1. Introduction
In the field of image processing and restoration, the simultaneous optimization of multiple objectives is often necessary to achieve desired results. For instance, when restoring historical paintings, balancing original brushstrokes while also removing degradation is crucial. To tackle this complex optimization problem, researchers have proposed various algorithms and techniques.
In what follows, we abstract the above issues into the following two types of problems and proceed to solve them. Let
and
be two positive numbers, and let
be a real Hilbert space with inner product
and associated norm
. We consider two classes of minimization problems
and
where each
is a lower semicontinuous, and proper and convex functions on
and
g is differentiable with the
-Lipschitz gradient. As described in [
1], the discussion can be conducted in a general setting where problems (
1) and (
2) are special instances of those that follow, respectively:
where
is the maximal monotone and
is firmly nonexpansive. The main purpose of this paper is to present algorithms to solve these inclusions and consider the convergence of the proposed methods.
Finding the zeros of an operator is a crucial task in operator theory [
1,
2]. A number of ideas have been proposed to address the issue of when the operator is decomposed into a sum of two. For instance, Ref. [
3] developed the forward–backward method to solve when one of the operator is the maximal monotone and the other is firmly nonexpansive. We remark that by this, we gain a solution (
3) in the case
. On the other hand, in the case of when each of the two operators is maximal monotone, the authors of [
4] presented the Douglas–Rachford method, while the authors of [
5] developed the splitting operator. Likewise, we remark that it happens to be the special case of problem (4) for the case of
Details of these two algorithms can be found in
Section 2.2.
It is rather difficult to extend the forward–backward method to address problem (
3) for a general
n. Nevertheless, in [
6], the authors achieved an algorithm. This proves to be covered by the four kinds of generalized forward–backward method given in this paper.
As for problem (4), extension variants of the Douglas–Rachford method have emerged. Most of them rely on reducing the original problem to the simple case of
by introducing auxiliary variables. In this paper, we point out that this problem can be solved by applying the generalized forward–backward method described above with
. Additionally, we prove that this method can also be explained as a reduction in the original problem for the case
. A benefit of this observation is that we can design splitting operators which allow us to solve (4) by the proximal point algorithm. The work of [
7] is also included in this paper. For recent improvements in this field, please refer to references [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24].
This paper is divided into five sections. The conception of the product space as well as some of the operators on it are introduced in
Section 2. To address the general problem, we will review in this section some classical methods for finding the zeros of a sum of two operators.
Section 3 is devoted to solving problem (
3) by deriving fixed-point equations. In this process, we obtain the so called generalized forward–backward splitting methods. In
Section 4, we solve problem (4) from two different perspectives. Firstly, we view it as an instance of (
3), so we obtain fixed-point equations as a consequence of the generalized forward–backward methods. Secondly, we point out by introducing new operators on the product space that this problem can be reduced to the case of
, which has been solved in
Section 2. Based on this, we can obtain fixed-point equations in another way and we can generate splitting operators with good properties. We provide in
Section 5 iterative schemes which produce convergent sequences, and these sequences eventually lead to solutions of either problem (
3) or (4).
4. Zeros of a Sum of n Maximal Monotone Operators
The main focus of this section is to present fixed-point equations as well as algorithms to solve problem (4).
4.1. An Application of the Generalized Forward–Backward Methods
Observe that it can be identified as a special instance of problem (
3) with
, so we can apply the methods given in Proposition 5 to obtain a solution. In this case, the operators
are simplified to
P, and
to
Q, respectively:
The following results are a direct consequence of Lemma 9 and Proposition 5.
Proposition 8. Let , then if and only if there exists such that is a fixed point of P or Q and .
Proposition 9. The operators P and Q are both firmly nonexpansive.
We also present some properties of the fixed points of the operators P and Q.
Lemma 10. Let be a fixed point of the operator P, then we have Proof. It suffices to prove that . By assumption, we have . So, we obtain that thus □
Lemma 11. Let be a fixed point of the operator Q, then we have Proof. It suffices to prove that . By assumption, we have . It can be induced from the linearity of the operator that , and thus, So, we obtain that Note that the first and second identities hold due two Proposition 5 and (ii) of Proposition 3, respectively. □
It seems that we have solved the problem perfectly. Nevertheless, we shall investigate what the intrinsic mode is. This research is conducted from a distinctive perspective.
4.2. Reformulation of Fixed-Point Equations
We shall begin with a crucial proposition which reveals that finding the zero of the sum of n maximal monotone operators can be reduced to the simple case of .
Proposition 10. Let then if and only if with .
Proof. We investigate problem (4) as follows:
Define
then we obtain an equivalent relation
□
Based on the above discussions, we can regain Proposition 8 by an alternative approach.
Proposition 11. Let , then if and only if there exists such that is a fixed point of P and .
Proof. Let
then by Lemma 7,
is equivalent to
Note that the second identity states that
, and thus, we conclude from Lemma 10 that
. □
Proposition 12. Let , then if and only if there exists such that is a fixed point of Q and .
Proof. Similar to the proof of the last proposition, let
then, by Lemma 7,
is equivalent to
The second identity states that . It follows from Lemma 11 that . And, we conclude from Lemma 10 that . □
Therefore, although the above fixed-point methods originate from applications of generalized forward–backward methods, they intrinsically rely on bringing back the original problem to the case of . As previously discussed, most methods can be explained as extensions of the Douglas–Rachford splitting. We remark that our work is an extension of algorithms described in Lemma 7.
4.3. Splitting Operators
In this subsection, we will design splitting operators for problem (4) with general
n. It is clear from Lemma 10 that we only have to investigate the next one:
Inspired by construction (
11), we can generate a splitting operator as
Note that here,
and
Write the identity
as
This, combined with identity (
53), implies that
Therefore, the operator in (
52) can be written equivalently as
With some changes in notations, we obtain the next splitting operator.
Definition 8. Let be any maximal monotone operator, and define Proposition 13. Let , then
Proof. By assumption, we have
A weighted sum on both sides of (58) leads to
Let
then, we claim that
Likewise, a weighted sum on both sides of (
57) implies that
. By (
59), we have completed the proof. □
Proposition 14. Let Q be given in (45), then . Proof. It is implied from the second identity of (
60) that
. Substituting this into the third identity of (
60), we obtain that
so
Thus, we have , which completes this proof. □
Proposition 15. is a maximal monotone operator.
Proof. Assume that
are arbitrary elements in
. That is,
Let
Then
Note that the last inequality holds due to the fact that and each operator is monotone. Thus, we have proved that is a monotone operator. By Proposition 1, to prove that it is also maximal, it remains to show that . This is quite easy since we have proved in Proposition 14 that is firmly nonexpansive. □
By exchanging the two operators
and
in (
52) and applying the same methods, we obtain a splitting operator
, which possesses properties similar to
.
Definition 9. Let be any maximal monotone operator, and then define Proposition 16. Let , then
Proposition 17. Let P be given in (45), then . Proposition 18. is a maximal monotone operator.
Recall that we have proved in Proposition 9 that are both firmly nonexpansive. In this subsection, we propose constructive approaches to obtain maximal monotone operators and such that
5. Iterative Algorithms and Convergence
The main focus of this section is to present iterative schemes which produce convergent sequences leading to a solution of problem (
3) or (4). Firstly, we recall three lemmas which can be used to find
, the set of fixed points of an averaged nonexpansive operator A.
Lemma 12. Let let A be a k-averaged nonexpansive operator on such that and let be a sequence in such that . Set , and define for ,then, converges weakly to a point in Fix(A). Under particular circumstances, we can use Picard iteration and obtain sequences which converge to a fixed point of A.
Lemma 13. Let and let C be a closed and convex set of . If is a k-averaged nonexpansive operator on such that set , and define for ,then converges weakly to a point in Fix(A). Lemma 14. Let A be a k-averaged nonexpansive operator on such that set , and define for ,then converges weakly to a point in Fix(A). Based on the above discussions, we present the following series of propositions. We remark that the corresponding iterative schemes lead to a solution of problem (
3) or (4). Therefore, we have developed efficient algorithms to solve the minimization problem (
1) and (
2).
Proposition 19. Suppose we have and Let be a sequence in such that . For each set , and define for ,then, the following hold: - (i)
For each there exists , such that .
- (ii)
For we have
- (iii)
For we have
Proof. This relation is due to Propositions 5 and 7 and Lemma 12. □
Similarly, Propositions 5 and 7 and Lemma 13 result in the next result.
Proposition 20. Suppose we have and For each set , and define for ,then, the following hold: - (i)
For each there exists , such that .
- (ii)
For we have
- (iii)
For we have
Proposition 21. Suppose we have and Set , and definethen, the following hold: - (i)
There exists , such that .
- (ii)
Proof. This relation follows from Propositions 8 and 9 and Lemma 14. □
6. Conclusions
In this paper, we have explored iterative schemes for solving problems (3) or (4) and introduced the generalized forward–backward splitting methods along with the introduction of new operators. By introducing new operators, we were able to simplify the problems to the case of n = 2 and obtain fixed-point equations and splitting operators with desirable properties in a different way.
Our research has shown that the generalized forward–backward splitting methods are highly beneficial. Through these methods, we can generate convergent sequences that eventually lead to the solution of the problems.
Additionally, the introduction of new operators has proved to be effective in simplifying the problems and enabling a better understanding and handling of complex situations. When the zeros of the sum of n operators are reduced to the zeros of just two operators, symmetry manifests in the potential alignment of properties and patterns across the operators. If the operators possess symmetry, such as invariance under certain transformations, the zeros of these operators may exhibit similar behaviors or locations. This symmetry allows us to generalize and simplify the problem by focusing on just two representative operators.