Newton Methods for Quasidifferentiable Equations and Their Convergence

The Newton method and the inexact Newton method for solving quasidifferentiable equations via the quasidifferential are investigated. The notion of Q-semismoothness for a quasidifferentiable function is proposed. The superlinear convergence of the Newton method proposed by Zhang and Xia is proved under the Q-semismooth assumption. An inexact Newton method is developed and its linear convergence is shown.Project sponsored by Shanghai Education Committee Grant 04EA01 and by Shanghai Government Grant T0502.     

三阶方向牛顿法的收敛性

通过递推关系,证明了解希尔伯特空间上的实系数非线性方程组的三阶方向牛顿法的半局部收敛性,给出了解的存在性以及先验误差界,最后计算出一些数值结果来证明我们的结论.     

牛顿法的改进及其在Banach空间中收敛性

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.     

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

Semismooth Newton Methods for Solving Semi-Infinite Programming Problems

In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given.     

Newton Methods for Solving Two Classes of Nonsmooth Equations

The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since they do not require an element of Clarke generalized Jacobian, of B-differential, or of b-differential, at each iteration point.     

Convergence Analysis of Gauss–Newton Methods for the Complementarity Problem

In a recent paper (Ref. 1), the first author presented a damped Gauss–Newton algorithm to solve the complementarity problem. Although the conclusions of the convergence theorems in the paper are valid, the proofs contain minor errors. The aim of this paper is to give correct proofs and also to show that they are valid under milder conditions.     

一类非光滑方程组的牛顿法及全局收敛性

借助于一种新的微分 - -微分 ,本文给出极大值函数及其光滑复合的非光滑方程组的牛顿法 .最后证明了该牛顿法具有全局收敛性 .     

一维凸函数牛顿法的全局收敛性及其应用

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。     

修正的三次收敛的牛顿迭代法

给出了牛顿迭代法的两种修正形式,证明了它们都是三阶收敛的,给出的相互比较的数值例子有力地说明了这一点.     

修正的三阶收敛的牛顿迭代法

给出了牛顿迭代法的两种修正形式,证明了它们是三阶收敛的,数值实验表明,与其它已知的三阶收敛的牛顿迭代法相比,修正的牛顿迭代法具有一定的优势.     

Directional Newton methods in variables

Directional Newton methods for functions of variables are shown to converge, under standard assumptions, to a solution of . The rate of convergence is quadratic, for near-gradient directions, and directions along components of the gradient of with maximal modulus. These methods are applied to solving systems of equations without inversion of the Jacobian matrix.

     

改进的PSB拟牛顿修正矩阵的收敛性

本文在已建立的一类新拟牛顿方程Ｂｋ＋１δｋ＝ｙｋ＝ｙｋ＋θｋ／δｋ＾Ｔｕ的基础上,证明了满足新拟牛顿方程的改进ＰＳＢ算法产生的拟牛顿修正矩阵序列在序列｛ｘｋ｝收敛于ｘ＾＊,｛δｋ｝一致性无关及二阶导数阵连续有界的条件下收敛于海色阵Ｇ（ｘ＾＊）。     

The convergence of the perturbed Newton method and its application for ill-conditioned problems

Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples.     

关于Brown方法的半局部收敛性

在本文中,我们讨论解非线性方程组的Brown方法的半局部收敛性。通过对Brown方法的算法结构作深入的分析,我们将Brown方法变换成带有特殊误差项的近似Newton法,基于这种等价变形,我们建立了Brown方法的半局部收敛定理,从而完善了Brown方法的收敛理论。     

Newton Linearized Methods for Semilinear Parabolic Equations

In this study, Newton linearized finite element methods are presented for solving semi-linear parabolic equations in two- and three-dimensions. The proposed scheme is a one-step, linearized and second-order method in temporal direction, while the usual linearized second-order schemes require at least two starting values. By using a temporal-spatial error splitting argument, the fully discrete scheme is proved to be convergent without time-step restrictions dependent on the spatial mesh size. Numerical examples are given to demonstrate the efficiency of the methods and to confirm the theoretical results.     

解奇异非光滑方程组的牛顿法和不精确牛顿法

本文主要解决奇异非光滑方程组的解法。应用一种新的次微分的外逆，我们提出了牛顿法和不精确牛顿法，它们的收敛性同时也得到了证明。这种方法能更容易在一引起实际应用中实现。这种方法可以看作是已存在的解非光滑方程组的方法的延伸。     

非光滑方程的方向牛顿法

通过引入广义梯度,将求解含n个未知量方程的方向牛顿法推广到非光滑的情形.证明了该方法在半光滑条件下的收敛性定理,给出了解的存在性以及先验误差界.     

Local Convergence of the Interior-Point Newton Method for General Nonlinear Programming

In this paper, a formulation for an interior-point Newton method of general nonlinear programming problems is presented. The formulation uses the Coleman-Li scaling matrix. The local convergence and the q-quadratic rate of convergence for the method are established under the standard assumptions of the Newton method for general nonlinear programming.     

m步非线性多重分裂Newton法的收敛性

本文讨论了多重分裂算法在求解一类非线性方程组的全局收敛性和单侧收敛性．当用研步Newton法来代替求得每个非线性多重分裂子问题的近似解时，同样给出相应收敛性结论．数值算例证实了算法的有效性．