奇异值函数的二阶方向导数

本文用一个直接的方法给出了奇异值函数的二阶方向导数公式. 作为应用, 利用这一公式建立了谱范数的上图集合与核范数的上图集合的切锥和二阶切集的具体表达式, 这些表达式在矩阵优化的一阶和二阶最优条件的研究中起着重要作用.     

一类不可微二次规划逆问题

本文求解了一类二次规划的逆问题,具体为目标函数是矩阵谱范数与向量无穷范数之和的最小化问题.首先将该问题转化为目标函数可分离变量的凸优化问题,提出用G-ADMM法求解.并结合奇异值阈值算法,Moreau-Yosida正则化算法,matlab优化工具箱的quadprog函数来精确求解相应的子问题.而对于其中一个子问题的精确求解过程中发现其仍是目标函数可分离变量的凸优化问题,由于其变量都是矩阵,所以采用适合多个矩阵变量的交替方向法求解,通过引入新的变量,使其每个子问题的解都具有显示表达式.最后给出采用的G-ADMM法求解本文问题的数值实验.数据表明,本文所采用的方法能够高效快速地解决该二次规划逆问题.     

负相协随机变量非随机和的差的精确大偏差

In this paper,we study precise large deviation for the non-random difference sum from j=1 to n_1(t) X_(1j)-sum from j=1 to n_2(t) X_(2j),where sum from j=1 to n_1(t) X_(1j) is the non-random sum of {X_(1j),j≥1} which is a sequence of negatively associated random variables with common distribution F_1(x),and sum from j=1 to n_2(t) X_(2j) is the non-random sum of {X_(2j),j≥1} which is a sequence of independent and identically distributed random variables,n_1(t) and n_2(t) are two positive integer functions.Under some other mild conditions,we establish the following uniformly asymptotic relation lim t→∞ sup x≥r(n_1(t))~(p+1)|(P(∑~(n_1(t)_(j=1)X_(1j)-∑~(n_2(t)_(j=1)X_(2j)-(μ_1n_1(t)-μ_2n_2(t)x))/(n_1(t)F_1(x))-1|=0.     

求解一类逆鲁棒优化问题的牛顿型扰动方法

逆优化问题是指通过调整目标函数和约束中的某些参数使得已知的一个解成为参数调整后的优化问题的最优解.本文考虑求解一类逆鲁棒优化问题.首先,我们将该问题转化为带有一个线性等式约束,一个二阶锥互补约束和一个线性互补约束的极小化问题;其次,通过一类扰动方法来对转化后的极小化问题进行求解,然后利用带Armijo线搜索的非精确牛顿法求解每一个扰动问题.最后,通过数值实验验证该方法行之有效.     

基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析

We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f~δ. Instead of the observation L~2 space, we perform the reconstruction of the derivative in a weighted L~2 space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.     

调和Bergman空间上Toeplitz算子的代数性质

In this paper, we first investigate the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols on the harmonic Bergman space. Next, we characterize finite rank commutators and semi-commutators of two Toeplitz operators with quasihomogeneous symbols.     

关于一个普遍本源公式及其应用

Here presented is a further investigation on a general source formula(GSF) that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author's recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31(3):260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308:3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the(Σ?D) formula class are given in the closing section.     

K-理论在一类算子逼近中的应用

构造了一类强不可约的Cowen—Douglas算子,刻画了其换位代数的K_0-群.我们证明了:对于复可分的Hilbert空间上的有界线性算子T及ε0,总可以找到由有限多个强不可约的Cowen-Douglas算子构成的直和算子S,使得||T-S||ε.     

数学机械化与数学道理化

以中国传统文化为底蕴、机械化和道理化为特色的中国传统数学,曾经取得光辉的成就.将这些思想用于计算机科学、孤子理论和量子场论,不仅可以系统地产生已有结果,还可以开辟新的研究方向,取得重大突破.     

非线性半定规划若干进展

讨论非线性半定规划的四个专题,包括半正定矩阵锥的变分分析、非凸半定规划问题的最优性条件、非凸半定规划问题的扰动分析和非凸半定规划问题的增广Lagrange方法．