scholarly journals On Bregman-Type Distances and Their Associated Projection Mappings

2021 ◽  
Author(s):  
Juan Enrique Martínez-Legaz ◽  
Maryam Tamadoni Jahromi ◽  
Eskandar Naraghirad
Keyword(s):  
Monotone Operators ◽  
Convergence Properties ◽  
Bregman Distances ◽  
Maximally Monotone Operators

AbstractWe investigate convergence properties of Bregman distances induced by convex representations of maximally monotone operators. We also introduce and study the projection mappings associated with such distances.

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.

An inexact algorithm with proximal distances for variational inequalities

2018 ◽  
Vol 52 (1) ◽  
pp. 159-176 ◽  
Author(s):  
E.A. Papa Quiroz ◽  
L. Mallma Ramirez ◽  
P.R. Oliveira
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Proximal Point Algorithm ◽  
Convergence Properties ◽  
Proximal Point ◽  
Extra Condition ◽  
Bregman Distances ◽  
Optim Theory ◽  
Inexact Proximal ◽  
Inexact Algorithm

In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generated by the algorithm is convergent for the pseudomonotone case and assuming an extra condition on the solution set we prove the convergence for the quasimonotone case. This approach unifies the results obtained by Auslender et al. [Math Oper. Res. 24 (1999) 644–688] and Brito et al. [J. Optim. Theory Appl. 154 (2012) 217–234] and extends the convergence properties for the class of φ-divergence distances and Bregman distances.

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 SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD

2020 ◽  
pp. 1-38
Author(s):  
SCOTT B. LINDSTROM ◽  
BRAILEY SIMS
Keyword(s):  
Normal Cone ◽  
Splitting Method ◽  
Monotone Operators ◽  
The Subject ◽  
Iterated Process ◽  
Theoretical Standpoint ◽  
Maximally Monotone Operators

The Douglas–Rachford method is a splitting method frequently employed for finding zeros of sums of maximally monotone operators. When the operators in question are normal cone operators, the iterated process may be used to solve feasibility problems of the following form: Find $x\in \bigcap _{k=1}^{N}S_{k}$ . The success of the method in the context of closed, convex, nonempty sets $S_{1},\ldots ,S_{N}$ is well known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less well understood, yet it is surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg and Thibault’s success in applying the method to solving sudoku puzzles, began an investigation of his own. We survey the current body of literature on the subject, and we summarize its history. We especially commemorate Professor Borwein’s celebrated contributions to the area.

scholarly journals Learning Maximally Monotone Operators for Image Recovery

2021 ◽  
Vol 14 (3) ◽  
pp. 1206-1237
Author(s):  
Jean-Christophe Pesquet ◽  
Audrey Repetti ◽  
Matthieu Terris ◽  
Yves Wiaux
Keyword(s):  
Monotone Operators ◽  
Image Recovery ◽  
Maximally Monotone Operators

Representative functions of maximally monotone operators and bifunctions

2016 ◽  
Vol 168 (1-2) ◽  
pp. 433-448
Author(s):  
Monica Bianchi ◽  
Nicolas Hadjisavvas ◽  
Rita Pini
Keyword(s):  
Monotone Operators ◽  
Maximally Monotone Operators
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