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

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.

Download Full-text

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

Abstract and Applied Analysis ◽

10.1155/2014/587865 ◽

2014 ◽

Vol 2014 ◽

pp. 1-26 ◽

Cited By ~ 1

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.

Download Full-text

Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

Journal of Function Spaces ◽

10.1155/2015/980352 ◽

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.

Download Full-text

On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions and systems of generalized equilibria

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.45.2014.1658 ◽

2014 ◽

Vol 45 (3) ◽

pp. 297-334 ◽

Cited By ~ 5

Author(s):

Jen-Chih Yao ◽

Cheng Lu-chuan

Keyword(s):

Variational Inequality ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Extragradient Method ◽

Solution Set ◽

Steepest Descent Method ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Variational Inclusions ◽

Mixed Equilibria

In this paper, we introduce and analyze a relaxed iterative algorithm by combining Korpelevich's extragradient method, hybrid steepest-descent method and Mann's iteration method. It is proven 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 a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions and the solution set of a system of generalized equilibrium problems (SGEP), 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 to solving a hierarchical variational inequality problem with constraints of the GMEP, the SGEP and finitely many variational inclusions.

Download Full-text

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

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02583-1 ◽

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.

Download Full-text

Generalized Mixed Equilibria, Variational Inclusions, and Fixed Point Problems

Abstract and Applied Analysis ◽

10.1155/2014/251065 ◽

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.

Download Full-text

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

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.46.2015.1917 ◽

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.

Download Full-text

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

Abstract and Applied Analysis ◽

10.1155/2014/132053 ◽

2014 ◽

Vol 2014 ◽

pp. 1-25 ◽

Cited By ~ 4

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.

Download Full-text

Hybrid Iterative Scheme for Triple Hierarchical Variational Inequalities with Mixed Equilibrium, Variational Inclusion, and Minimization Constraints

Abstract and Applied Analysis ◽

10.1155/2014/767109 ◽

2014 ◽

Vol 2014 ◽

pp. 1-22

Author(s):

Lu-Chuan Ceng ◽

Cheng-Wen Liao ◽

Chin-Tzong Pang ◽

Ching-Feng Wen

Keyword(s):

Variational Inequality ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Solution Set ◽

Steepest Descent Method ◽

Projection Algorithm ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Variational Inclusions ◽

Mixed Equilibrium

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid 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 a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also 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 to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

Download Full-text

An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2013-143 ◽

2013 ◽

Vol 2013 (1) ◽

Cited By ~ 1

Author(s):

Atid Kangtunyakarn

Keyword(s):

Variational Inequalities ◽

Fixed Points ◽

Iterative Algorithm ◽

Common Fixed Points ◽

Finite Family ◽

Common Element ◽

System Of Variational Inequalities

Download Full-text

Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2014/271208 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17

Author(s):

Poom Kumam ◽

Thanyarat Jitpeera

Keyword(s):

Iterative Algorithm ◽

Equilibrium Problems ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Variational Inclusion ◽

Common Element ◽

Point Problem ◽

Mixed Equilibrium Problems ◽

Mixed Equilibrium

We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.

Download Full-text