scholarly journals Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

2014 ◽  
Vol 2014 ◽  
pp. 1-26 ◽  
Author(s):  
A. E. Al-Mazrooei ◽  
A. Latif ◽  
J. C. Yao
Keyword(s):  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Generalized Mixed Equilibrium ◽  
Fréchet Differentiable ◽  
Implicit And Explicit ◽  
Mixed Equilibria

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.

scholarly journals Algorithms of Common Solutions for Generalized Mixed Equilibria, Variational Inclusions, and Constrained Convex Minimization

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Suliman Al-Homidan
Keyword(s):  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Common Element ◽  
Variational Inclusions ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Fréchet Differentiable ◽  
Implicit And Explicit ◽  
Mixed Equilibria

We introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.

scholarly journals Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-27 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Juei-Ling Ho
Keyword(s):  
Real Hilbert Space ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
Common Element ◽  
Monotone Mappings ◽  
Strong And Weak Convergence ◽  
Generalized Mixed Equilibrium ◽  
Hybrid Extragradient Method ◽  
Mixed Equilibrium ◽  
Fréchet Differentiable

We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

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 Generalized Mixed Equilibria, Variational Inclusions, and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
A. E. Al-Mazrooei ◽  
A. S. M. Alofi ◽  
A. Latif ◽  
J.-C. Yao
Keyword(s):  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Common Fixed Points ◽  
Common Element ◽  
Monotone Mappings ◽  
Variational Inclusions ◽  
Strong And Weak Convergence ◽  
Mixed Equilibria

We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.

scholarly journals Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 270 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Mihai Postolache ◽  
Ching-Feng Wen ◽  
Yonghong Yao
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Variational Inclusions ◽  
Minimization Problems ◽  
Convergence Results ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Implicit And Explicit ◽  
Mixed Equilibria

Multistep composite implicit and explicit extragradient-like schemes are presented for solving the minimization problem with the constraints of variational inclusions and generalized mixed equilibrium problems. Strong convergence results of introduced schemes are given under suitable control conditions.

scholarly journals Modified subgradient extragradient method for system of variational inclusion problem and finite family of variational inequalities problem in real Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Araya Kheawborisut ◽  
Atid Kangtunyakarn
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Finite Family ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Subgradient Extragradient Method ◽  
Generalized System

AbstractFor the purpose of this article, we introduce a modified form of a generalized system of variational inclusions, called the generalized system of modified variational inclusion problems (GSMVIP). This problem reduces to the classical variational inclusion and variational inequalities problems. Motivated by several recent results related to the subgradient extragradient method, we propose a new subgradient extragradient method for finding a common element of the set of solutions of GSMVIP and the set of a finite family of variational inequalities problems. Under suitable assumptions, strong convergence theorems have been proved in the framework of a Hilbert space. In addition, some numerical results indicate that the proposed method is effective.

scholarly journals Relaxed Iterative Algorithms for Generalized Mixed Equilibrium Problems with Constraints of Variational Inequalities and Variational Inclusions

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Chi-Ming Chen ◽  
Ching-Feng Wen ◽  
Chin-Tzong Pang
Keyword(s):  
Variational Inequalities ◽  
Finite Family ◽  
Generalized Mixed Equilibrium Problem ◽  
Monotone Mappings ◽  
Variational Inclusions ◽  
Strongly Monotone ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Strongly Monotone Mappings ◽  
Mixed Equilibrium

We introduce and analyze a relaxed extragradient-like viscosity iterative algorithm for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems which also solves a variational inequality problem.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

scholarly journals Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
S. Imnang
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Real Hilbert Space ◽  
General System ◽  
Projection Algorithm ◽  
Common Element ◽  
System Of Variational Inequalities ◽  
Demiclosedness Principle ◽  
Hybrid Projection Algorithm

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

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