scholarly journals Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Genaro López ◽  
Victoria Martín-Márquez ◽  
Fenghui Wang ◽  
Hong-Kun Xu
Keyword(s):  
Banach Spaces ◽  
Nonlinear Problems ◽  
Split Feasibility Problem ◽  
Splitting Methods ◽  
Feasibility Problem ◽  
Nonlinear Operator Equation ◽  
Accretive Operators ◽  
Nonlinear Operators ◽  
Constrained Convex Minimization Problem ◽  
The Split Feasibility Problem

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.

An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces

2021 ◽  
pp. 2250120
Author(s):  
Yan Tang ◽  
Pongsakorn Sunthrayuth
Keyword(s):  
Banach Spaces ◽  
Split Feasibility Problem ◽  
Null Point ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Inertial Algorithm ◽  
The Split Feasibility Problem ◽  
Inertial Technique ◽  
Split Feasibility

In this work, we introduce a modified inertial algorithm for solving the split common null point problem without the prior knowledge of the operator norms in Banach spaces. The strong convergence theorem of our method is proved under suitable assumptions. We apply our result to the split feasibility problem, split equilibrium problem and split minimization problem. Finally, we provide some numerical experiments including compressed sensing to illustrate the performances of the proposed method. The result presented in this paper improves and generalizes many recent important results in the literature.

scholarly journals On Solving of Constrained Convex Minimize Problem Using Gradient Projection Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Taksaporn Sirirut ◽  
Pattanapong Tianchai
Keyword(s):  
Projection Method ◽  
Iterative Scheme ◽  
Bounded Operator ◽  
Split Feasibility Problem ◽  
Gradient Projection ◽  
Data Availability ◽  
Gradient Projection Method ◽  
Feasibility Problem ◽  
Constrained Convex Minimization Problem ◽  
The Split Feasibility Problem

Let C and Q be closed convex subsets of real Hilbert spaces H1 and H2, respectively, and let g:C→R be a strictly real-valued convex function such that the gradient ∇g is an 1/L-ism with a constant L>0. In this paper, we introduce an iterative scheme using the gradient projection method, based on Mann’s type approximation scheme for solving the constrained convex minimization problem (CCMP), that is, to find a minimizer q∈C of the function g over set C. As an application, it has been shown that the problem (CCMP) reduces to the split feasibility problem (SFP) which is to find q∈C such that Aq∈Q where A:H1→H2 is a linear bounded operator. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the CCMP and the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (2017) and many others. The data availability for the proposed SFP is shown and the example of this problem is also shown through numerical results.

scholarly journals Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 916
Author(s):  
Anchalee Sripattanet ◽  
Atid Kangtunyakarn
Keyword(s):  
Variational Inequalities ◽  
Convergence Theorem ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
System Of Variational Inequalities ◽  
Generalized System ◽  
Constrained Convex Minimization Problem ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.

A halpern-type iteration for solving the split feasibility problem and the fixed point problem of Bregman relatively nonexpansive semigroup in Banach spaces

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3211-3227 ◽  
Author(s):  
Prasit Cholamjiak ◽  
Pongsakorn Sunthrayuth
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Nonexpansive Semigroup ◽  
Relatively Nonexpansive ◽  
The Split Feasibility Problem ◽  
Split Feasibility

We study the split feasibility problem (SFP) involving the fixed point problems (FPP) in the framework of p-uniformly convex and uniformly smooth Banach spaces. We propose a Halpern-type iterative scheme for solving the solution of SFP and FPP of Bregman relatively nonexpansive semigroup. Then we prove its strong convergence theorem of the sequences generated by our iterative scheme under implemented conditions. We finally provide some numerical examples and demonstrate the efficiency of the proposed algorithm. The obtained result of this paper complements many recent results in this direction.

scholarly journals Strong Convergence of Modified Algorithms Based on the Regularization for the Constrained Convex Minimization Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ming Tian ◽  
Jun-Ying Gong
Keyword(s):  
Minimization Problem ◽  
Iterative Algorithms ◽  
Split Feasibility Problem ◽  
Convex Minimization ◽  
Feasibility Problem ◽  
Convex Minimization Problem ◽  
Constrained Convex Minimization ◽  
Constrained Convex Minimization Problem ◽  
The Split Feasibility Problem ◽  
Implicit And Explicit

As is known, the regularization method plays an important role in solving constrained convex minimization problems. Based on the idea of regularization, implicit and explicit iterative algorithms are proposed in this paper and the sequences generated by the algorithms can converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality. As an application, we also apply the algorithm to solve the split feasibility problem.

Sign in / Sign up

Export Citation Format

Share Document