一个新的求解半正定规划问题的原始对偶内点算法

选择合适的核函数对设计求解线性规划与半正定规划的原始对偶内点算法以及复杂性分析都十分重要.Bai等针对线性规划提出三种核函数,并给出求解线性规划的大步迭代复杂界,但未给出数值算例验证算法的实际效果(Bai Y Q,Xie W,Zhang J.New parameterized kernel functions for linear optimization.J Global Optim,2012.DOI 10.1007/s10898-012-9934-z).基于这三种核函数设计了新的求解半正定规划问题的原始对内点算法.进一步分析了算法关于大步方法的计算复杂性界,同时通过数值算例验证了算法的有效性和核函数所带参数对计算复杂性的影响.     

二阶锥规划的基于自协调指数核函数的原始-对偶内点算法

基于一个自协调指数核函数, 设计求解二阶锥规划的原始-对偶内点算法. 根据自协调指数核函数的二阶导数与三阶导数的特殊关系, 在求解问题的中心路径时, 用牛顿方向代替了负梯度方向来确定搜索方向. 由于自协调指数核函数不具有``Eligible'性质, 在分析算法的迭代界时, 利用牛顿方法求解目标函数满足自协调性质的无约束优化问题的技术, 估计算法内迭代中自协调指数核函数确定的障碍函数的下降量, 得到原始-对偶内点算法大步校正的迭代界O(2N\frac{\log2N}{\varepsilon}), 这里N是二阶锥的个数. 这个迭代界与线性规划情形下的迭代界一致. 最后, 通过数值算例验证了算法的有效性.     

一个求解半正定规划问题的新原始-对偶内点算法

在原始对偶内点算法的设计和分析中,障碍函数对算法的搜索方法和复杂性起着重要的作用.本文由核函数来确定障碍函数,设计了一个求解半正定规划问题的原始-对偶内点算法.这个障碍函数即可以定义算法新的搜索方向,又度量迭代点与中心路径的距离,同时对算法的复杂性分析起着关键的作用.我们计算了算法的迭代界,得出了关于大步校正法和小步校正法的迭代界,它们分别是O(√n log n 10g n/ε)和O(√n log n/ε),这里n是半正定规划问题的维数.最后,我们根据一个算例,说明了算法的有效性以及对核函数的参数的敏感性.     

一个求解半正定规划问题的新原始一对偶内点算法

在原始对偶内点算法的设计和分析中,障碍函数对算法的搜索方法和复杂性起着重要的作用。本文由核函数来确定障碍函数,设计了一个求解半正定规划问题的原始。对偶内点算法。这个障碍函数即可以定义算法新的搜索方向,又度量迭代点与中心路径的距离,同时对算法的复杂性分析起着关键的作用。我们计算了算法的迭代界,得出了关于大步校正法和小步校正法的迭代界,它们分别是O（√n log n log n/c）和O（√n log n/ε）,这里n是半正定规划问题的维数。最后,我们根据一个算例,说明了算法的有效性以及对核函数的参数的敏感性。     

求解约束规划的一个非线性Lagrange函数

本文提出了一个求解不等式约束优化问题的非线性Lagrange函数,并构造了基于该函数的对偶算法.证明了当参数σ小于某一阈值σ_0时,由算法生成的原始-对偶点列是局部收敛的,并给出了原始-对偶解的误差估计.此外,建立了基于该函数的对偶理论.最后给出了算法的数值结果.     

二阶锥线性互补问题内点法的复杂性

本文提出一个二阶锥线性互补问题的长步原始对偶内点法,搜索方向由一个一般的核函数来定义.如果给出初始的严格内点,可以得到本算法的复杂性为O((1+2k)llog(lμ0/ε)).     

基于局部self-concordant障碍函数的全牛顿步多项式时间算法

由Nesterov和Nemirovski[4]创立的self-concordant障碍函数理论为解线性和凸优化问题提供了多项式时间内点算法.根据self-concordant障碍函数的参数,就可以分析内点算法的复杂性.在这篇文章中,我们介绍了基于核函数的局部self-concordant障碍函数,它在线性优化问题的中心路径及其邻域内满足self-concordant性质.通过求解此障碍函数的局部参数值,我们得到了求解线性规划问题的基于此局部Self-concordant障碍函数的纯牛顿步内点算法的理论迭代界.此迭代界与目前已知的最好的理论迭代界是一致的.     

关于求解DEA原始CCR模型中最优输入输出权重的方法

本文给出了求解DEA原始CCR模型中最优输入输出权重的简便方法：首先将原始CCR模型化为线性规划模型，然后从该线性规划模型的对偶模型入手，运用单纯形法，在得到决策单元最优效率评价指数时，根据线性规划的对偶理论，得到决策单元最优输入输出权重。该权重可用在逆DEA新算法中。     

使用自正则度量的凸二次规划的原始对偶内点法的多项式复杂性

最近Peng等人使用新的搜索方向和自正则度量为求解线性规划问题提出了一个原始对偶内点法.本文将这个长步法延伸到凸二次规划.在线性规划情形时,原始空间和对偶空间中的尺度Newton方向是正交的,而在二次规划情形时这是不成立的.本文将处理这个问题并且证明多项式复杂性,并且得到复杂性的上界为O(n√log n log (n/ε)).     

一种基于核心矩阵的原始对偶算法

本文通过对线性规划问题中的核心矩阵的分析，提出了一种基于核心矩阵的原始对偶算法。该算法以核心矩阵为运算单元，一方面呈现了存储空间小，计算量小的特点；另一方面，该算法采用了一种新的转轴规则的外点算法，在保持原始可行的基础上，不断改善对偶解使其可行。数值实验结果表明该算法在迭代次数、转轴效率和存储空间上都有一定的提高。     

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 REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

CONTENTS OF VOLUME 30

正~~     

Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].     

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.     

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.