scholarly journals Convergence Analysis of an Inexact Three-Operator Splitting Algorithm

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 563 ◽  
Author(s):  
Chunxiang Zong ◽  
Yuchao Tang ◽  
Yeol Cho
Keyword(s):  
Hilbert Spaces ◽  
Operator Splitting ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Monotone Inclusion ◽  
Splitting Algorithm ◽  
Inclusion Problems ◽  
Convergence Results ◽  
The Resolvent Operator ◽  
Monotone Inclusion Problem

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature.

scholarly journals Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

2020 ◽  
Vol 10 (1) ◽  
pp. 450-476
Author(s):  
Radu Ioan Boţ ◽  
Sorin-Mihai Grad ◽  
Dennis Meier ◽  
Mathias Staudigl
Keyword(s):  
Dynamical Systems ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Minimum Norm ◽  
Monotone Inclusion ◽  
Inclusion Problems ◽  
Monotone Inclusion Problem ◽  
Maximally Monotone Operators

Abstract In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

scholarly journals An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lingling Huang ◽  
Sanyang Liu ◽  
Weifeng Gao
Keyword(s):  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Computational Results ◽  
Monotone Inclusion ◽  
Proximal Point ◽  
Inclusion Problems ◽  
Correction Step

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.

scholarly journals Iterative approximation of split feasibility problem in real Hilbert spaces

2021 ◽  
Vol 7 (2) ◽  
pp. 30-47
Author(s):  
J. N. Ezeora ◽  
◽  
F. E. Bazuaye
Keyword(s):  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Monotone Inclusion ◽  
Weak And Strong Convergence ◽  
Iterative Approximation ◽  
Asymptotically Nonexpansive ◽  
Split Feasibility ◽  
Monotone Inclusion Problem

In this paper, we propose an iterative algorithm for finding solution of split feasibility problem involving a λ−strictly pseudo-nonspreading map and asymptotically nonexpansive semigroups in two real Hilbert spaces. We prove weak and strong convergence theorems using the sequence obtained from the proposed algorithm. Finally, we applied our result to solve a monotone inclusion problem and present a numerical example to support our result.

scholarly journals Dynamical System Related to Primal–Dual Splitting Projection Methods

2021 ◽  
Author(s):  
E. M. Bednarczuk ◽  
R. N. Dhara ◽  
K. E. Rutkowski
Keyword(s):  
Dynamical System ◽  
Time Discretization ◽  
Monotone Operators ◽  
Projection Methods ◽  
Inclusion Problem ◽  
Monotone Inclusion ◽  
The Best Approximation ◽  
Primal Dual ◽  
Monotone Inclusion Problem ◽  
Maximally Monotone Operators

AbstractWe introduce a dynamical system to the problem of finding zeros of the sum of two maximally monotone operators. We investigate the existence, uniqueness and extendability of solutions to this dynamical system in a Hilbert space. We prove that the trajectories of the proposed dynamical system converge strongly to a primal–dual solution of the considered problem. Under explicit time discretization of the dynamical system we obtain the best approximation algorithm for solving coupled monotone inclusion problem.

scholarly journals Tikhonov Regularization Terms for Accelerating Inertial Mann-Like Algorithm with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 554
Author(s):  
Hasanen A. Hammad ◽  
Habib ur Rehman ◽  
Hassan Almusawa
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Fixed Points ◽  
Tikhonov Regularization ◽  
Hilbert Spaces ◽  
Variational Inequality Problem ◽  
Nonexpansive Mappings ◽  
Inclusion Problem ◽  
Monotone Inclusion ◽  
Monotone Inclusion Problem

In this manuscript, we accelerate the modified inertial Mann-like algorithm by involving Tikhonov regularization terms. Strong convergence for fixed points of nonexpansive mappings in real Hilbert spaces was discussed utilizing the proposed algorithm. Accordingly, the strong convergence of a forward–backward algorithm involving Tikhonov regularization terms was derived, which counts as finding a solution to the monotone inclusion problem and the variational inequality problem. Ultimately, some numerical discussions are presented here to illustrate the effectiveness of our algorithm.

Multivariate Monotone Inclusions in Saddle Form

2021 ◽  
Author(s):  
Minh N. Bùi ◽  
Patrick L. Combettes
Keyword(s):  
Large Scale ◽  
Operator Splitting ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Minimization Problems ◽  
Large Scale Problems ◽  
Novel Approach ◽  
Primal Dual ◽  
Monotone Inclusion Problem ◽  
Monotonicity Preserving

We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. Under investigation is a highly structured multivariate monotone inclusion problem involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. This model encompasses most formulations found in the literature. A limitation of existing primal-dual algorithms is that they operate in a product space that is too small to achieve full splitting of our problem in the sense that each operator is used individually. To circumvent this difficulty, we recast the problem as that of finding a zero of a saddle operator that acts on a bigger space. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them. The latter feature is of critical importance in large-scale problems. The weak convergence of the main algorithm is established, as well as the strong convergence of a variant. Various applications are discussed, and instantiations of the proposed framework in the context of variational inequalities and minimization problems are presented.

scholarly journals Modified Tseng’s Method with Inertial Viscosity Type for Solving Inclusion Problems and Its Application to Image Restoration Problems

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1104
Author(s):  
Nattakarn Kaewyong ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Image Restoration ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Monotone Inclusion ◽  
Inclusion Problems ◽  
Computational Performance ◽  
Convex Minimization Problems ◽  
Tseng’S Method ◽  
Monotone Inclusion Problem ◽  
Restoration Problem

In this paper, we study a monotone inclusion problem in the framework of Hilbert spaces. (1) We introduce a new modified Tseng’s method that combines inertial and viscosity techniques. Our aim is to obtain an algorithm with better performance that can be applied to a broader class of mappings. (2) We prove a strong convergence theorem to approximate a solution to the monotone inclusion problem under some mild conditions. (3) We present a modified version of the proposed iterative scheme for solving convex minimization problems. (4) We present numerical examples that satisfy the image restoration problem and illustrate our proposed algorithm’s computational performance.

scholarly journals Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 444
Author(s):  
Praveen. Agarwal ◽  
Doaa Filali ◽  
M. Akram ◽  
M. Dilshad
Keyword(s):  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Existence And Uniqueness Results ◽  
Variational Inclusion Problem ◽  
Uniqueness Results ◽  
Convergence Results ◽  
The Resolvent Operator

This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H(·,·)-compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.

scholarly journals An inertially constructed forward–backward splitting algorithm in Hilbert spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yasir Arfat ◽  
Poom Kumam ◽  
Muhammad Aqeel Ahmad Khan ◽  
Parinya Sa Ngiamsunthorn ◽  
Attapol Kaewkhao
Keyword(s):  
Hilbert Spaces ◽  
Fixed Point Problem ◽  
Projection Algorithm ◽  
Inclusion Problem ◽  
Point Problem ◽  
Splitting Algorithm ◽  
Common Solution ◽  
Monotone Inclusion Problem ◽  
Shrinking Projection Algorithm ◽  
Theoretical Results

AbstractIn this paper, we develop an iterative algorithm whose architecture comprises a modified version of the forward–backward splitting algorithm and the hybrid shrinking projection algorithm. We provide theoretical results concerning weak and strong convergence of the proposed algorithm towards a common solution of the fixed point problem associated to a finite family of demicontractive operators, the split equilibrium problem and the monotone inclusion problem in Hilbert spaces. Moreover, we compute a numerical experiment to show the efficiency of the proposed algorithm. As a consequence, our results improve various existing results in the current literature.

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