Douglas–Rachford splitting algorithm for solving state-dependent maximal monotone inclusions

2021 ◽  
Author(s):  
S. Adly ◽  
B. K. Le
Keyword(s):  
Maximal Monotone ◽  
Splitting Algorithm ◽  
Monotone Inclusions ◽  
State Dependent

scholarly journals Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1395
Author(s):  
Charles Castaing ◽  
Christiane Godet-Thobie ◽  
Le Xuan Truong
Keyword(s):  
Fractional Order ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Separable Hilbert Space ◽  
Maximal Monotone ◽  
Fractional Flow ◽  
Order Problem ◽  
Absolutely Continuous ◽  
State Dependent ◽  
Fractional Differential

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

Inertial viscosity forward–backward splitting algorithm for monotone inclusions and its application to image restoration problems

2019 ◽  
Vol 97 (1-2) ◽  
pp. 482-497 ◽  
Author(s):  
Duangkamon Kitkuan ◽  
Poom Kumam ◽  
Juan Martínez-Moreno ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Image Restoration ◽  
Splitting Algorithm ◽  
Monotone Inclusions

A weakly convergent fully inexact Douglas-Rachford method with relative error tolerance

2019 ◽  
Vol 25 ◽  
pp. 57
Author(s):  
Benar Fux Svaiter
Keyword(s):  
Relative Error ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Error Tolerance ◽  
Inclusion Problem ◽  
Splitting Algorithm ◽  
Sequential Solution

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Each of its iterations requires the sequential solution of two proximal subproblems. The aim of this work is to present a fully inexact version of Douglas-Rachford method wherein both proximal subproblems are solved approximately within a relative error tolerance. We also present a semi-inexact variant in which the first subproblem is solved exactly and the second one inexactly. We prove that both methods generate sequences weakly convergent to the solution of the underlying inclusion problem, if any.

A novel adaptive method for operator inclusions

2021 ◽  
pp. 33-35
Author(s):  
Vladimir V. Semenov ◽  
Serhii Denysov ◽  
Yana Vedel
Keyword(s):  
Banach Space ◽  
Lipschitz Constant ◽  
Maximal Monotone ◽  
Continuous Operator ◽  
Adaptive Method ◽  
Step Size ◽  
Splitting Algorithm ◽  
Operator Inclusion ◽  
Lipschitz Continuous ◽  
Lipschitz Continuous Operator

A novel splitting algorithm for solving operator inclusion with the sum of the maximal monotone operator and the monotone Lipschitz continuous operator in the Banach space is proposed and studied. The proposed algorithm is an adaptive variant of the forward-reflected-backward algorithm, where the rule used to update the step size does not require knowledge of the Lipschitz constant of the operator. For operator inclusions in 2-uniformly convex and uniformly smooth Banach space, the theorem on the weak convergence of the method is proved.

scholarly journals A stochastic inertial forward–backward splitting algorithm for multivariate monotone inclusions

Optimization ◽  
2016 ◽  
Vol 65 (6) ◽  
pp. 1293-1314 ◽  
Author(s):  
Lorenzo Rosasco ◽  
Silvia Villa ◽  
Băng Công Vũ
Keyword(s):  
Splitting Algorithm ◽  
Monotone Inclusions

scholarly journals On a second-order functional evolution problem with time and state dependent maximal monotone operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Soumia Saïdi
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Uniqueness Result ◽  
Second Order ◽  
Absolutely Continuous ◽  
Lipschitz Continuous ◽  
State Dependent ◽  
Real Separable Hilbert Space ◽  
Order State

<p style='text-indent:20px;'>The present paper proposes, in a real separable Hilbert space, to analyze the existence of solutions for a class of perturbed second-order state-dependent maximal monotone operators with a finite delay. The dependence of the operators is -in some sense- absolutely continuous (or bounded continuous) variation in time, and Lipschitz continuous in state. The approach to solve our problem is based on a discretization scheme. The uniqueness result is applied to optimal control.</p>

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