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 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.

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 Primal-Dual Splitting Algorithms for Solving Structured Monotone Inclusion with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2415
Author(s):  
Jinjian Chen ◽  
Xingyu Luo ◽  
Yuchao Tang ◽  
Qiaoli Dong
Keyword(s):  
Numerical Experiments ◽  
Monotone Operators ◽  
Simple Calculation ◽  
Monotone Inclusion ◽  
Splitting Algorithm ◽  
Splitting Algorithms ◽  
Parallel Sum ◽  
Primal Dual ◽  
Cocoercive Operator ◽  
Maximally Monotone Operators

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward–backward splitting algorithm, and the other is the forward–backward–half-forward splitting algorithm. Both algorithms have a simple calculation framework. In particular, the single-valued operators are processed via explicit steps, while the set-valued operators are computed by their resolvents. Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms.

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.

A viscosity-type algorithm for an infinitely countable family of (f,g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces

Analysis ◽  
2020 ◽  
Vol 40 (1) ◽  
pp. 19-37 ◽  
Author(s):  
Kazeem O. Aremu ◽  
Hammed Abass ◽  
Chinedu Izuchukwu ◽  
Oluwatosin T. Mewomo
Keyword(s):  
Integral Equation ◽  
Fixed Point Problem ◽  
Countable Family ◽  
Inclusion Problem ◽  
Point Problem ◽  
Monotone Inclusion ◽  
Nonlinear Volterra Integral Equation ◽  
Common Solution ◽  
Type Algorithm ◽  
Monotone Inclusion Problem

AbstractIn this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of {(f,g)}-generalized k-strictly pseudononspreading mappings in a {\mathrm{CAT}(0)} space. We obtain a strong convergence of the proposed algorithm to the aforementioned problems in a complete {\mathrm{CAT}(0)} space. Furthermore, we give an application of our result to a nonlinear Volterra integral equation and a numerical example to support our main result. Our results complement and extend many recent results in literature.

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 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.

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