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1.
MUHAMMAD ASLAM NOOR 《Journal of Global Optimization》2000,18(1):75-89
In this paper, we suggest and analyze a number of resolvent-splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement. 相似文献
2.
We suggest and analyze some new splitting type projection methods for solving general variational inequalities by using the updating technique of the solution. The convergence analysis of these new methods is considered and the proof of convergence is very simple. These new methods are versatile. 相似文献
3.
《Applied Mathematics Letters》2001,14(2):231-236
We consider and analyze some new splitting methods for solving quasi-monotone mixed variational inequalities by using the technique of updating the solution. The modified methods converge for quasi-monotone continuous operators. The new splitting methods differ from the existing splitting methods. Proof of convergence is very simple. 相似文献
4.
《Optimization》2012,61(1-2):29-44
We consider some new iterative methods for solving quasimonotone mixed variational inequalities by updating the solution. These algorithms are based on combining extrapolation and splitting techniques. The convergence analysis of these new methods is considered. These new methods are versatile and are easy to implement. Our method of proof of convergence is very simple and uses either monotonicity or quasimonotonicity of the operator. 相似文献
5.
《Mathematical and Computer Modelling》1999,29(7):1-9
We consider some new iterative methods for solving general monotone mixed variational inequalities by using the updating technique of the solution. The convergence analysis of these new methods is considered and the proof of convergence is very simple. These new methods are versatile and are easy to implement. Our results differ from those of He [1,2], Solodov and Tseng [3], and Noor [4–6] for solving the monotone variational inequalities. 相似文献
6.
一类求解单调变分不等式的隐式方法 总被引:6,自引:0,他引:6
1.引言变分不等式是一个非常有趣。非常困难的数学问题["].它具有广泛的应用(例如,数学规划中的许多基本问题都可以归结为一个变分不等式问题),因而得到深入的研究并有了不少算法[1,2,5-8,17-21].对线性单调变分不等式,我们最近提出了一系列投影收缩算法Ig-13].本文考虑求解单调变分不等式其中0CW是一闭凸集,F是从正p到自身的一个单调算子,一即有我们用比(·)表示到0上的投影.求解单调变分不等式的一个简单方法是基本投影法[1,6],它的迭代式为然而,如果F不是仿射函数,只有当F一致强单调且LIPSChitZ连续… 相似文献
7.
代宏霞 《应用泛函分析学报》2008,10(1):39-43
在算子的分裂技巧基础上介绍了求解伪单调广义混合变分不等式的改进五步预解算法,算法的收敛性只要求算子的g-伪单调和g—Lipschitz连续性,算子的伪单调比单调更弱.本文的新算法推广了文献中某些已有的结果. 相似文献
8.
Muhammad Aslam Noor 《Journal of Mathematical Analysis and Applications》2003,277(2):379-394
In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. 相似文献
9.
D. Han 《Journal of Optimization Theory and Applications》2007,132(2):227-243
The Peaceman-Rachford and Douglas-Rachford operator splitting methods are advantageous for solving variational inequality
problems, since they attack the original problems via solving a sequence of systems of smooth equations, which are much easier
to solve than the variational inequalities. However, solving the subproblems exactly may be prohibitively difficult or even
impossible. In this paper, we propose an inexact operator splitting method, where the subproblems are solved approximately
with some relative error tolerance. Another contribution is that we adjust the scalar parameter automatically at each iteration
and the adjustment parameter can be a positive constant, which makes the methods more practical and efficient. We prove the
convergence of the method and present some preliminary computational results, showing that the proposed method is promising.
This work was supported by the NSFC grant 10501024. 相似文献
10.
In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued variational inequalities. The convergence of the proposed methods requires only the partially-relaxed strong monotonicity of the operator, which is weaker than cocoercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities. 相似文献
11.
M. A. Noor 《Mathematical and Computer Modelling》2000,31(13):139-19
In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems. 相似文献
12.
Muhammad Aslam Noor 《Journal of Applied Mathematics and Computing》2010,32(1):83-95
In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. 相似文献
13.
Muhammad Aslam Noor 《Computational Mathematics and Modeling》2010,21(1):97-108
It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent
alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational
inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class
of Wiener – Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener – Hopf
equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence
analysis of these iterative methods. Our method of proofs is very simple compared to other techniques. 相似文献
14.
Proximal-point algorithms (PPAs) are classical solvers for convex optimization problems and monotone variational inequalities
(VIs). The proximal term in existing PPAs usually is the gradient of a certain function. This paper presents a class of PPA-based
methods for monotone VIs. For a given current point, a proximal point is obtained via solving a PPA-like subproblem whose
proximal term is linear but may not be the gradient of any functions. The new iterate is updated via an additional slight
calculation. Global convergence of the method is proved under the same mild assumptions as the original PPA. Finally, profiting
from the less restrictions on the linear proximal terms, we propose some parallel splitting augmented Lagrangian methods for
structured variational inequalities with separable operators.
B.S. He was supported by NSFC Grant 10571083 and Jiangsu NSF Grant BK2008255. 相似文献
15.
A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods. 相似文献
16.
Muhammad Aslam Noor Khalida Inayat Noor Zhenyu Huang 《Journal of Applied Mathematics and Computing》2011,35(1-2):595-605
In this paper, we introduce and consider a new class of variational inequalities, which is called the bifunction hemivariational inequality. This new class includes several classes of variational inequalities as special cases. A number of iterative methods for solving bifunction hemivariational inequalities are suggested and analyzed by using the auxiliary principle technique. We also study the convergence analysis of these iterative methods under some mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique. 相似文献
17.
In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities. 相似文献
18.
Convergence rate analysis of iteractive algorithms for solving variational inequality problems 总被引:3,自引:0,他引:3
M.V. Solodov 《Mathematical Programming》2003,96(3):513-528
We present a unified convergence rate analysis of iterative methods for solving the variational inequality problem. Our results
are based on certain error bounds; they subsume and extend the linear and sublinear rates of convergence established in several
previous studies. We also derive a new error bound for $\gamma$-strictly monotone variational inequalities. The class of algorithms
covered by our analysis in fairly broad. It includes some classical methods for variational inequalities, e.g., the extragradient,
matrix splitting, and proximal point methods. For these methods, our analysis gives estimates not only for linear convergence
(which had been studied extensively), but also sublinear, depending on the properties of the solution. In addition, our framework
includes a number of algorithms to which previous studies are not applicable, such as the infeasible projection methods, a
separation-projection method, (inexact) hybrid proximal point methods, and some splitting techniques. Finally, our analysis
covers certain feasible descent methods of optimization, for which similar convergence rate estimates have been recently obtained
by Luo [14].
Received: April 17, 2001 / Accepted: December 10, 2002
Published online: April 10, 2003
RID="⋆"
ID="⋆" Research of the author is partially supported by CNPq Grant 200734/95–6, by PRONEX-Optimization, and by FAPERJ.
Key Words. Variational inequality – error bound – rate of convergence
Mathematics Subject Classification (2000): 90C30, 90C33, 65K05 相似文献
19.
On General Mixed Quasivariational Inequalities 总被引:5,自引:0,他引:5
In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed. 相似文献
20.
M. A. Noor 《Journal of Optimization Theory and Applications》2004,121(2):385-395
In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets. 相似文献