两个半光滑函数之和的非光滑方程组解法

对两个半光滑函数之和F（x）=F1（x）＋F2（x），其中F1，F2均为半光滑函数，给出了求解F（x）=0的一种广义牛顿法．算法在每一迭代点处分别计算中一个元素，而不需计算中元素．     

非光滑方程组牛顿法的全局收敛性分析

通过定义一种新的*-微分,本文给出了局部Lipschitz非光滑方程组的牛顿法,并对其全局收敛性进行了研究.该牛顿法结合了非光滑方程组的局部收敛性和全局收敛性.最后,我们把这种牛顿法应用到非光滑函数的光滑复合方程组问题上,得到了较好的收敛性.     

二阶锥约束随机变分不等式问题的数值方法研究

本文研究二阶锥约束随机变分不等式(SOCCSVI)问题,运用样本均值近似(SAA)方法结合光滑Fischer-Burmeister互补函数来求解该问题.首先,将SOCCSVI问题的Karush-Kuhn-Tucker系统转化为与之等价的方程组,并证明了该方程组的雅可比矩阵的非奇异性.其次,构造了光滑牛顿算法求解该方程组.最后,文章给出了两个数值实验证明了算法的有效性.     

求解圆锥规划的光滑牛顿法

圆锥规划是一类重要的非对称锥优化问题.基于一个光滑函数,将圆锥规划的最优性条件转化成一个非线性方程组,然后给出求解圆锥规划的光滑牛顿法.该算法只需求解一个线性方程组和进行一次线搜索.运用欧几里得约当代数理论,证明该算法具有全局和局部二阶收敛性.最后数值结果表明算法的有效性.     

求解半光滑方程组的近似Newton法

本文提出了求解半光滑方程组的近似Newton法，并证明了该算法的局部超线性收敛性。数值结果表明 该算法是有效的。     

复合凸优化的快速邻近点算法

在大数据时代,随着数据采集手段的不断提升,大规模复合凸优化问题大量的出现在包括统计数据分析,机器与统计学习以及信号与图像处理等应用中.本文针对大规模复合凸优化问题介绍了一类快速邻近点算法.在易计算的近似准则和较弱的平稳性条件下,本文给出了该算法的全局收敛与局部渐近超线性收敛结果.同时,我们设计了基于对偶原理的半光滑牛顿法来高效稳定求解邻近点算法所涉及的重要子问题.最后,本文还讨论了如何通过深入挖掘并利用复合凸优化问题中由非光滑正则函数所诱导的非光滑二阶信息来极大减少半光滑牛顿算法中求解牛顿线性系统所需的工作量,从而进一步加速邻近点算法.     

解约束优化的分段线性有理NCP函数

本文给出新的NCP函数,这些函数是分段线性有理正则伪光滑的,且具有良好的性质.把这些NCP函数应用到解非线性优化问题的方法中.例如,把求解非线性约束优化问题的KKT点问题分别用QP-free方法,乘子法转化为解半光滑方程组或无约束优化问题.然后再考虑用非精确牛顿法或者拟牛顿法来解决该半光滑方程组或无约束优化问题.这个方法是可实现的,且具有全局收敛性.可以证明在一定假设条件下,该算法具有局部超线性收敛性.     

一类非线性Schrodinger-Maxwell方程组半经典解的存在性

本文将研究如下非线性Schrodinger—Maxwell方程组问题 {-ε^2△u＋V（x）u＋K（x）Фu=｜u｜^p-2u, x∈R^3, -△Ф=4πK（x）u^2, x∈R^3. 当势函数V（x）和电量函数K（x）满足一定假设条件时,作者利用变分法证明了ε充分小时,该方程组半经典解的存在性．     

一类二次规划逆问题的交替方向数值方法

考虑求解一类二次规划逆问题的交替方向数值算法．首先给出矩阵变量子问题解的显示表达式,而后构造了两个求解向量变量子问题近似解的数值算法,其中一个算法基于不动点原理,另一算法则应用半光滑牛顿法．数值实验表明,所提出的算法能够快速高效地求解二次规划逆问题．     

样本线性相关系数的几何解释

经全国中小学教材审定委员会2003年审查通过的人教版全日制高中教科书《数学》第三册（选修Ⅱ）之“§1．6线性回归”中给出了样本相关系数r=^n∑i=1（xi-^-x）（yi-^-y）/^n∑i=1（xi-^-x）^2^n∑i=1（yi-^-y）^2，并指出其性质“｜r｜≤1，且｜r｜越接近1，相关程度越大；｜r｜越接近于0，相关程度越小。     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.