General -monotone operators and perturbed iterations for nonlinear set-valued relaxed cocoercive operator inclusion problems

2009 ◽  
Vol 215 (4) ◽  
pp. 1583-1592 ◽  
Author(s):  
Heng-you Lan ◽  
Le-cai Cai ◽  
Zi-shan Liu
Keyword(s):  
Monotone Operators ◽  
Operator Inclusion ◽  
Inclusion Problems ◽  
Cocoercive Operator

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 Approximation solvability for a system of implicit nonlinear variational inclusions with Η-monotone operators

2018 ◽  
Vol 51 (1) ◽  
pp. 241-254
Author(s):  
Jong Kyu Kim ◽  
Muhammad Iqbal Bhat
Keyword(s):  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Monotone Operators ◽  
Inner Product ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Inclusion Problems ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator ◽  
New System

AbstractIn this paper, we introduce and study a new system of variational inclusions which is called a system of nonlinear implicit variational inclusion problems with A-monotone and H-monotone operators in semi-inner product spaces. We define the resolvent operator associated with A-monotone and H-monotone operators and prove its Lipschitz continuity. Using resolvent operator technique, we prove the existence and uniqueness of solution for this new system of variational inclusions. Moreover, we suggest an iterative algorithm for approximating the solution of this system and discuss the convergence analysis of the sequences generated by the iterative algorithm under some suitable conditions.

scholarly journals An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lingling Huang ◽  
Sanyang Liu ◽  
Weifeng Gao
Keyword(s):  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Computational Results ◽  
Monotone Inclusion ◽  
Proximal Point ◽  
Inclusion Problems ◽  
Correction Step

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.

scholarly journals Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Inclusion Problems ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

scholarly journals Existence and Stability of Iterative Algorithm for a System of Random Set-Valued Variational Inclusion Problems Involving (A,m,η)-Generalized Monotone Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Jittiporn Suwannawit ◽  
Narin Petrot
Keyword(s):  
Iterative Algorithm ◽  
Monotone Operator ◽  
Existence Of Solutions ◽  
Monotone Operators ◽  
Variational Inclusion ◽  
Random Set ◽  
Inclusion Problems ◽  
Existence And Stability ◽  
The Stability

We introduce and study a class of a system of random set-valued variational inclusion problems. Some conditions for the existence of solutions of such problems are provided, when the operators are contained in the classes of generalized monotone operators, so-called (A,m,η)-monotone operator. Further, the stability of the iterative algorithm for finding a solution of the considered problem is also discussed.

scholarly journals Existence of solution and iterative approximation of a system of generalized variational-like inclusion problems in semi-inner product spaces

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6051-6070
Author(s):  
Mohd Bhat ◽  
Bisma Zahoor
Keyword(s):  
Iterative Algorithm ◽  
Monotone Operators ◽  
Existence Of Solution ◽  
Inner Product ◽  
Operator Technique ◽  
Product Spaces ◽  
Iterative Approximation ◽  
Generalized Resolvent ◽  
Inclusion Problems ◽  
Inner Product Spaces

In this paper, we consider the system of generalized variational-like inclusion problems in semi-inner product spaces. We define a class of (H,?)-?-monotone operators and its associated class of generalized resolvent operators. Further, using generalized resolvent operator technique, we give the existence of solution of the generalized variational-like inclusion problems. Furthermore, we suggest an iterative algorithm and give the convergence analysis of the sequences generated by the iterative algorithm. The results presented in this paper extend and unify the related known results in the literature.

scholarly journals Split variational inclusions for Bregman multivalued maximal monotone operators

2020 ◽  
Author(s):  
Mujahid Abbas ◽  
Faik Gürsoy ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan
Keyword(s):  
Strong Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Strong Convergence Theorem ◽  
Variational Inclusions ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

scholarly journals Generalized Yosida Approximations Based on RelativelyA-Maximalm-Relaxed Monotonicity Frameworks

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Heng-you Lan
Keyword(s):  
Banach Space ◽  
Resolvent Operator ◽  
Evolution Equations ◽  
Maximal Monotone ◽  
Monotone Operators ◽  
Monotone Mappings ◽  
Evolution Inclusions ◽  
Inclusion Problems ◽  
First Order ◽  
New Class

We introduce and study a new notion of relativelyA-maximalm-relaxed monotonicity framework and discuss some properties of a new class of generalized relatively resolvent operator associated with the relativelyA-maximalm-relaxed monotone operator and the new generalized Yosida approximations based on relativelyA-maximalm-relaxed monotonicity framework. Furthermore, we give some remarks to show that the theory of the new generalized relatively resolvent operator and Yosida approximations associated with relativelyA-maximalm-relaxed monotone operators generalizes most of the existing notions on (relatively) maximal monotone mappings in Hilbert as well as Banach space and can be applied to study variational inclusion problems and first-order evolution equations as well as evolution inclusions.

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.

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