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1.
Smoothing Newton methods for nonlinear complementarity problems NCP(F) often require F to be at least a P
0-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP(F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points. 相似文献
2.
Inexact Interior-Point Method 总被引:2,自引:0,他引:2
S. Bellavia 《Journal of Optimization Theory and Applications》1998,96(1):109-121
In this paper, we introduce an inexact interior-point algorithm for a constrained system of equations. The formulation of the problem is quite general and includes nonlinear complementarity problems of various kinds. In our convergence theory, we interpret the inexact interior-point method as an inexact Newton method. This enables us to establish a global convergence theory for the proposed algorithm. Under the additional assumption of the invertibility of the Jacobian at the solution, the superlinear convergence of the iteration sequence is proved. 相似文献
3.
C. Durazzi 《Journal of Optimization Theory and Applications》2000,104(1):73-90
Interior-point methods have been developed largely for nonlinear programming problems. In this paper, we generalize the global Newton interior-point method introduced in Ref. 1 and we establish a global convergence theory for it, under the same assumptions as those stated in Ref. 1. The generalized algorithm gives the possibility of choosing different descent directions for a merit function so that difficulties due to small steplength for the perturbed Newton direction can be avoided. The particular choice of the perturbation enables us to interpret the generalized method as an inexact Newton method. Also, we suggest a more general criterion for backtracking, which is useful when the perturbed Newton system is not solved exactly. We include numerical experimentation on discrete optimal control problems. 相似文献
4.
徐勤亚 《应用数学与计算数学学报》2002,16(2):68-72
牛顿法是求解非线性方程F(x)=0的一种经典方法。在一般假设条件下,牛顿法只具有局部收敛性。本文证明了一维凸函数牛顿法的全局收敛性,并且给出了它在全局优化积分水平集方法中的应用。 相似文献
5.
Newton‐HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive‐definite Jacobian matrices (J. Comp. Math. 2010; 28 :235–260). In that paper, only local convergence was proved. In this paper, we prove a Kantorovich‐type semilocal convergence. Then we introduce Newton‐HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton‐HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
6.
牛顿法是求解非线性方程(组)的一种经典方法,本文在Banach空间中对经典牛顿法加以了改进,研究了其收敛性,改进后的牛顿法具有更广泛的应用前景. 相似文献
7.
We study the convergence of a general perturbation of the Newton method for solving a nonlinear system of equations. As an application, we show that the augmented Lagrangian successive quadratic programming is locally and q-quadratically convergent in the variable x to the solution of an equality constrained optimization problem, under a mild condition on the penalty parameter and the choice of the Lagrange multipliers. 相似文献
8.
9.
关于广义Newton法的收敛性问题 总被引:4,自引:0,他引:4
本文在较弱的条件下,证明了B-可微方程组的广义Newton法的局部超线性收敛性,为该算法直接应用于非线性规划问题、变分不等问题以及非线性互补问题等提供了理论依据。最后,本文给出了广义Newton法付之实践的具体策略。数值结果表明,算法是行之有效的。 相似文献
10.
On the formulation and theory of the Newton interior-point method for nonlinear programming 总被引:16,自引:0,他引:16
A. S. El-Bakry R. A. Tapia T. Tsuchiya Y. Zhang 《Journal of Optimization Theory and Applications》1996,89(3):507-541
In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.The first two authors were supported in part by NSF Cooperative Agreement No. CCR-8809615, by Grants AFOSR 89-0363, DOE DEFG05-86ER25017, ARO 9DAAL03-90-G-0093, and the REDI Foundation. The fourth author was supported in part by NSF DMS-9102761 and DOE DE-FG02-93ER25171. The authors would like to thank Sandra Santos for painstakingly proofreading an earlier verion of this paper. 相似文献
11.
关于多元非线性方程的Broyden方法 总被引:2,自引:0,他引:2
本文提出了求解多元非线性方程的Broyden方法,讨论了该方法的局部与半局部收敛性,并估计了其超线性收敛速度.数值实验表明,新方法是可行有效的,并且其计算效率高于方向Newton法和方向割线法. 相似文献
12.
The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step. 相似文献
13.
Liqun Qi 《Mathematical Programming》1994,64(1-3):277-294
In the literature, the proof of superlinear convergence of approximate Newton or SQP methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions. Sometimes, the second-order derivatives of those functions are required to be Lipschitzian. In this paper, we present approximate Newton or SQP methods for solving nonlinear programming problems whose objective and constraint functions have locally Lipschitzian derivatives, and establishQ-superlinear convergence of these methods under the assumption that these derivatives are semismooth. This assumption is weaker than the second-order differentiability. The extended linear-quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives.This work is supported by the Australian Research Council.This paper is dedicated to Professor O.L. Mangasarian on the occasion of his 60th birthday. 相似文献
14.
Ioannis K. Argyros 《Numerical Functional Analysis & Optimization》2019,40(3):303-318
The convergence set for Newton’s method is small in general using Lipschitz-type conditions. A center-Lipschitz-type condition is used to determine a subset of the convergence set containing the Newton iterates. The rest of the Lipschitz parameters and functions are then defined based on this subset instead of the usual convergence set. This way the resulting parameters and functions are more accurate than in earlier works leading to weaker sufficient semi-local convergence criteria. The novelty of the paper lies in the observation that the new Lipschitz-type functions are special cases of the ones given in earlier works. Therefore, no additional computational effort is required to obtain the new results. The results are applied to solve Hammerstein nonlinear integral equations of Chandrasekhar type in cases not covered by earlier works. 相似文献
15.
一个四阶收敛的牛顿类方法 总被引:2,自引:0,他引:2
A fourth-order convergence method of solving roots for nonlinear equation,which is a variant of Newton's method given.Its convergence properties is proved.It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end,numerical tests are given and compared with other known Newton and Newtontype methods.The results show that the proposed method has some more advantages than others.It enriches the methods to find the roots of non-linear equations and it ... 相似文献
16.
一类四阶牛顿变形方法 总被引:1,自引:0,他引:1
给出非线性方程求根的一类四阶方法,也是牛顿法的变形方法.证明了方法收敛性,它们至少四次收敛到单根,线性收敛到重根.文末给出数值试验,且与牛顿法及其它牛顿变形法做了比较.结果表明方法具有很好的优越性,它丰富了非线性方程求根的方法,在理论上和应用上都有一定的价值. 相似文献
17.
Recursive quadratic programming is a family of techniques developed by Bartholomew-Biggs and other authors for solving nonlinear programming problems. The first-order optimality conditions for a local minimizer of the augmented Lagrangian are transformed into a nonlinear system where both primal and dual variables appear explicitly. The inner iteration of the algorithm is a Newton-like procedure that updates simultaneously primal variables and Lagrange multipliers. In this way, as observed by Gould, the implementation of the Newton method becomes stable, in spite of the possibility of having large penalization parameters. In this paper, the inner iteration is analyzed from a different point of view. Namely, the size of the convergence region and the speed of convergence of the inner process are considered and it is shown that, in some sense, both are independent of the penalization parameter when an adequate version of the Newton method is used. In other words, classical Newton-like iterations are improved, not only in relation to stability of the linear algebra involved, but also with regard to the ovearll convergence of the nonlinear process. Some numerical experiments suggset that, in fact, practical efficiency of the methods is related to these theoretical results. 相似文献
18.
19.
Z. J. Huang 《Journal of Optimization Theory and Applications》1992,75(2):331-344
In this paper, by a further investigation of the algorithm structure of the nonlinear block scaled ABS methods, we convert it into an inexact Newton method. Based on this equivalent version, we establish the semilocal convergence theorem of the nonlinear block scaled ABS methods and obtain convergence conditions that mainly depend on the behavior of the mapping at the initial point. This complements the convergence theory of the nonlinear block scaled ABS methods. 相似文献
20.
In this paper, we study the convergence and the convergence rates of an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. Opposed to the traditional methods, we analyze an inexact Newton–Landweber iteration depending on the Hölder continuity of the inverse mapping when the data are not contaminated by noise. With the namely Hölder-type stability and the Lipschitz continuity of DF, we prove convergence and monotonicity of the residuals defined by the sequence induced by the iteration. Finally, we discuss the convergence rates. 相似文献