关于Newton法的Kantorovich型定理及其优函数

本文研究Newton法的Kantorovich型定理的特点及其对Newton法的半局部收敛性研究的思想方法,论述广义Lipschitz条件下的Kantorovich型定理的概括性和统一性.同时,在理论上当x_0取定时,针对每一个满足广义Lipschitz条件的光滑算子,给出优函数的一个构造方法.     

Auerbach 与 Banach 的一个定理的推广

Auerbach 与 Banach 曾证明,当0<σ<τ≤1时,在满足σ阶 Lipschitz 条件的函数中,存在函数 f(x)使关系式(?)处处成立.本文将推广这个定理,并从而得到如下的推论:设φ(x)是定义在[0,1]上的增函数,(?)φ(x)=0,如果φ(x)是比 x 较低阶的无穷小,则在连续模ω_f(δ)≤φ(δ)的函数 f(x)所组成的类中,存在处处不可微的函数.     

一类拟线性椭圆方程的很弱解的唯一性

利用以极大函数表示的有关Sobolev函数的逐点不等式来构造全局的Lipschitz型检验函数,得到了,在一定条件下,拟线性椭圆方程-div A(x,u,Du)=f(x)在grand Sobolev空间W_0~(θ,p)(Ω)中的很弱解是唯一的．     

Lipschitz函数近似次微分的凸性

A.D.IOFFE 在研究不可微优化中针对一类 Lipschitz 函数提出了近似次微分的概念.有许多问题有待解决.本文主要讨论了 Lipschitz 函数近似次微分的凸性,并在一维的情况下给出了一个充分条件.为方便起见,我们用“f∈L.(R~m→R~1)”来表示“f 是 R~m 上的 Lipschitz 函数”.定义1 设 R~n 为 n 维欧氏空间.f:R~n→R~1,|f(x)|<+∞,定义 f(x)的 Dini 导数为     

一类拟线性椭圆方程的很弱解的唯一性

利用以极大函数表示的有关Sobolev函数的逐点不等式来构造全局的Lipschitz型检验函数,得到了在一定条件下,拟线性椭圆方程-divA(x, u, Du) = f(x)在grand sobolev空间W0θ,p)(Ω)中的很弱解是唯一的.     

新题征展(72)

A题组新编1.下列条件对于函数f(x)定义域中的每一个x都成立,其中(a≠0,k≠0,a,b,k∈R):(1)条件1f(x)-f(-x)=0;条件2f(a x)=f(a-x);条件3f(kx b)=f(-kx-b);条件4f(x)=(x-a)0.其中判断函数f(x)是偶函数的条件是.(2)条件1f(a x)=f(a-x);2f(x)=f(2a-x);3f(3a-x)=f(x-a);4f(x)=(x-a)     

A Newton Method for Minimizing One-Order Lipschitz Functions

In this paper, we consider the following unconstrained optimization problem (P) min{f(x)|x∈R~n}, where f(x) is a one order Lipschitz function on R~n, i.e., g(x)—the gradient of f(x)—is Lipschitzian. We will represent a kind of Newton method for solving the problem (P).     

推广的Lipschitz类函数的Fourier乘子

本文首先推广 Lipschitz函数类如下: 如果ω(t)是一个满足附加条件的连续模,而f(x)∈L_p[-π,π],它的r阶中心差分△_t~rf(t)满足‖△_t~rf(x)‖=O(ω(t)),1≤p≤∞,r是一个由ω的属性所决定的正整数。则称f(x)为推广的Lipschitz类函数,记为f(x)∈Lip(ω,p)。本文讨论了这类函数的性质,以及常用的各类函数到Lip(ω,p)的Fourier乘子问题,并得出一系列充要条件。     

高阶半线性椭园方程Dirichlct问题解的存在性和正则性

对于2m阶半线性椭园型方程 Lu=f(x,u,Du,…,D~mu)要求f(x,y_1,…,y_k)满足Lipschitz条件,目前尚未有深入的讨论,众所周知,满足Lips chitz条件的函数是一类范围很广的函数,本文准备就这一条件,给出半线性椭园型方程的广义解的定义,且证明了其解的存在性和正则性。     

求解广义互补问题

1 引言 设为一闭凸锥,f是R~n到自身的一映射.广义互补问题,记作GCP(K,f),即找一向量x满足 GCP(K,f) x∈K,f(x)∈且x~Tf(x)=0,(1) 其中,是K的对偶锥(即对任一K中向量x,满足x~Ty≤0的所有y的集合).该问题首先 由Habetler和Price提出.当K=R_+~n(R~n空间的正卦限),此问题就是一般的互补问题.许多作者已经提出了很多求解线性或非线性互补问题的方法.例如:Dafermos,Fukushima,Harker和Price以及其它如参考文献所列.近年来,何针对单调线性变分不等式提出了一些投影收缩算法. Fang在函数是Lipschitz连续及强单调的条件下,在[3]给出一简单的迭代投影法,在[4]中给出一线性化方法去求解广义互补问题(1).在[3]中,他的迭代模式是     

Convergence behaviour of inexact Newton methods under weak Lipschitz condition

Under weak Lipschitz condition, local convergence properties of inexact Newton methods and Newton-like methods for systems of nonlinear equations are established in an arbitrary vector norm. Processes with modified relative residual control are considered; the results easily provide an estimate of convergence ball for inexact methods. For a special case, the results are affine invariant. Some applications are given.     

A Newton-like method for generalized operator equations in Banach spaces

In this paper, we are concerned with the semilocal convergence analysis of a Newton-like method discussed by Bartle (Amer Math Soc 6: 827–831, 1955) to solve the generalized operator equations containing nondifferentiatble term in Banach spaces. This method has also been studied by Rheinboldt (SIAM J Numer Anal 5: 42–63, 1968). The aim of the paper is to discuss the convergence analysis under local Lipschitz condition \(\|F'_{x}-F'_{x_{0}}\|\le \omega (\|x-x_{0}\|)\) for a given point \(x_{0}\) . Our results extend and improve the previous ones in the sense of local Lipschitz conditions. We apply our results to solve the Fredholm-type operator equations.     

A convergence theorem for the Newton-like methods under some kind of weak Lipschitz conditions

We provide sufficient conditions for the convergence of the Newton-like methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. Consequently, some important convergence theorems follow from our main result in this paper.     

Toward a unified theory for third R-order iterative methods for operators with unbounded second derivative

In this paper, we provide a semilocal convergence analysis for a family of Newton-like methods, which contains the best-known third-order iterative methods for solving a nonlinear equation F(x)=0 in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable and F^″ satisfies a Lipschitz type condition but it is unbounded. By using majorant sequences, we provide sufficient convergence conditions to obtain cubic semilocal convergence. Results on existence and uniqueness of solutions, and error estimates are also given. Finally, a numerical example is provided.     

Local properties of algorithms for minimizing nonsmooth composite functions

This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems.This work was done while the author was a Research fellow at the Mathematical Sciences Research Centre, Australian National University.     

Relaxing the convergence conditions for Newton-like methods

Local as well as semilocal convergence theorems for Newton-like methods have been given by us and other authors [1]—[8] using various Lipschitz type conditions on the operators involved. Here we relax these conditions by introducing weaker center-Lipschitz type conditions. This way we can cover a wider range of problems than before in the semilocal case, where as in the local case a larger convergence radius can be obtained in some cases.     

用黎曼求和法(R,1)求和時的基卜斯現象

<正> 假設級數滿足下面兩個條件,即:(甲)在原點的某一鄰域內,對於h(≠0)的一切值級數收斂;     

Local convergence analysis of inexact Newton-like methods under majorant condition

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.     

Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method

In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed graph.We establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of f.Indeed,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular.An application of this method to variational inequality is given.In addition,a numerical experiment is given which illustrates the theoretical result.     

Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces

The aim of this paper is to establish the semilocal convergence of a multipoint third order Newton-like method for solving F(x)=0 in Banach spaces by using recurrence relations. The convergence of this method is studied under the assumption that the second Fréchet derivative of F satisfies Hölder continuity condition. This continuity condition is milder than the usual Lipschitz continuity condition. A new family of recurrence relations are defined based on the two new constants which depend on the operator F. These recurrence relations give a priori error bounds for the method. Two numerical examples are worked out to demonstrate the applicability of the method in cases where the Lipschitz continuity condition over second derivative of F fails but Hölder continuity condition holds.