scholarly journals Proximal extragradient methods for pseudomonotone variational inequalities

2006 ◽  
Vol 37 (2) ◽  
pp. 109-116
Author(s):  
Muhammad Aslam Noor ◽  
Abdellah Bnouhachem
Keyword(s):  
Variational Inequalities ◽  
Iterative Methods ◽  
Proximal Methods ◽  
Pseudomonotone Operators ◽  
Extragradient Methods ◽  
Special Cases

We consider and analyze some new proximal extragradient type methods for solving variational inequalities. The modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. These new iterative methods include the projection, extragradient and proximal methods as special cases.

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 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.

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 Extended Extragradient Methods for Generalized Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yonghong Yao ◽  
Yeong-Cheng Liou ◽  
Cun-Lin Li ◽  
Hui-To Lin
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Extragradient Methods ◽  
Generalized Variational Inequalities ◽  
Mild Conditions ◽  
Convergence Results ◽  
Special Cases

We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed.

scholarly journals Iterative Schemes for a Class of Mixed Trifunction Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Iterative Methods ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

scholarly journals Some Iterative Methods for Solving Nonconvex Bifunction Equilibrium Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Saira Zainab
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Iterative Methods ◽  
Equilibrium Problems ◽  
Implicit Method ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Proof Of Convergence

We introduce and consider a new class of equilibrium problems and variational inequalities involving bifunction, which is called the nonconvex bifunction equilibrium variational inequality. We suggest and analyze some iterative methods for solving the nonconvex bifunction equilibrium variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only monotonicity. Some special cases are also considered. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

scholarly journals Implicit Schemes for Solving Extended General Nonconvex Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Zhenyu Huang ◽  
Eisa Al-Said
Keyword(s):  
Variational Inequalities ◽  
Iterative Methods ◽  
Projection Technique ◽  
Implicit Schemes ◽  
Special Cases

We suggest and analyze some implicit iterative methods for solving the extended general nonconvex variational inequalities using the projection technique. We show that the convergence of these iterative methods requires only thegh-pseudomonotonicity, which is a weaker condition thangh-monotonicity. We also discuss several special cases. Our method of proof is very simple as compared with other techniques.

scholarly journals On New Proximal Point Methods for Solving the Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Proximal Point ◽  
Equivalent Formulation ◽  
Extragradient Methods ◽  
Proximal Point Methods ◽  
New Methods ◽  
Special Cases

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

scholarly journals A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1855 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Methods ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
High Order ◽  
Convergence Theorems ◽  
New Family ◽  
Special Cases ◽  
Iteration Functions ◽  
Iteration Function ◽  
Local Convergence Analysis

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

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