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 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 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 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.

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.

A new forward-backward penalty scheme and its convergence for solving monotone inclusion problems

2019 ◽  
Vol 35 (3) ◽  
pp. 349-363
Author(s):  
NATTHAPHON ARTSAWANG ◽  
◽  
KASAMSUK UNGCHITTRAKOOL ◽  
◽  
Keyword(s):  
Monotone Operator ◽  
Penalty Method ◽  
Normal Cone ◽  
Zero Point ◽  
Inclusion Problem ◽  
Maximally Monotone Operator ◽  
Monotone Inclusion ◽  
Inclusion Problems ◽  
J Coupling ◽  
Monotone Inclusion Problem

The purposes of this paper are to establish an alternative forward-backward method with penalization terms called new forward-backward penalty method (NFBP) and to investigate the convergence behavior of the new method via numerical experiment. It was proved that the proposed method (NFBP) converges in norm to a zero point of the monotone inclusion problem involving the sum of a maximally monotone operator and the normal cone of the set of zeros of another maximally monotone operator. Under the observation of some appropriate choices for the available properties of the considered functions and scalars, we can generate a suitable method that weakly ergodic converges to a solution of the monotone inclusion problem. Further, we also provide a numerical example to compare the new forward-backward penalty method with the algorithm introduced by Attouch [Attouch, H., Czarnecki, M.-O. and Peypouquet, J., Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities, SIAM J. Optim., 21 (2011), 1251-1274].

scholarly journals Strong convergence theorem for family of minimization and monotone inclusion problems in Hadamard spaces

2021 ◽  
Vol 40 (2) ◽  
pp. 525-559
Author(s):  
Chinedu Izuchukwu ◽  
Godwin C. Ugwunnadi ◽  
Oluwatosin Temitope Mewomo
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Minimization Problem ◽  
Fixed Point Problem ◽  
Finite Family ◽  
Inclusion Problem ◽  
Point Problem ◽  
Monotone Inclusion ◽  
Common Solution ◽  
Monotone Inclusion Problem

In this paper, we introduce a modified Ishikawa-type proximal point algorithm for approximating a common solution of minimization problem, monotone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of minimization problem, finite family of monotone inclusion problem and fixed point problem for asymptotically demicontractive mapping in Hadamard spaces. Numerical example is given to illustrate the applicability of our main result. Our results complement and extend some recent results in literature.

scholarly journals Common Solution for a Finite Family of Equilibrium Problems, Quasi-Variational Inclusion Problems and Fixed Points on Hadamard Manifolds

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1161
Author(s):  
Jinhua Zhu ◽  
Jinfang Tang ◽  
Shih-sen Chang ◽  
Min Liu ◽  
Liangcai Zhao
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Finite Family ◽  
Variational Inclusion ◽  
Point Problem ◽  
Hadamard Manifolds ◽  
Inclusion Problems ◽  
Common Solution

In this paper, we introduce an iterative algorithm for finding a common solution of a finite family of the equilibrium problems, quasi-variational inclusion problems and fixed point problem on Hadamard manifolds. Under suitable conditions, some strong convergence theorems are proved. Our results extend some recent results in literature.

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