General proximal point algorithm for monotone operators
We introduce a new general proximal point algorithm for an infinite family of monotone operators in a real Hilbert space. We establish strong convergence of the iterative process to a common zero point of the infinite family of monotone operators. Our result generalizes and improves numerous results...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507828859437056 |
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| author | Eslamian, M. Vahidi, J. Есламіан, М. Вахіді, Й. |
| author_facet | Eslamian, M. Vahidi, J. Есламіан, М. Вахіді, Й. |
| author_sort | Eslamian, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:32:19Z |
| description | We introduce a new general proximal point algorithm for an infinite family of monotone operators in a real Hilbert space.
We establish strong convergence of the iterative process to a common zero point of the infinite family of monotone
operators. Our result generalizes and improves numerous results in the available literature. |
| first_indexed | 2026-03-24T02:15:31Z |
| format | Article |
| fulltext |
UDC 517.9
M. Eslamian*
(Dep. Math., Univ. Sci. and Technology Mazandaran, Behshahr and School Math., Inst. Res. Fundam. Sci., Tehran, Iran),
J. Vahidi (Dep. Math., Iran Univ. Sci. and Technology, Tehran, Iran)
GENERAL PROXIMAL POINT ALGORITHM FOR MONOTONE OPERATORS
ЗАГАЛЬНИЙ АЛГОРИТМ НАЙБЛИЖЧОЇ ТОЧКИ
ДЛЯ МОНОТОННИХ ОПЕРАТОРIВ
We introduce a new general proximal point algorithm for an infinite family of monotone operators in a real Hilbert space.
We establish strong convergence of the iterative process to a common zero point of the infinite family of monotone
operators. Our result generalizes and improves numerous results in the available literature.
Введено новий загальний алгоритм найближчої точки для нескiнченної сiм’ї монотонних операторiв у дiйсному
гiльбертовому просторi. Встановлено сильну збiжнiсть iтерацiйного процесу до спiльної нульової точки нескiнчен-
ної сiм’ї монотонних операторiв. Отриманий результат узагальнює та покращує численнi результати, що вiдомi з
лiтературних джерел.
1. Introduction. Let H be a real Hilbert space with scalar product \langle ., .\rangle and A : D(A) \subset H \rightarrow H be
a set-valued operator. Recall that A is called monotone if \langle u - v, x - y\rangle \geq 0, for any [x, u], [y, v] \in
\in G(A), where
G(A) =
\bigl\{
(x, u) : x \in D(A), u \in A(x)
\bigr\}
.
A monotone operator A is said to be maximal monotone if its graph G(A) is not properly contained
in the graph of any other monotone operator. Monotone operators have proven to be a key class of
objects in modern Optimization and Analysis (see, e.g., the books [1 – 4] and the references therein).
On the other hand, a variety of problems, including convex programming and variational inequalities,
can be formulated as finding zeros of monotone operators. Consequently the problem of finding a
solution z \in H of 0 \in Az has been investigated by many researchers. A popular method used to
solve iteratively 0 \in Az is the proximal point algorithm of Rockafellar [5], which is recognized as a
powerful and successful algorithm in finding zeros of monotone operators. Starting from any initial
guess x0 \in H, this proximal point algorithm generates a sequence \{ xn\} given by
xn+1 = JA
cn(xn + en), (1.1)
where JA
r = (I + rA) - 1 for all r > 0 is the resolvent of A and \{ en\} is a sequence of errors.
Rockafellar proved the weak convergence of the algorithm (1.1). However, as shown by Güler [6],
the proximal point algorithm does not necessarily converge strongly. Since then, many authors
have conducted worthwhile research on modifying the proximal point algorithm so that the strong
convergence is guaranteed (see, for instance, [7 – 10]). In particular, Xu [11] introduced the following
iterative scheme:
xn+1 = tnx0 + (1 - tn)J
A
rnxn + en, (1.2)
* The research of the first author was in part supported by a grant from IPM (No. 94470071).
c\bigcirc M. ESLAMIAN, J. VAHIDI, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11 1483
1484 M. ESLAMIAN, J. VAHIDI
where x0 is the starting point and \{ en\} is the error sequence. For \{ en\} summable, it was proved that
\{ xn\} is strongly convergent if rn \rightarrow \infty , and \{ tn\} \subset (0, 1) with \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty tn = 0,
\sum \infty
n=0
tn = \infty .
Boikanyo and Morosanu [12] generalized this algorithm (1.2) with error sequences in lp for 1 \leq p < 2.
Recently, Xu [13] proposed the following regularization for the proximal point algorithm:
xn+1 = JA
rn(tnx0 + (1 - tn)xn + en) (1.3)
which essentially includes the so called prox-Tikhonov algorithm introduced by Lehdili and Mou-
dafi [14] as a special cases. Boikanyo and Morosanu [15] noted that the proximal point algorithm (1.3)
is equivalent to algorithm (1.2). These algorithms have been further studied and analyzed by many
authors (see [16 – 23]).
In this work we introduce a general proximal point algorithm for finding a common zero point for
an infinite family of monotone operators. We establish strong convergence of the iterative process to
a common zero of the family of monotone operators. Our result generalizes some results of Xu [11],
Tian and Song [17], Boikanyo and Morosanu [16], Yao and Noor [23], and many others.
2. Preliminaries. Let H be a real Hilbert space with inner product \langle ., .\rangle and induced norm \| .\| .
We write xn \rightharpoonup x to indicate that the sequence \{ xn\} converge weakly to x, and xn \rightarrow x to indicate
that the sequence \{ xn\} converges strongly to x. Let K be a nonempty, closed and convex subset of
H. Then, for any x \in H, there exists a unique nearest point in K, denoted by PKx, such that
\| x - PKx\| \leq \| x - y\| \forall y \in K.
Operator PK is called the metric projection of H onto K. We also know that for x \in H and z \in K,
z = PKx if and only if
\langle x - z, y - z\rangle \leq 0 \forall y \in K.
It is known that H satisfies Opial’s condition, i.e., for any sequence \{ xn\} with xn \rightharpoonup x, the inequality
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\| xn - x\| < \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\| xn - y\|
holds for every y \in H with y \not = x. We will use the following notions on S : K \rightarrow H.
(i) S is nonexpansive if
\| Sx - Sy\| \leq \| x - y\| \forall x, y \in K.
(ii) S is firmly nonexpansive if
\| Sx - Sy\| 2 \leq \| x - y\| 2 - \| (x - Sx) - (y - Sy)\| 2 \forall x, y \in K.
It is well known that PK is a nonexpansive mapping.
The resolvent operator has the following properties:
Lemma 2.1 [1]. For a \lambda > 0,
(i) A is monotone if and only if the resolvent JA
\lambda of A is single valued and firmly nonexpansive;
(ii) A is maximal monotone if and only if JA
\lambda of A is single valued and firmly nonexpansive and
its domain is all of H;
(iii) 0 \in A(x \star ) \Leftarrow \Rightarrow x \star \in \mathrm{F}\mathrm{i}\mathrm{x}(JA
\lambda ), where \mathrm{F}\mathrm{i}\mathrm{x}(JA
\lambda ) denotes the fixed point set of JA
\lambda .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
GENERAL PROXIMAL POINT ALGORITHM FOR MONOTONE OPERATORS 1485
Since the fixed point set of a nonexpansive operator is closed convex, the projection onto the
solution set Z = A - 1(0) = \{ x \in D(A) : 0 \in Ax\} is well defined whenever Z \not = \varnothing . For more
details, see [24].
Lemma 2.2 [1] (The resolvent identity). For \lambda , \mu > 0, there holds the identity
JA
\lambda x = JA
\mu
\Bigl( \mu
\lambda
x+
\Bigl(
1 - \mu
\lambda
\Bigr)
JA
\lambda x
\Bigr)
, x \in H.
Let B be a strongly positive bounded linear operator on H, that is, there is a constant \gamma > 0
such that
\langle Bx, x\rangle \geq \gamma \| x\| 2 \forall x \in H.
A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive
mapping S :
\mathrm{m}\mathrm{i}\mathrm{n}
x\in F (S)
1
2
\langle Bx, x\rangle - \langle x, b\rangle .
Marino and Xu [25] introduced the following iterative process for finding a fixed point of a nonex-
pansive mapping based on the viscosity approximation method introduced by Moudafi [26]:
xn+1 = an\gamma f(xn) + (I - anB)Sxn \forall n \geq 0. (2.1)
They proved that under some appropriate condition imposed on the parameters, the sequence \{ xn\}
generated by (2.1) converges strongly to the unique solution of the variational inequality
\langle (B - \gamma f)x \star , x - x \star \rangle \geq 0 \forall x \in F (S),
which is the optimality condition for the minimization problem
\mathrm{m}\mathrm{i}\mathrm{n}
x\in F (S)
1
2
\langle Bx, x\rangle - h(x),
where h is a potential function for \gamma f
\bigl(
i.e., h\prime (x) = \gamma f(x) \forall x \in H
\bigr)
.
Lemma 2.3 [25]. Assume that B is a strongly positive bounded linear operator on a Hilbert
space H with coefficient \gamma > 0 and 0 < \rho \leq \| B\| - 1. Then \| I - \rho B\| \leq 1 - \rho \gamma .
Lemma 2.4. There holds the following inequality in a Hilbert space H :
\| x+ y\| 2 \leq \| x\| 2 + 2\langle y, x+ y\rangle , \forall x, y \in H.
Lemma 2.5 [27]. Let H be a Hilbert space and \{ xn\} be a sequence in H. Then for any given
\{ \lambda n\} \infty n=1 \subset (0, 1) with
\sum \infty
n=1
\lambda n = 1 and for any positive integer i, j with i < j,\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
n=1
\lambda nxn
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
\leq
\infty \sum
n=1
\lambda n\| xn\| 2 - \lambda i\lambda j\| xi - xj\| 2.
Lemma 2.6 [11]. Assume that \{ \alpha n\} is a sequence of nonnegative real numbers such that
\alpha n+1 \leq (1 - \gamma n)\alpha n + \gamma n\delta n + \beta n, n \geq 0,
where \{ \gamma n\} , \{ \beta n\} and \{ \delta n\} satisfy the conditions:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1486 M. ESLAMIAN, J. VAHIDI
(i) \gamma n \subset [0, 1],
\sum \infty
n=1
\gamma n = \infty ,
(ii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty \delta n \leq 0 or
\sum \infty
n=1
| \gamma n\delta n| < \infty ,
(iii) \beta n \geq 0 for all n \geq 0 with
\sum \infty
n=0
\beta n < \infty .
Then \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \alpha n = 0.
Lemma 2.7 [28]. Let \{ tn\} be a sequence of real numbers that does not decrease at infinity, in
the sense that there exists a subsequence \{ tni\} of \{ tn\} such that tni \leq tni+1 for all i \geq 0. For
sufficiently large numbers n \in \BbbN , define an integer sequence \{ \tau (n)\} as
\tau (n) = \mathrm{m}\mathrm{a}\mathrm{x}\{ k \leq n : tk < tk+1\} .
Then \tau (n) \rightarrow \infty as n \rightarrow \infty and
\mathrm{m}\mathrm{a}\mathrm{x}\{ t\tau (n), tn\} \leq t\tau (n)+1.
3. Main result. Now, we state our main result.
Theorem 3.1. Let Ai, i \in \BbbN , be an infinite family of monotone operators of a Hilbert space H
with Z =
\bigcap \infty
i=1A
- 1
i (\{ 0\} ) \not = \varnothing . Assume that K is a nonempty closed convex subset of H such that\bigcap \infty
i=1D(Ai) \subset K \subset
\bigcap \infty
i=1R(I + rAi) for all r > 0. Assume that f is a b-contraction of K into
itself and B is a strongly positive bounded linear operator on H with coefficient \gamma and 0 < \gamma <
\gamma
b
.
Let \{ xn\} be a sequence generated by x0 \in H and
yn = \alpha n,0xn +
\infty \sum
i=1
\alpha n,iJ
Ai
rn xn, n \geq 0,
xn+1 = \beta n\gamma f(xn) + (I - \beta nB)yn \forall n \geq 0,
where
\sum \infty
i=0
\alpha n,i = 1 and \{ \alpha n,i\} and \{ \beta n\} satisfy the following conditions:
(i) \{ \beta n\} \subset (0, 1), \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0,
\sum \infty
n=1
\beta n = \infty ,
(ii) \{ rn\} \subset (0,\infty ) and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty rn > 0,
(iii) \{ \alpha n,i\} \subset (0, 1) and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty \alpha n,0\alpha n,i > 0 for all i \in \BbbN .
Then the sequence \{ xn\} converges strongly to z \in Z, which solves the variational inequality;
\langle (B - \gamma f)z, x - z\rangle \geq 0 \forall x \in Z.
Proof. Since Z =
\bigcap \infty
i=1A
- 1
i (\{ 0\} ) is closed and convex, we have the projection PZ is well
defined. Since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0, we can assume that \beta n \in (0, \| B\| - 1) for all n \geq 0. Applying
Lemma 2.3 we have
\| I - \beta nB\| \leq 1 - \beta n\gamma . (3.1)
Next, we show that \{ xn\} is bounded. By Lemma 2.1, the operators JAi
rn are nonexpansive and hence
we get
\| yn - z\| \leq \| \alpha n,0xn +
\infty \sum
i=1
\alpha n,iJ
Ai
rn xn - z\| \leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
GENERAL PROXIMAL POINT ALGORITHM FOR MONOTONE OPERATORS 1487
\leq \alpha n,0\| xn - z\| +
\infty \sum
i=1
\alpha n,i\| JAi
rn xn - z\| \leq
\leq \alpha n,0\| xn - z\| +
\infty \sum
i=1
\alpha n,i\| xn - z\| \leq \| xn - z\| .
By using inequality (3.1) we obtain
\| xn+1 - z\| = \| \beta n(\gamma f(xn) - Bz) + ((I - \beta nB)(yn - z)\| \leq
\leq \beta n\| \gamma f(xn) - Bz\| + \| I - \beta nB\| \| yn - z\| \leq
\leq \beta n\gamma \| f(xn) - f(z)\| + \beta n\| \gamma f(z) - Bz\| + (1 - \beta n\gamma )\| xn - z\| \leq
\leq \beta n\gamma b\| xn - z\| + \beta n\| \gamma f(z) - Bz\| + (1 - \beta n\gamma )\| xn - z\| \leq
\leq (1 - \beta n(\gamma - \gamma b))\| xn - z\| + \beta n\| \gamma f(z) - Bz\| \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| xn - z\| , 1
\gamma - \gamma b
\| \gamma f(z) - Bz\|
\biggr\}
.
It follows by induction that
\| xn - z\| \leq \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| x0 - z\| , 1
\gamma - \gamma b
\| \gamma f(z) - Bz\|
\biggr\}
\forall n \geq 0.
This shows that \{ xn\} is bounded and so is \{ f(xn)\} . Next, we show that for each i \in \BbbN ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| xn - JAi
rn xn\| = 0.
By using Lemma 2.5, for each i \in \BbbN we get
\| yn - z\| 2 \leq \| \alpha n,0xn +
\infty \sum
i=1
\alpha n,iJ
Ai
rn xn - z\| 2 \leq
\leq \alpha n,0\| xn - z\| 2 +
\infty \sum
i=1
\alpha n,i\| JAi
rn xn - z\| 2 - \alpha n,0\alpha n,i\| JAi
rn xn - xn\| 2 \leq
\leq \alpha n,0\| xn - z\| 2 +
\infty \sum
i=1
\alpha n,i\| xn - z\| 2 - \alpha n,0\alpha n,i\| JAi
rn xn - xn\| 2 \leq
\leq \| xn - z\| 2 - \alpha n,0\alpha n,i\| JAi
rn xn - xn\| 2. (3.2)
Consequently, we have
\| xn+1 - z\| 2 = \| \beta n(\gamma f(xn) - Bz) + (I - \beta nB)(yn - z)\| 2 \leq
\leq \| \beta n(\gamma f(xn) - Bz) + (I - \beta nB)(yn - z)\| 2 \leq
\leq \beta 2
n\| \gamma f(xn) - Bz\| 2 + (1 - \beta n\gamma )
2\| yn - z\| 2+
+2\beta n(1 - \beta n\gamma )\| \gamma f(xn) - Bz\| \| yn - z\| \leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1488 M. ESLAMIAN, J. VAHIDI
\leq \beta 2
n\| \gamma f(xn) - Bz\| 2 + 2\beta n(1 - \beta n\gamma )\| \gamma f(xn) - Bz\| \| xn - z\| +
+(1 - \beta n\gamma )
2\| xn - z\| 2 - (1 - \beta n\gamma )
2\alpha n,0\alpha n,i\| JAi
rn xn - xn\| 2. (3.3)
So, we have for every i \in \BbbN
(1 - \beta n\gamma )
2\alpha n,0\alpha n,i\| JAi
rn xn - xn\| 2 \leq
\leq \| xn - z\| 2 - \| xn+1 - z\| 2 + 2\beta n(1 - \beta n\gamma )\| \gamma f(xn) - Bz\| \| xn - z\| +
+\beta 2
n\| \gamma f(xn) - Bz\| 2. (3.4)
We note that the Banach contraction mapping principle guarantees that P\scrZ (I - B+\gamma f) has a unique
fixed point z which is the unique solution of the variational inequality
\langle (B - \gamma f)z, x - z\rangle \geq 0 \forall x \in Z.
We finally analyze the inequality (3.4) by considering the following two cases.
Case 1. Assume that
\bigl\{
\| xn - z\|
\bigr\}
is a monotone sequence. In other words, for n0 large enough,
\{ \| xn - z\| \} n\geq n0 is either nondecreasing or nonincreasing. Since \| xn - z\| is bounded, we have
\| xn - z\| is convergent. Since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0, and \{ f(xn)\} and \{ xn\} are bounded, from (3.4) we
get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
(1 - \beta n\gamma )
2\alpha n,0\alpha n,i\| JAi
rn xn - xn\| 2 = 0,
which implies that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| JAi
rn xn - xn\| = 0.
Using the resolvent identity (Lemma 2.2), for each r > 0 we have\bigm\| \bigm\| xn - JAi
r xn
\bigm\| \bigm\| \leq
\bigm\| \bigm\| xn - JAi
rn xn
\bigm\| \bigm\| +
\bigm\| \bigm\| JAi
rn xn - JAi
r xn
\bigm\| \bigm\| \leq
\leq
\bigm\| \bigm\| xn - JAi
rn xn
\bigm\| \bigm\| +
\bigm\| \bigm\| \bigm\| \bigm\| JAi
r
\biggl(
r
rn
xn +
\biggl(
1 - r
rn
\biggr)
JAi
rn xn
\biggr)
- JAi
r xn
\bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq \| xn - JAi
rn xn\| +
\bigm\| \bigm\| \bigm\| \bigm\| r
rn
xn +
\biggl(
1 - r
rn
\biggr)
JAi
rn xn - xn
\bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq
\bigm\| \bigm\| xn - JAi
rn xn
\bigm\| \bigm\| +
\bigm| \bigm| \bigm| \bigm| 1 - r
rn
\bigm| \bigm| \bigm| \bigm| \bigm\| \bigm\| JAi
rn xn - xn
\bigm\| \bigm\| \rightarrow 0, n \rightarrow \infty .
Next we show that \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty \langle (B - \gamma f)z, z - xn\rangle \leq 0. We can choose a subsequence \{ xni\} of
\{ xn\} such that
\mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
(\langle B - \gamma f)z, z - xni\rangle = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
(\langle B - \gamma f)z, z - xn\rangle .
Since \{ xni\} is bounded, there exists a subsequence \{ xnij
\} of \{ xni\} which converges weakly to x\ast .
Without loss of generality, we can assume that xni \rightharpoonup x\ast . We show that x\ast \in Z. Indeed,
\| xni - JAi
r x\ast \| \leq \| xni - JAi
r xni\| + \| JAi
r xni - JAi
r x\ast \| \leq
\leq \| xni - JAi
r xni\| + \| xni - x\ast \| ,
which implies that
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GENERAL PROXIMAL POINT ALGORITHM FOR MONOTONE OPERATORS 1489
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
i\rightarrow \infty
\| xni - JAi
r x\ast \| \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
i\rightarrow \infty
\| xni - x\ast \| .
By the Opial property of Hilbert space H we obtain x\ast = JAi
r x\ast , i \in \BbbN . Hence x\ast \in Z. Therefore,
from z = PZ(I - B + \gamma f)z and x\ast \in \Psi , it follows that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\langle (B - \gamma f)z, z - xn\rangle = \mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
(\langle B - \gamma f)z, z - xni\rangle = (\langle B - \gamma f)z, z - x\ast \rangle \leq 0.
Since
xn+1 - z = \beta n(\gamma f(xn) - Bz) + (I - \beta nB)(yn - z),
by using Lemma 2.4 and the inequality (3.2) we have
\| xn+1 - z\| 2 \leq \| (I - \beta nB)(yn - z)\| 2 + 2\beta n\langle \gamma f(xn) - Bz, xn+1 - z\rangle \leq
\leq (1 - \beta n\gamma )
2\| xn - z\| 2+
+2\beta n\gamma \langle f(xn) - f(z), xn+1 - z\rangle + 2\beta n\langle \gamma f(z) - Bz, xn+1 - z\rangle \leq
\leq (1 - \beta n\gamma )
2\| xn - z\| 2 + 2\beta nb\gamma \| xn - z\| \| xn+1 - z\| +
+2\beta n\langle \gamma f(z) - Bz, xn+1 - z\rangle \leq
\leq (1 - \beta n\gamma )
2\| xn - z\| 2 + \beta nb\gamma (\| xn - z\| 2 + \| xn+1 - z\| 2)+
+2\beta n\langle \gamma fz - Bz, xn+1 - z\rangle \leq
\leq
\bigl(
(1 - \beta n\gamma )
2 + \beta nb\gamma
\bigr)
\| xn - z\| 2 + \beta n\gamma b\| xn+1 - z\| 2+
+2\beta n\langle \gamma f(z) - Bz, xn+1 - z\rangle .
This implies that
\| xn+1 - z\| 2 \leq 1 - 2\beta n\gamma + (\beta n\gamma )
2 + \beta n\gamma b
1 - \beta n\gamma b
\| xn - z\| 2+
+
2\beta n
1 - \eta n\gamma b
\langle \gamma fz - Bz, xn+1 - z\rangle =
=
\biggl(
1 - 2(\gamma - \gamma b)\beta n
1 - \beta n\gamma b
\biggr)
\| xn - z\| 2 + (\beta n\gamma )
2
1 - \eta n\gamma b
\| xn - z\| 2+
+
2\beta n
1 - \beta n\gamma b
\langle \gamma fz - Bz, xn+1 - z\rangle \leq
\biggl(
1 - 2(\gamma - \gamma b)\beta n
1 - \beta n\gamma b
\biggr)
\| xn - z\| 2+
+
2(\gamma - \gamma b)\beta n
1 - \beta n\gamma b
\biggl(
(\beta n\gamma
2)P
2(\gamma - \gamma b)
+
1
\gamma - \gamma b
\biggr)
\langle \gamma fz - Bz, xn+1 - z\rangle =
= (1 - \gamma n)\| xn - z\| 2 + \gamma n\delta n
where P = \mathrm{s}\mathrm{u}\mathrm{p}\{ \| xn - z\| 2 : n \geq 0\} , \gamma n =
2(\gamma - \gamma b)\beta n
1 - \beta n\gamma b
and
\delta n =
(\beta n\gamma
2)P
2(\gamma - \gamma b)
+
1
\gamma - \gamma b
\langle \gamma fz - Bz, xn+1 - z\rangle .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1490 M. ESLAMIAN, J. VAHIDI
It is easy to see that \gamma n \rightarrow 0,
\sum \infty
n=1
\gamma n = \infty and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty \delta n \leq 0. Now by applying Lemma 2.6
we conclude that the sequence \{ xn\} converges strongly to z.
Case 2. Assume that
\bigl\{
\| xn - z\|
\bigr\}
is not a monotone sequence. Then, we can define an integer
sequence \{ \tau (n)\} for all n \geq n0 (for some n0 large enough) by
\tau (n) = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
k \in \BbbN ; k \leq n : \| xk - z\| < \| xk+1 - z\|
\bigr\}
.
Clearly, \tau (n) is a nondecreasing sequence such that \tau (n) \rightarrow \infty as n \rightarrow \infty and for all n \geq n0,
\| x\tau (n) - z\| < \| x\tau (n)+1 - z\| .
Now, it follows from (3.3) that
\| xn+1 - z\| 2 - \| xn - z\| 2 \leq \beta 2
n\| \gamma f(xn) - Bz\| 2 + ((\beta n\gamma )
2 - 2\beta n\gamma )\| xn - z\| 2+
+2\beta n(1 - \beta n\gamma )\| \gamma f(xn) - Bz\| \| xn - z\| .
Since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0 and \{ f(xn)\} and \{ xn\} are bounded, we derive
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bigl(
\| x\tau (n)+1 - z\| 2 - \| x\tau (n) - z\| 2
\bigr)
= 0. (3.5)
By the similar argument as in Case 1 we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bigm\| \bigm\| JAi
rn x\tau (n) - x\tau (n)
\bigm\| \bigm\| = 0
and
\| x\tau (n)+1 - z\| 2 \leq (1 - \gamma \tau (n))\| x\tau (n) - z\| 2 + \gamma \tau (n)\delta \tau (n),
where \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty \delta \tau (n) \leq 0. Since \| x\tau (n) - z\| \leq \| x\tau (n)+1 - z\| , we have
\gamma \tau (n)\| x\tau (n) - z\| 2 \leq \gamma \tau (n)\delta \tau (n).
Since \gamma \tau (n) > 0 we deduce
\| x\tau (n) - z\| 2 \leq \delta \tau (n).
From \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty \delta \tau (n) \leq 0 we get \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| x\tau (n) - z\| = 0. This together with (3.5), implies that
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| x\tau (n)+1 - z\| = 0. Thus by Lemma 2.7 we have
0 \leq \| xn - z\| \leq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\| x\tau (n) - z\| , \| xn - z\|
\bigr\}
\leq \| x\tau (n)+1 - z\| .
Therefore \{ xn\} converges strongly to z = P\scrZ (I - B + \gamma f)z.
Theorem 3.1 is proved.
Theorem 3.2. Let Ai, i \in \BbbN , be an infinite family of maximal monotone operators of a real
Hilbert space H with Z =
\bigcap \infty
i=1A
- 1
i (\{ 0\} ) \not = \varnothing . Assume that f is a b-contraction of H into itself
and A is a strongly positive bounded linear operator on H with coefficient \gamma and 0 < \gamma <
\gamma
b
. Let
\{ xn\} be a sequence generated by x0 \in H and
yn = \alpha n,0xn +
\infty \sum
i=1
\alpha n,iJ
Ai
rn xn, n \geq 0,
xn+1 = \beta n\gamma f(xn) + (I - \beta nB)yn \forall n \geq 0,
where
\sum \infty
i=0
\alpha n,i = 1 and \{ \alpha n,i\} and \{ \beta n\} satisfy the following conditions:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
GENERAL PROXIMAL POINT ALGORITHM FOR MONOTONE OPERATORS 1491
(i) \{ \beta n\} \subset (0, 1), \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0,
\sum \infty
n=1
\beta n = \infty ,
(ii) \{ rn\} \subset (0,\infty ) and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty rn > 0,
(iii) \{ \alpha n,i\} \subset (0, 1) and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty \alpha n,0\alpha n,i > 0 for all i \in \BbbN .
Then the sequence \{ xn\} converges strongly to z \in Z, which solves the variational inequality
\langle (B - \gamma f)z, x - z\rangle \geq 0 \forall x \in Z.
Proof. Since Ai are maximal monotones, then Ai are monotone and satisfy the condition
R(I + rAi) = H for all r > 0. Putting K = H in Theorem 3.1, the desired result follows.
Putting B = I and \gamma = 1 in Theorem 3.1, for a finite family of monotone operators we obtain
immediately the following result.
Corollary 3.1. Let Ai, i = 1, 2, . . . ,m, be a finite family of monotone operators of a Hilbert
space H with Z =
\bigcap m
i=1A
- 1
i (\{ 0\} ) \not = \varnothing . Assume that K is a nonempty closed convex subset of H
such that
\bigcap m
i=1D(Ai) \subset K \subset
\bigcap m
i=1R(I + rAi) for all r > 0. Assume that f is a b-contraction of
K into itself. Let \{ xn\} be a sequence generated by x0 \in H and
yn = \alpha n,0xn +
m\sum
i=1
\alpha n,iJ
Ai
rn xn, n \geq 0,
xn+1 = \beta nf(xn) + (1 - \beta n)yn \forall n \geq 0,
where
\sum m
i=0
\alpha n,i = 1 and \{ \alpha n,i\} , \{ \beta n\} satisfy the following conditions:
(i) \{ \beta n\} \subset (0, 1), \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0,
\sum \infty
n=1
\beta n = \infty ,
(ii) \{ rn\} \subset (0,\infty ) and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty rn > 0,
(iii) \{ \alpha n,i\} \subset (0, 1) and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty \alpha n,0\alpha n,i > 0 for i = 1, 2, . . . ,m.
Then the sequence \{ xn\} converges strongly to z \in Z, which solves the variational inequality
\langle z - fz, x - z\rangle \geq 0 \forall x \in Z.
References
1. Brezis H. Operateurs maximaux monotones et semi-groups de contractions dans les espaces de Hilbert. – Amsterdam:
North-Holland, 1973.
2. Morosanu G. Nonlinear evolution equations and applications. – D. Reidel, 1988.
3. Pardalos P. M., Rassias T. M., Khan A. A. Nonlinear analysis and variational problems. – Berlin: Springer, 2010.
4. Burachik R. S., IusemA. N. Set-valued mappings and enlargements of monotone operators. – New York: Springer,
2008.
5. Rockafellar R. T. Monotone operators and the proximal point algorithm // SIAM J. Contr. Optim. – 1976. – 14. –
P. 877 – 898.
6. Güler O. On the convergence of the proximal point algorithm for convex minimization // SIAM J. Contr. Optim. –
1991. – 29. – P. 403 – 419.
7. Solodov M. V., Svaiter B. F. Forcing strong convergence of proximal point iterations in a Hilbert space // Math. Progr.
Ser. A. – 2000. – 87. – P. 189 – 202.
8. Kamimura S., Takahashi W. Approximating solutions of maximal monotone operators in Hilbert spaces // J. Approxim.
Theory. – 2000. – 106. – P. 226 – 240.
9. Kamimura S., Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space // SIAM J. Optim. –
2002. – 13. – P. 938 – 945.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1492 M. ESLAMIAN, J. VAHIDI
10. Takahashi W. Viscosity approximation methods for resolvents of accretive operators in Banach spaces // J. Fixed
Point Theory and Appl. – 2007. – 1. – P. 135 – 147.
11. Xu H. K. Iterative algorithms for nonlinear operators // J. London Math. Soc. – 2002. – 66. – P. 240 – 256.
12. Boikanyo O. A., Morosanu G. Modified Rockafellar’s algorithms // Math. Sci. Res. J. – 2009. – 12. – P. 101 – 122.
13. Xu H. K. A regularization method for the proximal point algorithm // J. Global. Optim. – 2006. – 36. – P. 115 – 125.
14. Lehdili N., Moudafi A. Combining the proximal algorithm and Tikhonov method // Optimization. – 1996. – 37. –
P. 239 – 252.
15. Boikanyo O. A., Morosanu G. A proximal point algorithm converging strongly for general errors // Optim. Lett. –
2010. – 4. – P. 635 – 641.
16. Boikanyo O. A., Morosanu G. Inexact Halpern-type proximal point algorithm // J. Global Optim. – 2011. – 51. –
P. 11 – 26.
17. Tian C. A., Song Y. Strong convergence of a regularization method for Rockafellar’s proximal point algorithm // J.
Global Optim. / DOI 10.1007/s10898-011-9827-6.
18. Yao Y., Shahzad N. Strong convergence of a proximal point algorithm with general errors // Optim. Lett./DOI
10.1007/s11590-011-0286-2.
19. Song Y., Yang C. A note on a paper “A regularization method for the proximal point algorithm” // J. Global Optim. –
2009. – 43. – P. 171 – 174.
20. Wang F. A note on the regularized proximal point algorithm // J. Global Optim. – 2011. – 50. – P. 531 – 535.
21. Chai X., Li B., Song Y. Strong and weak convergence of the modified proximal point algorithms in Hilbert space //
Fixed Point Theory and Appl. – 2010. – 2010. – Article ID 240450. – 11 p.
22. Wang F., Huanhuan C. On the contraction-proximal point algorithms with multi-parameters // J. Global Optim./DOI
10.1007/s10898-011-9772-4.
23. Yao Y., Noor M. A. On convergence criteria of generalized proximal point algorithms // J. Comput. Appl. Math. –
2008. – 217. – P. 46 – 55.
24. Takahashi W. Nonlinear functional analysisfixed point theory and its applications. – Yokohama: Yokohama Publ.
Inc., 2000.
25. Marino G., Xu H. K. A general iterative method for nonexpansive mappings in Hilbert spaces // J. Math. Anal. and
Appl. – 2006. – 318. – P. 43 – 52.
26. Moudafi A. Viscosity approximation methods for fixed-point problems // J. Math. Anal. and Appl. – 2000. – 241. –
P. 46 – 55.
27. Chang S. S., Kim J. K., Wang X. R. Modified block iterative algorithm for solving convex feasibility problems in
Banach spaces // J. Inequal. Appl. – 2010. – 2010, № 14. – Article ID 869684.
28. Mainge P. E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimiza-
tion // Set-Valued Analysis. – 2008. – 16. – P. 899 – 912.
Received 06.07.13,
after revision — 11.08.16
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| id | umjimathkievua-article-1935 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:31Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9a/ec671124578e42d1f8c056842a31fe9a.pdf |
| spelling | umjimathkievua-article-19352019-12-05T09:32:19Z General proximal point algorithm for monotone operators Загальний алгоритм найближчої точки для монотонних операторiв Eslamian, M. Vahidi, J. Есламіан, М. Вахіді, Й. We introduce a new general proximal point algorithm for an infinite family of monotone operators in a real Hilbert space. We establish strong convergence of the iterative process to a common zero point of the infinite family of monotone operators. Our result generalizes and improves numerous results in the available literature. Введено новий загальний алгоритм найближчої точки для нескiнченної сiм’ї монотонних операторiв у дiйсному гiльбертовому просторi. Встановлено сильну збiжнiсть iтерацiйного процесу до спiльної нульової точки нескiнченної сiм’ї монотонних операторiв. Отриманий результат узагальнює та покращує численнi результати, що вiдомi з лiтературних джерел. Institute of Mathematics, NAS of Ukraine 2016-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1935 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 11 (2016); 1483-1492 Український математичний журнал; Том 68 № 11 (2016); 1483-1492 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1935/917 Copyright (c) 2016 Eslamian M.; Vahidi J. |
| spellingShingle | Eslamian, M. Vahidi, J. Есламіан, М. Вахіді, Й. General proximal point algorithm for monotone operators |
| title | General proximal point algorithm for monotone operators |
| title_alt | Загальний алгоритм найближчої точки для монотонних операторiв |
| title_full | General proximal point algorithm for monotone operators |
| title_fullStr | General proximal point algorithm for monotone operators |
| title_full_unstemmed | General proximal point algorithm for monotone operators |
| title_short | General proximal point algorithm for monotone operators |
| title_sort | general proximal point algorithm for monotone operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1935 |
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