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 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 Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1548
Author(s):  
Yuanheng Wang ◽  
Mingyue Yuan ◽  
Bingnan Jiang
Keyword(s):  
Projection Method ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Inertial Method ◽  
Inclusion Problems ◽  
Variational Inclusion Problem ◽  
Convex Minimization Problems ◽  
Tseng’S Method

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems.

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 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 An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wanna Sriprad ◽  
Somnuk Srisawat
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Finite Family ◽  
Variational Inequality Problems ◽  
Variational Inclusion ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Convex Minimization Problems

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.

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.

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