
2
+
∥
∇
l
(
x
n
)
∥
2 is bounded, we deduce immediately that
(31)
lim
⁡
n
→
∞
(
h
(
x
n
)
+
l
(
x
n
)
)
=
0
.
Therefore,
(32)
lim
⁡
n
→
∞
h
(
x
n
)
=
lim
⁡
n
→
∞
l
(
x
n
)
=
0
.
Next, we prove
(33)
limsup
⁡
n
→
∞
〈
u
-
z
,
x
n
-
z
〉
≤
0
.
Since {
x
n
} is bounded, there exists a subsequence {
x
n
i
} satisfying x
n
i
⇀
z
† and
(34)
limsup
⁡
n
→
∞
〈
u
-
z
,
x
n
-
z
〉
=
lim
⁡
i
→
∞
〈
u
-
z
,
x
n
i
-
z
〉
.
By the lower semicontinuity of h, we get
(35)
0
≤
h
(
z
†
)
≤
liminf
⁡
i
→
∞
h
(
x
n
i
)
=
lim
⁡
n
→
∞
h
(
x
n
)
=
0
.
So,
(36)
h
(
z
†
)
=
1
2
∥
(
I
-
pro
x
λ
g
)
A
z
†
∥
=
0
.
That is, A
z
† is a fixed point of the proximal mapping of g or equivalently 0
∈
∂
g
(
A
z
†
). In other words, A
z
† is a minimizer of g.
Similarly, from the lower semicontinuity ofl, we get
(37)
0
≤
l
(
z
†
)
≤
liminf
⁡
i
→
∞
l
(
x
n
i
)
=
lim
⁡
n
→
∞
l
(
x
n
)
=
0
.
Therefore,
(38)
l
(
z
†
)
=
1
2
∥
(
I
-
pro
x
μ
n
λ
f
)
z
†
∥
=
0
.
That is, z
† is a fixed point of the proximal mapping of f or equivalently 0
∈
∂
f
(
z
†
). In other words, z
† is a minimizer of f. Hence, z
†
∈
Γ. Therefore,
(39)
limsup
⁡
n
→
∞
〈
u
-
z
,
x
n
-
z
〉
=
lim
⁡
i
→
∞
〈
u
-
z
,
x
n
i
-
z
〉
=
〈
u
-
z
,
z
†
-
z
〉
≤
0
.
From (22), we have
(40)
∥
x
n
+
1
-
z
∥
2
≤
∥
μ
n
1
-
α
n
α
n
(
u
-
z
)
+
(
1
-
α
n
)
h
h
l
l
×
[
x
n
-
μ
n
1
-
α
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
z
]
∥
2
=
(
1
-
α
n
)
2
∥
x
n
-
μ
n
1
-
α
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
z
∥
2
+
α
n
2
∥
u
-
z
∥
2
+
2
α
n
(
1
-
α
n
)
×
〈
x
n
-
μ
n
1
-
α
n
A
*
(
I
-
pro
x
λ
g
)
A
x
n
-
z
,
u
-
z
〉
≤
(
1
-
α
n
)
2
∥
x
n
-
z
∥
2
+
α
n
2
∥
u
-
z
∥
2
+
2
α
n
(
1
-
α
n
)
〈
x
n
-
z
,
u
-
z
〉
-
2
α
n
μ
n
〈
∇
h
(
x
n
)
,
u
-
z
〉
≤
(
1
-
α
n
)
∥
x
n
-
z
∥
2
+
α
n
(
α
n
∥
u
-
z
∥
2
+
2
(
1
-
α
n
)
〈
x
n
-
z
,
u
-
z
〉
hhhlhhh
+
2
μ
n
∥
∇
h
(
x
n
)
∥
∥
u
-
z
∥
∥
u
-
z
∥
2
)
.
Since ∇
h is Lipschitz continuous with Lipschitzian constant ∥
A
∥
2 and ∇
l is nonexpansive, ∇
h
(
u
n
),∇
l
(
u
n
), and θ
2
(
x
n
)
=
∥
∇
h
(
x
n
)
∥
2
+
∥
∇
l
(
x
n
)
∥
2 are bounded. Note that μ
n
∥
∇
h
(
x
n
)
∥
=
ρ
n
(
(
h
(
x
n
)
+
l
(
x
n
)
)
/
θ
2
(
x
n
)
)
∥
∇
h
(
x
n
)
∥. Thus, μ
n
∥
∇
h
(
x
n
)
∥
→
0 by (32). From Lemma 2, (39), and (40) we deduce that x
n
→
z.
Case  2. There exists a subsequence {
∥
x
n
j
-
z
∥
} of {
∥
x
n
-
z
∥
} such that
(41)
∥
x
n
j
-
z
∥
<
∥
x
n
j
+
1
-
z
∥
,
for all j
≥
1. By Lemma 3, there exists a strictly increasing sequence {
m
k
} of positive integers such that li
m
k
→
∞
m
k
=
+
∞ and the following properties are satisfied by all numbers k
∈
N:
(42)
∥
x
m
k
-
z
∥
≤
∥
x
m
k
+
1
-
z
∥
,
∥
x
k
-
z
∥
≤
∥
x
m
k
+
1
-
z
∥
.
Consequently,
(43)
0
≤
lim
⁡
k
→
∞
(
∥
x
m
k
+
1
-
z
∥
-
∥
x
m
k
-
z
∥
)
≤
limsup
⁡
n
→
∞
(
∥
x
n
+
1
-
z
∥
-
∥
x
n
-
z
∥
)
≤
limsup
⁡
n
→
∞
(
α
n
∥
u
-
z
∥
+
(
1
-
α
n
)
∥
x
n
-
z
∥
-
∥
x
n
-
z
∥
)
=
limsup
⁡
n
→
∞
α
n
(
∥
u
-
z
∥
-
∥
x
n
-
z
∥
)
=
0
.
Hence,
(44)
lim
⁡
k
→
∞
(
∥
x
m
k
+
1
-
z
∥
-
∥
x
m
k
-
z
∥
)
=
0
.
By a similar argument as that of Case 1, we can prove that
(45)
limsup
⁡
k
→
∞
〈
u
-
z
,
x
m
k
-
z
〉
≤
0
,
∥
x
m
k
+
1
-
z
∥
2
≤
(
1
-
α
m
k
)
∥
x
m
k
-
z
∥
2
+
α
m
k
σ
m
k
,
where σ
m
k
=
α
m
k
∥
u
-
z
∥
2
+
2
(
1
-
α
m
k
)
〈
x
m
k
-
z
,
u
-
z
〉
+
2
μ
m
k
∥
∇
h
(
x
m
k
)
∥
∥
u
-
z
∥.
In particular, we get(46)
α
m
k
∥
x
m
k
-
z
∥
2
≤
∥
x
m
k
-
z
∥
2
-
∥
x
m
k
+
1
-
z
∥
+
α
m
k
σ
m
k
≤
α
m
k
σ
m
k
.
Then,
(47)
limsup
⁡
k
→
∞
∥
x
m
k
-
z
∥
2
≤
limsup
⁡
k
→
∞
⁡
σ
m
k
≤
0
.
Thus, from (42) and (44), we conclude that
(48)
limsup
⁡
k
→
∞
∥
x
k
-
z
∥
≤
limsup
⁡
k
→
∞
∥
x
m
k
+
1
-
z
∥
=
0
.
Therefore, x
n
→
z. This completes the proof.Remark 6.
Note that problem (13) was considered, for example, in [12, 13]; however, the iterative methods proposed to solve it need to know a priori the norm of the bounded linear operator A.Remark 7.
We would like also to emphasize that by takingf
=
δ
C, g
=
δ
Q the indicator functions of two nonempty closed convex sets C, Q of H
1 and H
2 respectively, our algorithm (22) reduces to
(49)
x
n
+
1
=
pro
j
C
[
α
n
u
+
(
1
-
α
n
)
x
n
-
μ
n
A
*
(
I
-
pro
j
Q
)
A
x
n
]
,
hhhhhhlhhhhhhhhhhhhhhhhhhhhhhhhhhh
∀
n
≥
0
.
We observe that (49) is simpler than the one in [14].



---

*Source: 894272-2014-07-02.xml*