scholarly journals On Self-Adaptive Method for General Mixed Variational Inequalities

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Abdellah Bnouhachem ◽  
Muhammad Aslam Noor ◽  
Eman H. Al-Shemas
Keyword(s):  
Global Convergence ◽  
Variational Inequalities ◽  
Complementarity Problems ◽  
Adaptive Method ◽  
Computational Results ◽  
Quasivariational Inequalities ◽  
Special Cases ◽  
General Variational Inequalities ◽  
Mixed Variational Inequalities ◽  
Self Adaptive

We suggest and analyze a new self-adaptive method for solving general mixed variational inequalities, which can be viewed as an improvement of the method of (Noor 2003). Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given to illustrate the efficiency of the proposed method. Since the general mixed variational inequalities include general variational inequalities, quasivariational inequalities, and nonlinear (implicit) complementarity problems as special cases, results proved in this paper continue to hold for these problems.

scholarly journals Parametric Extended General Mixed Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Complementarity Problems ◽  
Equivalent Formulation ◽  
Resolvent Equations ◽  
Special Cases ◽  
General Variational Inequalities ◽  
Significant Extension ◽  
Mixed Variational Inequalities ◽  
The Given

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

scholarly journals Iterative resolvent methods for general mixed variational inequalities

2003 ◽  
Vol 16 (3) ◽  
pp. 283-294
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Complementarity Problems ◽  
Splitting Methods ◽  
Pseudomonotone Operators ◽  
Special Cases ◽  
Mixed Variational Inequalities ◽  
Proof Of Convergence ◽  
Self Adaptive

In this paper, we use the technique of updating the solution to suggest and analyze a class of new self-adaptive splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Proof of convergence is very simple. Since general mixed variational include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.

ON A NEW NUMERICAL METHOD FOR SOLVING GENERAL VARIATIONAL INEQUALITIES

2011 ◽  
Vol 25 (32) ◽  
pp. 4443-4455 ◽  
Author(s):  
ABDELLAH BNOUHACHEM ◽  
MUHAMMAD ASLAM NOOR ◽  
MOHAMED KHALFAOUI ◽  
ZHAOHAN SHENG
Keyword(s):  
Lower Bound ◽  
Global Convergence ◽  
Variational Inequalities ◽  
Complementarity Problems ◽  
Extragradient Method ◽  
Mild Conditions ◽  
Modified Method ◽  
Adaptive Technique ◽  
Special Cases ◽  
General Variational Inequalities

In this paper, we suggest and analyze a new extragradient method for solving the general variational inequalities involving two operators. We also prove the global convergence of the proposed modified method under certain mild conditions. We used a self-adaptive technique to adjust parameter ρ at each iteration. It is proved theoretically that the lower-bound of the progress obtained by the proposed method is greater than that by the extragradient method. An example is given to illustrate the efficiency and its comparison with the extragradient method. Since the general variational inequalities include the classical variational inequalities and complementarity problems as special cases, our results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this field.

MODIFIED PROJECTION METHOD FOR GENERAL VARIATIONAL INEQUALITIES

2011 ◽  
Vol 25 (27) ◽  
pp. 3595-3610 ◽  
Author(s):  
ABDELLAH BNOUHACHEM ◽  
MUHAMMAD ASLAM NOOR ◽  
ZHAOHAN SHENG
Keyword(s):  
Global Convergence ◽  
Variational Inequalities ◽  
Projection Method ◽  
Complementarity Problems ◽  
Descent Direction ◽  
Step Size ◽  
Special Cases ◽  
General Variational Inequalities ◽  
Modified Projection Method

In this paper, we suggest and analyze a modified descent-projection method for solving general variational inequalities. The method makes use of a descent direction to produce the new iterate and can be viewed as an improvement of the descent-projection method by using a new step size. We also prove the global convergence of the proposed method. An example is given to illustrate the efficiency and its comparison with other methods. Since the general variational inequalities include quasi variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.

scholarly journals A Self-Adaptive Method for Solving a System of Nonlinear Variational Inequalities

2007 ◽  
Vol 2007 ◽  
pp. 1-7
Author(s):  
Chaofeng Shi
Keyword(s):  
Iterative Method ◽  
Variational Inequalities ◽  
Adaptive Method ◽  
New Method ◽  
Computational Results ◽  
Computation Cost ◽  
Self Adaptive

The system of nonlinear variational inequalities (SNVI) is a useful generalization of variational inequalities. Verma (2001) suggested and analyzed an iterative method for solving SNVI. In this paper, we present a new self-adaptive method, whose computation cost is less than that of Verma's method. The convergence of the new method is proved under the same assumptions as Verma's method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.

scholarly journals Forward-backward resolvent splitting methods for general mixed variational inequalities

2003 ◽  
Vol 2003 (43) ◽  
pp. 2759-2770
Author(s):  
Muhammad Aslam Noor ◽  
Muzaffar Akhter ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Complementarity Problems ◽  
Splitting Methods ◽  
Pseudomonotone Operators ◽  
Special Cases ◽  
Mixed Variational Inequalities

We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known three-step forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for solving various classes of variational inequalities and complementarity problems. Since general mixed variational inequalities include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.

scholarly journals Some Proximal Methods for Solving Mixed Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Muhammad Aslam Noor ◽  
Zhenyu Huang
Keyword(s):  
Variational Inequalities ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Proximal Methods ◽  
Proximal Point ◽  
Equivalent Formulation ◽  
New Methods ◽  
Special Cases ◽  
Mixed Variational Inequalities

It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest some new proximal point methods for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered under some suitable conditions. Our method of constructing these iterative methods is very simple. Results proved in this paper may stimulate further research in this direction.

scholarly journals A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2039 ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Adaptive Method ◽  
Convergence Result ◽  
Null Point ◽  
Fixed Point Problems ◽  
Bounded Linear ◽  
Contraction Method ◽  
Projection And Contraction Method ◽  
Self Adaptive

We introduce a new projection and contraction method with inertial and self-adaptive techniques for solving variational inequalities and split common fixed point problems in real Hilbert spaces. The stepsize of the algorithm is selected via a self-adaptive method and does not require prior estimate of norm of the bounded linear operator. More so, the cost operator of the variational inequalities does not necessarily needs to satisfies Lipschitz condition. We prove a strong convergence result under some mild conditions and provide an application of our result to split common null point problems. Some numerical experiments are reported to illustrate the performance of the algorithm and compare with some existing methods.

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