Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

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.

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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.

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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.

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

Abstract and Applied Analysis ◽

10.1155/2014/208717 ◽

2014 ◽

Vol 2014 ◽

pp. 1-22 ◽

Cited By ~ 1

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.

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Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

Abstract and Applied Analysis ◽

10.1155/2014/264965 ◽

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.

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Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros ofm-Accretive Operators in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2013/852760 ◽

2013 ◽

Vol 2013 ◽

pp. 1-21

Author(s):

Lu-Chuan Ceng ◽

Ching-Feng Wen

Keyword(s):

Variational Inequalities ◽

Extragradient Method ◽

General System ◽

Accretive Operator ◽

Descent Method ◽

Steepest Descent ◽

Steepest Descent Method ◽

Descent Methods ◽

Common Element ◽

Hybrid Steepest Descent Method

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of anm-accretive operatorAin the setting of real uniformly convex and 2-uniformly smooth Banach spaceX. Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the setA-1(0)of zeros ofAinX. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Journal of Applied Mathematics ◽

10.1155/2012/282094 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17

Author(s):

Rabian Wangkeeree ◽

Panatda Boonman

Keyword(s):

Hilbert Space ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Solution Set ◽

Common Element ◽

Fixed Point Set ◽

Mixed Equilibrium Problems ◽

Point Set ◽

Mixed Equilibrium

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptoticTJmapping and the solution set of SMEP in a real Hilbert space.

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A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

Abstract and Applied Analysis ◽

10.1155/2013/204948 ◽

2013 ◽

Vol 2013 ◽

pp. 1-24 ◽

Cited By ~ 1

Author(s):

Haitao Che ◽

Xintian Pan

Keyword(s):

Variational Inequality ◽

Equilibrium Problems ◽

Infinite Family ◽

Descent Method ◽

Steepest Descent ◽

Steepest Descent Method ◽

Common Element ◽

Hybrid Steepest Descent Method ◽

Pseudocontractive Mappings ◽

Strictly Pseudocontractive Mappings

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

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Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

Journal of Applied Mathematics ◽

10.1155/2014/129379 ◽

2014 ◽

Vol 2014 ◽

pp. 1-25

Author(s):

Lu-Chuan Ceng ◽

Cheng-Wen Liao ◽

Chin-Tzong Pang ◽

Ching-Feng Wen

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Variational Inequalities ◽

Convex Minimization ◽

Solution Set ◽

Projection Algorithm ◽

Common Element ◽

Point Problem ◽

Hierarchical Fixed Point ◽

Mixed Equilibria

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and 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 infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), 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 solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

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Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium Problems, a General System of Variational Inequalities and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

Journal of Applied Mathematics ◽

10.1155/2012/572326 ◽

2012 ◽

Vol 2012 ◽

pp. 1-18

Author(s):

Qiao-Li Dong ◽

Yan-Ni Guo ◽

Fang Su

Keyword(s):

Iterative Method ◽

Variational Inequalities ◽

Equilibrium Problems ◽

Nonexpansive Mappings ◽

Extragradient Method ◽

General System ◽

Countable Family ◽

Common Element ◽

Strong Convergence Theorem ◽

System Of Variational Inequalities

Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others.

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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.

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