scholarly journals Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

2015 ◽  
Vol 46 (4) ◽  
pp. 453-503
Author(s):  
Lu-Chuan Ceng
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Steepest Descent Method ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Special Cases ◽  
Mixed Equilibrium ◽  
Split Feasibility

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

scholarly journals Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petrusel ◽  
Mu-Ming Wong ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Finite Family ◽  
Steepest Descent Method ◽  
Common Element ◽  
Monotone Mappings ◽  
Mixed Equilibria

We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method.

scholarly journals Iterative Methods for Finding Solutions of a Class of Split Feasibility Problems over Fixed Point Sets in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1012
Author(s):  
Suthep Suantai ◽  
Narin Petrot ◽  
Montira Suwannaprapa
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Split Feasibility Problem ◽  
Inverse Image ◽  
Feasibility Problem ◽  
Point Sets ◽  
Fixed Point Set ◽  
Split Feasibility ◽  
Fixed Point Sets

We consider the split feasibility problem in Hilbert spaces when the hard constraint is common solutions of zeros of the sum of monotone operators and fixed point sets of a finite family of nonexpansive mappings, while the soft constraint is the inverse image of a fixed point set of a nonexpansive mapping. We introduce iterative algorithms for the weak and strong convergence theorems of the constructed sequences. Some numerical experiments of the introduced algorithm are also discussed.

scholarly journals Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

2015 ◽  
Vol 2015 ◽  
pp. 1-22
Author(s):  
L. C. Ceng ◽  
A. Latif ◽  
C. F. Wen ◽  
A. E. Al-Mazrooei
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Steepest Descent ◽  
Solution Set ◽  
Steepest Descent Method ◽  
Common Element ◽  
Variational Inclusions

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.

scholarly journals New Iterative Algorithm for Solving Constrained Convex Minimization Problem and Split Feasibility Problem

2021 ◽  
Vol 1 (2) ◽  
pp. 106-132
Author(s):  
Austine Efut Ofem ◽  
Unwana Effiong Udofia ◽  
Donatus Ikechi Igbokwe
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Almost Contraction ◽  
Contraction Mappings ◽  
Constrained Convex Minimization Problem ◽  
Split Feasibility

The purpose of this paper is to introduce a new iterative algorithm to approximate the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Also, we show that our proposed iterative algorithm converges weakly and strongly to the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Furthermore, it is proved analytically that our new iterative algorithm converges faster than one of the leading iterative algorithms in the literature for almost contraction mappings. Some numerical examples are also provided and used to show that our new iterative algorithm has better rate of convergence than all of S, Picard-S, Thakur and M iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings. Again, we show that the proposed iterative algorithm is stable with respect to T and data dependent for almost contraction mappings. Some applications of our main results and new iterative algorithm are considered. The results in this article are improvements, generalizations and extensions of several relevant results existing in the literature.

scholarly journals Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-18
Author(s):  
Lu-Chuan Ceng ◽  
Saleh A. Al-Mezel ◽  
Abdul Latif
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Steepest Descent Method ◽  
Point Problem ◽  
Hierarchical Fixed Point Problem ◽  
Hierarchical Fixed Point

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings inX, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

scholarly journals Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Projection Algorithm ◽  
Fixed Point Set

We introduce and analyze a hybrid steepest-descent algorithm by combining Korpelevich’s extragradient method, the steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to the unique solution of a triple hierarchical constrained optimization problem (THCOP) over the common fixed point set of finitely many nonexpansive mappings, with constraints of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and a convex minimization problem (CMP) in a real Hilbert space.

scholarly journals The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Himanshu Gupta ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
General System ◽  
Solution Set ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Split Feasibility

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution setΞof a general system of variational inequalities, the solution setΓof a split feasibility problem, and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of the Hilbert spaces. These iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge weakly to an element ofFix(S)∩Ξ∩Γunder mild conditions.

Iterative algorithm for a split equilibrium problem and fixed problem for finite asymptotically nonexpansive mappings in Hlbert space

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1423-1434 ◽  
Author(s):  
Sheng Wang ◽  
Min Chen
Keyword(s):  
Equilibrium Problem ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Fixed Point Set ◽  
Split Equilibrium Problem ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Point Set ◽  
The Common

In this paper, we propose an iterative algorithm for finding the common element of solution set of a split equilibrium problem and common fixed point set of a finite family of asymptotically nonexpansive mappings in Hilbert space. The strong convergence of this algorithm is proved.

scholarly journals An iterative algorithm for the system of split mixed equilibrium problem

2020 ◽  
Vol 53 (1) ◽  
pp. 309-324
Author(s):  
Ibrahim Karahan ◽  
Lateef Olakunle Jolaoso
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Equilibrium System ◽  
Convergence Conditions ◽  
Mixed Equilibrium Problems ◽  
Split Variational Inequality ◽  
Mixed Equilibrium ◽  
Split Mixed Equilibrium Problem

AbstractIn this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

scholarly journals Some Proximal Methods for Solving Mixed Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Muhammad Aslam Noor ◽  
Zhenyu Huang
Keyword(s):  
Variational Inequalities ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Proximal Methods ◽  
Proximal Point ◽  
Equivalent Formulation ◽  
New Methods ◽  
Special Cases ◽  
Mixed Variational Inequalities

It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest some new proximal point methods for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered under some suitable conditions. Our method of constructing these iterative methods is very simple. Results proved in this paper may stimulate further research in this direction.

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