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 Strong Convergence Theorems for Maximal Monotone Operators, Fixed-Point Problems, and Equilibrium Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng ◽  
De-ning Qu
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
New Iterative Method

We present a new iterative method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces. We also apply our results to variational inequality and optimization problems.

scholarly journals Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
Author(s):  
Kriengsak Wattanawitoon ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Monotone Operators ◽  
Common Element ◽  
Proximal Point ◽  
Strong And Weak Convergence ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song (2008) and many authors.

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 Approximating Iterations for Nonexpansive and Maximal Monotone Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Zhangsong Yao ◽  
Sun Young Cho ◽  
Shin Min Kang ◽  
Li-Jun Zhu
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Minimum Norm ◽  
Nonexpansive Operator

We present two algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive operator in Hilbert spaces. We show that these two algorithms converge strongly to the minimum norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive operator.

scholarly journals A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-39 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

2021 ◽  
pp. 2250106
Author(s):  
A. A. Mebawondu ◽  
L. O. Jolaoso ◽  
H. A. Abass ◽  
O. K. Narain
Keyword(s):  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Inertial Method ◽  
Split Feasibility

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

scholarly journals Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1836
Author(s):  
Yaqian Jiang ◽  
Rudong Chen ◽  
Luoyi Shi
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Linear Convergence ◽  
Monotone Operators ◽  
Null Point ◽  
Point Problem ◽  
Bounded Linear ◽  
Zero Points

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results.

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