正则局部Lipschitz函数可微性的几何条件

Ｂａｎａｃｈ空间的范数可微性可用它的单位球面来刻画．本文把这些范数可微性的几何条件推广到正则局部Ｌｉｐｓｃｈｉｔｚ函数情形．     

微分模式

１．论述ｄｘ与△ｘ的关系；ｄｘ是从△ｘ→０转化而来．比△ｘ更为抽象；２对多元函数引出几．种可微性概念，更加细致地刻划其可微性。     

关于二元函数可微性的判定

欲判定二元函数f（x，y）在一点处的可做性，首先要搞清可微性以及可微与连续、偏导数存在及偏导数连续等特性之间的相互关系，可用框图表示如下：其中记号“A一B”表示“A可以推出B”或“A蕴涵B”；记号“A～B”表示“由A不能推出B”或“A不蕴涵B”。因此，通常检查一个函数是否可微，先看它是否连续，如不连续，则不可做。例1讨论函数在（0，0）处的可微性。＿＿I，工y卜天又是、。。，。。。，，。、、。，。。解：因h平上六一timdri一／七，其值因k而异，极限limf（，y）不存在，所—”一；GH“＋v‘二二台（十人旬又‘1十月’，…     

线性空间上凸函数的可微性与次可微性

讨论了定义在线性空间上的广义实值凸函数的可做性和次可微性．     

Banach空间中广义集值拟变分包含的灵敏性分析

研究了Banach空间中一类广义集值拟变分包含问题的灵敏性分析．利用预解算子的技巧,在对给定条件没有假设可微性和单调性下,建立了这类问题与广义预解方程类的等价性．     

关于Norton问题

本文证明了ＮｏｒｔｏｎＡ．所提出的猜测．在映射具有适当可微性的假设下,对 秩为ｒ的集合,刻画了它的像的Ｈａｕｓｄｏｒｆｆ测度．     

新超复结构和Clifford分析

本文在Ｈ．Ｍａｌｏｎｅｋ［１］新的超复结构的意义下，研究Ｃｌｉｆｆｏｒｄ分析中Ａ值函数超复可微性及其一种新的锁链规则和微分法则．     

关于多元函数可微性的一个注记

对全微分的存在性进行了讨论．给出了某类多元函数z=f（P）在点Po处可微的一个判别方法     

可微cantor型函数的构造

可微Ｃａｎｔｏｒ型函数的存在性仅是从理论上给出证明，本文给出一个具体的构造，以使我们可以更好了解这种函数的本质．     

强不变原理与完全收敛性的统一形式

本文建立了强不变原理与完全收效性的统一形式，得到了用强不变原理研究完全收敛性的方法，前人许多关于强不交原理和完全收敛性的结论是本文的推论．     

线性方程组逼近双障碍问题

针对双障碍问题,本文提出了与其等价的B可微方程的类Broyden算法,并 在一定条件下证明了该算法的全局收敛性和超线性收敛性.经数值实验表明该算法是 有效的.     

INEXACT DAMPED NEWTON METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

In this paper, we propose an inexact clamped Newton method for solving nonlinear complementarity problems based on the equivalent B-differentiable equations.Global convergence and locally quadratic convergence are obtained,and numerical results are given.     

关于广义Newton法的收敛性问题

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。     

Newton's method for the nonlinear complementarity problem: A B-differentiable equation approach

This paper describes a damped-Newton method for solving the nonlinear complementarity problem when it is formulated as a system of B-differentiable equations through the use of the Minty-map. This general Newton algorithm contains a one-dimensional line search and possesses a global convergence property under certain conditions; modifications and heuristic implementations of the algorithm for the case when these conditions do not hold are also discussed. The numerical experiments show that, in general, this new scheme is more efficient and robust than the traditional Josephy-Newton algorithm.     

A B-differentiable equation-based,globally and locally quadratically convergent algorithm for nonlinear programs,complementarity and variational inequality problems

This paper presents a globally convergent, locally quadratically convergent algorithm for solving general nonlinear programs, nonlinear complementarity and variational inequality problems. The algorithm is based on a unified formulation of these three mathematical programming problems as a certain system of B-differentiable equations, and is a modification of the damped Newton method described in Pang (1990) for solving such systems of nonsmooth equations. The algorithm resembles several existing methods for solving these classes of mathematical programs, but has some special features of its own; in particular, it possesses the combined advantage of fast quadratic rate of convergence of a basic Newton method and the desirable global convergence induced by one-dimensional Armijo line searches. In the context of a nonlinear program, the algorithm is of the sequential quadratic programming type with two distinct characteristics: (i) it makes no use of a penalty function; and (ii) it circumvents the Maratos effect. In the context of the variational inequality/complementarity problem, the algorithm provides a Newton-type descent method that is guaranteed globally convergent without requiring the F-differentiability assumption of the defining B-differentiable equations.This work was based on research supported by the National Science Foundation under Grant No. ECS-8717968.     

求解非线性互补问题的一种序列线性方程组方法

１　引　言 设Ｆ：Ｒｎ→Ｒｎ．则非线性互补问题ＮＣＰ（Ｆ）的形式如下：求ｘ∈ＲＮ,使ＮＣＰ（Ｆ）是如下变分不等式ＶＩ（Ｆ,Ｘ）的一种重要形式：求ｘ∈Ｘ　Ｒ　使当Ｘ＝Ｒｎ＋时,ＶＩ（Ｆ,Ｘ）即为ＮＣＰ（Ｆ）．由于ＮＣＰ和ＶＩ在工程和经济等领域中有广泛的应用,因而,对其研究受到了很大的重视．目前,关于（１．２）的求解已发展了一系列算法,线性化方法是常用的一类算法．线性化方法的局部收敛性研究已有了许多好的结果（见［９,１０］等）．全局收敛性成为了当前研究ＶＩ（Ｆ,Ｘ）算法的一个热门课题．并在Ｎｅｗｔｏ…     

Local convergence of quasi-Newton methods for B-differentiable equations

We study local convergence of quasi-Newton methods for solving systems of nonlinear equations defined by B-differentiable functions. We extend the classical linear and superlinear convergence results for general quasi-Newton methods as well as for Broyden's method. We also show how Broyden's method may be applied to nonlinear complementarity problems and illustrate its computational performance on two small examples.     

On the convergence of quasi-newton methods for nonsmooth problems

We develop a theory of quasi-New ton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that Fcan be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.     

On the qualitative approximation of Lipschitzian functions

Problem of topological equivalence between a function and certain local approximations is studied. The study is carried out in a neighbourhood of a critical point with the concept of critical point of Clarke's theory. The function belongs to a particular class of non B-differentiable functions. The local approximations are positively homogeneous maps. Using the concept of topological equivalence we establish the existence of a local coordinate transformation between the original function and the positively homogeneous function. As a consequence we obtain sufficient conditions for the existence of local extremes for the initial map.     

Directional derivatives of the solution of a parametric nonlinear program

Consider a parametric nonlinear optimization problem subject to equality and inequality constraints. Conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known. We show, under the additional assumption of constant rank of the active constraint gradients, that the optimal solution is actually piecewise smooth, hence B-differentiable. We show, for the first time to our knowledge, a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique.This author's research was supported by the Australian Research Council.