scholarly journals The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Haiwen Xu
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Hybrid Steepest Descent Method ◽  
Descent Directions

We establish the strong convergence of prediction-correction and relaxed hybrid steepest-descent method (PRH method) for variational inequalities under some suitable conditions that simplify the proof. And it is to be noted that the proof is different from the previous results and also is not similar to the previous results. More importantly, we design a set of practical numerical experiments. The results demonstrate that the PRH method under some descent directions is more slightly efficient than that of the modified and relaxed hybrid steepest-descent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.

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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