General system of nonconvex variational inequalities and parallel projection method (original) (raw)
Fixed Point Theory and Applications, 2012
In this article, we introduce and consider a new system of general nonconvex variational inequalities defined on uniformly prox-regular sets. We establish the equivalence between the new system of general nonconvex variational inequalities and the fixed point problems to analyze an explicit projection method for solving this system. We also consider the convergence of the projection method under some suitable conditions. Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems. MSC (2000): 47J20; 47N10; 49J30.
Projection Method Approach for General Regularized Non-convex Variational Inequalities
Journal of Optimization Theory and Applications, 2013
In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied.
Journal of Inequalities and Applications, 2012
The purpose of this paper is to introduce a new system of general nonlinear regularized nonconvex variational inequalities and verify the equivalence between the proposed system and fixed point problems. By using the equivalent formulation, the existence and uniqueness theorems for solutions of the system are established. Applying two nearly uniformly Lipschitzian mappings S 1 and S 2 and using the equivalent alternative formulation, we suggest and analyze a new perturbed p-step projection iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S 1 , S 2) which is the unique solution of the system of general nonlinear regularized nonconvex variational inequalities. We also discuss the convergence analysis of the proposed iterative algorithm under some suitable conditions.
Solvability of a system of nonconvex general variational inequalities
Sarajevo Journal of Mathematics, 2013
Using the prox-regularity notion, we introduce a system of nonconvex general variational inequalities and a general three step algorithm for approximate solvability of this system. We establish the convergence of three-step projection method for a general system of nonconvex variational inequality problem. We obtain as a particular case some known results.
Journal of Applied Mathematics, 2012
A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.
System of Generalized Nonlinear Regularized Nonconvex Variational Inequalities
Korean Journal of Mathematics, 2016
In this work, we suggest a new system of generalized nonlinear regularized nonconvex variational inequalities in a real Hilbert space and establish an equivalence relation between this system and fixed point problems. By using the equivalence relation we suggest a new perturbed projection iterative algorithms with mixed errors for finding a solution set of system of generalized nonlinear regularized nonconvex variational inequalities.
System of implicit nonconvex variationl inequality problems A projection method approach
Journal of Nonlinear Sciences and Applications, 2013
In this paper, we consider a new system of implicit nonconvex variational inequality problems in setting of prox-regular subsets of two different Hilbert spaces. Using projection method, we establish the equivalence between the system of implicit nonconvex variational inequality problems and a system of relations. Using this equivalence formulation, we suggest some iterative algorithms for finding the approximate solution of the system of implicit nonconvex variational inequality problems and its special case. Further, we establish some theorems for the existence and iterative approximation of the solutions of the system of implicit nonconvex variational inequality problems and its special case. The results presented in this paper are new and different form the previously known results for nonconvex variational inequality problems. These results also generalize, unify and improve the previously known results of this area.
Projection algorithms for solving a system of general variational inequalities
Nonlinear Analysis: Theory, Methods & Applications, 2009
In this paper, we introduce and consider a new system of general variational inequalities involving four different operators. Using the projection operator technique, we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving three operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.
Projection Methods for Monotone Variational Inequalities
Journal of Mathematical Analysis and Applications, 1999
In this paper, we study some new iterative methods for solving monotone variational inequalities by using the updating technique of the solution. It is shown that the convergence of the new methods requires the monotonicity and pseudomonotonicity of the operator. The new methods are very versatile and are easy to implement. The techniques include the splitting and extragradient methods as special cases.
A General System of Regularized Non-convex Variational Inequalities
Journal of Applied & Computational Mathematics, 2014
In this communication, we introduced a general system of regularized non-convex variational inequalities (GSRNVI) and established an equivalence between this system and fixed point problems. By using this equivalence we define a projection iterative algorithm for solving GSRNVI, we also proved existence and uniqueness of GSRNVI. The convergence analysis of the suggested iterative algorithm is studied.
Applied Mathematics and Computation, 2007
In this paper, we introduce and consider a new system of variational inequalities involving two different operators. Using the projection technique, we suggest and analyze new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving the single operator, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.
Projection-proximal methods for general variational inequalities
Journal of Mathematical Analysis and Applications, 2006
In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.
Strong Convergence Theorems of Iterative Algorithm for Nonconvex Variational Inequalities
Thai Journal of Mathematics, 2016
In this work, we suggest and analyze an iterative scheme for solving the system of nonconvex variational inequalities by using projection technique. We prove strong convergence of iterative scheme to the solution of the system of nonconvex variational inequalities requires to the modified mapping T which is Lipschitz continuous but not strongly monotone mapping. Our result can be viewed and improvement the result of N. Petrot [18]
An extragradient algorithm for solving general nonconvex variational inequalities
Applied Mathematics Letters, 2010
In this work, we suggest and analyze an extragradient method for solving general nonconvex variational inequalities using the technique of the projection operator. We prove that the convergence of the extragradient method requires only pseudomonotonicity, which is a weaker condition than requiring monotonicity. In this sense, our result can be viewed as an improvement and refinement of the previously known results. Our method of proof is very simple as compared with other techniques.
Some Parallel Algorithms for a New System of Quasi Variational Inequalities
Applied Mathematics & Information Sciences, 2013
In this paper, we introduce a new system of quasi variational inequalities. The projection technique is used to establish the equivalence between this new system of quasi variational inequalities and the fixed point problem. The fixed point formulation enables us to suggest some parallel projection iterative methods for solving the system of quasi variational inequalities. Convergence analysis of the proposed methods is investigated. Several special cases are discussed. Results proved in this paper continue to hold for these problems.
Extragradient methods for solving nonconvex variational inequalities
Journal of Computational and Applied Mathematics, 2011
In this paper, we introduce and consider a new class of variational inequalities, which are called the nonconvex variational inequalities. Using the projection technique, we suggest and analyze an extragradient method for solving the nonconvex variational inequalities. We show that the extragradient method is equivalent to an implicit iterative method, the convergence of which requires only pseudo-monotonicity, a weaker condition than monotonicity. This clearly improves on the previously known result. Our method of proof is very simple as compared with other techniques.
Applicable Analysis, 2014
This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777-786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1-10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76-85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292-2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples.