scholarly journals Modified Inertial Hybrid and Shrinking Projection Algorithms for Solving Fixed Point Problems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 236 ◽  
Author(s):  
Bing Tan ◽  
Shanshan Xu ◽  
Songxiao Li
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Location Theory ◽  
Fixed Point Problems ◽  
Projection Algorithm ◽  
Feasibility Problem ◽  
Projection Algorithms ◽  
Convex Feasibility ◽  
Mann Algorithm

In this paper, we introduce two modified inertial hybrid and shrinking projection algorithms for solving fixed point problems by combining the modified inertial Mann algorithm with the projection algorithm. We establish strong convergence theorems under certain suitable conditions. Finally, our algorithms are applied to convex feasibility problem, variational inequality problem, and location theory. The algorithms and results presented in this paper can summarize and unify corresponding results previously known in this field.

scholarly journals New Double Projection Algorithm for Solving Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Lian Zheng
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Projection Algorithm ◽  
Iterative Sequence ◽  
Local Error ◽  
Projection Algorithms ◽  
Local Error Bound ◽  
Globally Convergent

We propose a class of new double projection algorithms for solving variational inequality problem, which can be viewed as a framework of the method of Solodov and Svaiter by adopting a class of new hyperplanes. By the separation property of hyperplane, our method is proved to be globally convergent under very mild assumptions. In addition, we propose a modified version of our algorithm that finds a solution of variational inequality which is also a fixed point of a given nonexpansive mapping. If, in addition, a certain local error bound holds, we analyze the convergence rate of the iterative sequence. Numerical experiments prove that our algorithms are efficient.

CONVERGENCE ANALYSIS ON HYBRID PROJECTION ALGORITHMS FOR EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS

2009 ◽  
Vol 14 (3) ◽  
pp. 335-351 ◽  
Author(s):  
Xiaolong Qin ◽  
Yeol Je Cho ◽  
Shin Min Kang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Equilibrium Problems ◽  
Variational Inequality Problems ◽  
Fixed Point Problems ◽  
Projection Algorithms

In this paper, we consider an iterative method for equilibrium problems, fixed point problems and variational inequality problems in the framework of Banach space. The results presented in this paper improve and extend the corresponding results announced by many others.

scholarly journals Tseng Type Methods for Inclusion and Fixed Point Problems with Applications

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1175 ◽  
Author(s):  
Raweerote Suparatulatorn ◽  
Anchalee Khemphet
Keyword(s):  
Fixed Point ◽  
Numerical Experiments ◽  
Fixed Point Problems ◽  
Feasibility Problem ◽  
Signal Recovery ◽  
Viscosity Approximation ◽  
Inclusion Problems ◽  
Convex Feasibility ◽  
Previous Algorithm ◽  
Viscosity Approximation Methods

An algorithm is introduced to find an answer to both inclusion problems and fixed point problems. This algorithm is a modification of Tseng type methods inspired by Mann’s type iteration and viscosity approximation methods. On certain conditions, the iteration obtained from the algorithm converges strongly. Moreover, applications to the convex feasibility problem and the signal recovery in compressed sensing are considered. Especially, some numerical experiments of the algorithm are demonstrated. These results are compared to those of the previous algorithm.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

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