scholarly journals The Hybrid Steepest Descent Method for Split Variational Inclusion and Constrained Convex Minimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Jitsupa Deepho ◽  
Poom Kumam
Keyword(s):  
Descent Method ◽  
Convex Minimization ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Constrained Convex Minimization ◽  
Hybrid Steepest Descent Method ◽  
Variational Inclusion Problem

We introduced an implicit and an explicit iteration method based on the hybrid steepest descent method for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem.

scholarly journals Adaptive hybrid steepest descent algorithms involving an inertial extrapolation term for split monotone variational inclusion problems

2021 ◽  
Author(s):  
Zheng Zhou ◽  
Bing Tan ◽  
Songxiao Li
Keyword(s):  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Variational Inclusion ◽  
Iterative Sequence ◽  
Inclusion Problem ◽  
Step Size ◽  
Hybrid Steepest Descent Method ◽  
Variational Inclusion Problem ◽  
Size Criterion

In this paper, we discuss the split monotone variational inclusion problem and propose two new inertial algorithms in infinite-dimensional Hilbert spaces. As well as, the iterative sequence by the proposed algorithms converges strongly to the solution of a certain variational inequality with the help of the hybrid steepest descent method. Furthermore, an adaptive step size criterion is considered in suggested algorithms to avoid the difficulty of calculating the operator norm. Finally, some numerical experiments show that our algorithms are realistic and summarize the known results.

scholarly journals 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

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
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.

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

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.

scholarly journals Hybrid Steepest Descent Viscosity Method for Triple Hierarchical Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
L.-C. Ceng ◽  
Q. H. Ansari ◽  
C.-F. Wen
Keyword(s):  
Approximate Solution ◽  
Variational Inequality Problem ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Viscosity Method ◽  
Hybrid Steepest Descent Method ◽  
Triple Hierarchical Variational Inequalities ◽  
Triple Hierarchical Variational Inequality ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (in short, THVIP). By combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP.

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