scholarly journals An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Minh N. Dao ◽  
Hung M. Phan
Keyword(s):  
Monotone Operators ◽  
The Other ◽  
Generalized Monotonicity ◽  
Splitting Algorithm ◽  
Splitting Algorithms ◽  
Cocoercive Operator

AbstractSplitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

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.

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.

ABOUT THE SPLITTING ALGORITHM FOR BOLTZMANN AND B.G.K. EQUATIONS

1996 ◽  
Vol 06 (08) ◽  
pp. 1079-1101 ◽  
Author(s):  
L. DESVILLETTES ◽  
S. MISCHLER
Keyword(s):  
Splitting Algorithm ◽  
Splitting Algorithms

We prove the convergence of splitting algorithms for transport and collision operators for Boltzmann and B.G.K. equations.

scholarly journals H(⋅,⋅)-Cocoercive Operator and an Application for Solving Generalized Variational Inclusions

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Rais Ahmad ◽  
Mohd Dilshad ◽  
Mu-Ming Wong ◽  
Jen-Chin Yao
Keyword(s):  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Lipschitz Continuity ◽  
Monotone Operators ◽  
Variational Inclusions ◽  
The Resolvent Operator ◽  
Cocoercive Operator

The purpose of this paper is to introduce a newH(⋅,⋅)-cocoercive operator, which generalizes many existing monotone operators. The resolvent operator associated withH(⋅,⋅)-cocoercive operator is defined, and its Lipschitz continuity is presented. By using techniques of resolvent operator, a new iterative algorithm for solving generalized variational inclusions is constructed. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. For illustration, some examples are given.

scholarly journals The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Oganeditse Aaron Boikanyo
Keyword(s):  
Real Hilbert Space ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Viscosity Approximation ◽  
Convergence Results ◽  
Bounded Below ◽  
Inexact Algorithm ◽  
Cocoercive Operator

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operatorAand maximal monotone operatorsBwithD(B)⊂H:xn+1=αnf(xn)+γnxn+δn(I+rnB)-1(I-rnA)xn+en, forn=1,2,…,for givenx1in a real Hilbert spaceH, where(αn),(γn), and(δn)are sequences in(0,1)withαn+γn+δn=1for alln≥1,(en)denotes the error sequence, andf:H→His a contraction. The algorithm is known to converge under the following assumptions onδnanden: (i)(δn)is bounded below away from 0 and above away from 1 and (ii)(en)is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i)(δn)is bounded below away from 0 and above away from 3/2 and (ii)(en)is square summable in norm; and we still obtain strong convergence results.

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.

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