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1.
In this paper, we extend the auxiliary principle (Cohen in J. Optim. Theory Appl. 49:325–333, 1988) to study a class of Lions-Stampacchia variational inequalities in Hilbert spaces. Our method consists in approximating,
in the subproblems, the nonsmooth convex function by a sequence of piecewise linear and convex functions, as in the bundle
method for nonsmooth optimization. This makes the subproblems more tractable. We show the existence of a solution for this
Lions-Stampacchia variational inequality and explain how to build a new iterative scheme and a new stopping criterion. This
iterative scheme and criterion are different from those commonly used in the special case of nonsmooth optimization. We study
also the convergence of iterative sequences generated by the algorithm.
This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for
the Doctoral Program of Higher Education (20060610005), the National Natural Science Foundation of Sichuan Education Department
of China (07ZB068) and the Open Fund (PLN0703) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest
Petroleum University). 相似文献
2.
Yonghong Yao Muhammad Aslam Noor Khalida Inayat Noor Yeong-Cheng Liou Huma Yaqoob 《Acta Appl Math》2010,110(3):1211-1224
In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence
between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative
equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational
inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed
iterative method under some suitable conditions. 相似文献
3.
We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous
variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal
methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme. 相似文献
4.
M. A. Noor 《Journal of Optimization Theory and Applications》2009,143(3):619-624
In this paper, we suggest and analyze an implicit iterative method for solving nonconvex variational inequalities using the technique of the projection operator. We also discuss the convergence of the iterative method under suitable weaker conditions. Our method of proof is very simple as compared with other techniques. 相似文献
5.
StrongConvergenceTheoremsforPerturbedMaximalMonotoneOperatorsinBanachSpacesWangWeimin(王为民)ZhaoYichun(赵义纯)(DepartmentofMathema... 相似文献
6.
Convergence of Hybrid Steepest-Descent Methods for Generalized Variational Inequalities 总被引:3,自引:0,他引:3
Liu Chuan ZENG N. C. Wong J. C. YAO 《数学学报(英文版)》2006,22(1):1-12
In this paper, we consider the generalized variational inequality GVI(F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI(F,g, C). Strong convergence results are established and applications to constrained generalized pseudo-inverse are included. 相似文献
7.
Xin Tao YE Chong LI 《数学学报(英文版)》2007,23(1):65-76
The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi-continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided. 相似文献
8.
We analyze a primal-dual pair of problems generated via a duality theory introduced by Svaiter. We propose a general algorithm and study its convergence properties. The focus is a general primal-dual principle for strong convergence of some classes of algorithms. In particular, we give a different viewpoint for the weak-to-strong principle of Bauschke and Combettes and unify many results concerning weak and strong convergence of subgradient type methods. 相似文献
9.
We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain
the rate of convergence n
−1/2
β
n
log 3/2
n in which β
n
can be particularly taken as (log log n)1/σ
with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n
−1/2(log log n)1/2log 2
n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n
−1/2
β
n
log 1/2
n for the above β
n
under the given covariance function, which improves the relevant one n
−1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results,
which extend and improve some known results. 相似文献
10.
B. S. Goh 《Journal of Optimization Theory and Applications》2010,144(1):43-55
It is desirable that an algorithm in unconstrained optimization converges when the guessed initial position is anywhere in
a large region containing a minimum point. Furthermore, it is useful to have a measure of the rate of convergence which can
easily be computed at every point along a trajectory to a minimum point. The Lyapunov function method provides a powerful tool to study convergence of
iterative equations for computing a minimum point of a nonlinear unconstrained function or a solution of a system of nonlinear
equations. It is surprising that this popular and powerful tool in the study of dynamical systems is not used directly to
analyze the convergence properties of algorithms in optimization. We describe the Lyapunov function method and demonstrate
how it can be used to study convergence of algorithms in optimization and in solutions of nonlinear equations. We develop
an index which can measure the rate of convergence at all points along a trajectory to a minimum point and not just at points
in a small neighborhood of a minimum point. Furthermore this index can be computed when the calculations are being carried
out. 相似文献
11.
Li Ping ZHANG Wen Xun XING 《数学学报(英文版)》2007,23(9):1553-1562
Based on the techniques used in non-smooth Newton methods and regularized smoothing Newton methods, a Newton-type algorithm is proposed for solving the P0 affine variational inequality problem. Under mild conditions, the algorithm can find an exact solution of the P0 affine variational inequality problem in finite steps. Preliminary numerical results indicate that the algorithm is promising. 相似文献
12.
We deal with a common fixed point problem for a family of quasinonexpansive mappings defined on a Hilbert space with a certain
closedness assumption and obtain strongly convergent iterative sequences to a solution to this problem. We propose a new type
of iterative scheme for this problem. A feature of this scheme is that we do not use any projections, which in general creates
some difficulties in practical calculation of the iterative sequence. We also prove a strong convergence theorem by the shrinking
projection method for a family of such mappings. These results can be applied to common zero point problems for families of
monotone operators. 相似文献
13.
Boundedness of Solutions for Elliptic Variational InequalitiesBoundednessofSolutionsforEllipticVariationalInequalities¥YeRuif... 相似文献
14.
Torsten Hein 《Numerical Functional Analysis & Optimization》2013,34(10):1158-1184
We introduce and discuss an iterative method of modified Landweber type for regularization of nonlinear operator equations in Banach spaces. Under smoothness and convexity assumptions on the solution space we present convergence and stability results. Furthermore, we will show that under the so-called approximate source conditions convergence rates may be achieved by a proper a-priori choice of the parameter of the presented algorithm. We will illustrate these theoretical results with a numerical example. 相似文献
15.
To solve a class of variational inequalities with separable structures, some classical methods such as the augmented Lagrangian
method and the alternating direction methods require solving two subvariational inequalities at each iteration. The most recent
work (B.S. He in Comput. Optim. Appl. 42(2):195–212, 2009) improved these classical methods by allowing the subvariational inequalities arising at each iteration to be solved in parallel,
at the price of executing an additional descent step. This paper aims at developing this strategy further by refining the
descent directions in the descent steps, while preserving the practical characteristics suitable for parallel computing. Convergence
of the new parallel descent-like method is proved under the same mild assumptions on the problem data. 相似文献
16.
Ha|YunZHOU 《数学学报(英文版)》2004,20(5):829-836
In this article, we will investigate the properties of iterative sequence for non-expansive mappings and present several strong and weak convergence results of successive approximations to fixed points of non-expansive mappings in uniformly convex Banach spaces. The results presented in this article generalize and improve various ones concerned with constructive techniques for the fixed points of non-expansive mappings. 相似文献
17.
In this paper,we establish a Rosenthal-type inequality of partial sums for ρ~mixing random variables.As its applications,we get the complete convergence rates in the strong laws for ρ~-mixing random variables.The result obtained extends the corresponding result. 相似文献
18.
H. Zegeye 《Numerical Functional Analysis & Optimization》2013,34(6):799-816
In this article, we introduce an algorithm which has strong convergence for solving the variational inequality problem for η-inverse strongly accretive mappings in the set of common fixed points of finite family of λ-strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. 相似文献
19.
In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of
bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results
are established and applications to variational inequalities are given.
In this research, the first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program
Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality
(075105118). The third author was partially supported by Grant NSC 96-2628-E-110-014-MY3. 相似文献
20.
The purpose of this article is to introduce a class of total quasi-φ-asymptotically nonexpansive nonself mappings. Strong convergence theorems for common fixed points of a countable family of total quasi-φ-asymptotically nonexpansive mappings are established in the framework of Banach spaces based on modified Halpern and Mann-type iteration algorithm. The main results presented in this article extend and improve the corresponding results of many authors. 相似文献