KS张量互补问题的稀疏解（英文）

针对KS张量互补问题,本文研究了该问题的稀疏解.由于l₀范数的非凸性和非连续性,求解KS张量互补问题的稀疏解是一个NP难问题.为了解决这一问题,我们将其转化为一个带约束的多项式优化问题,然后用序列二次规划(SQP)算法求解转化的问题.数值结果表明,该算法能有效地求解KS张量互补问题的稀疏解.     

稀疏优化在数独中的应用

数独是一个难以求解的整数规划问题,可以通过实数编码的方式去除整数约束的限制,将整数规划模型转化为一个l0范数极小化模型.已有算法大多是求解松弛的l1范数极小化模型,只能求解部分数独问题.本文证明对于数独这样一个特殊的问题,lq(0＜q＜1)范数极小化模型等价于l0范数极小化模型,同时用l1/2-SLP(sequenti...     

基于正弦型光滑打磨函数对0-1规划问题的连续化求解方法

传统的求解0-1规划问题方法大多属于直接离散的解法.现提出一个包含严格转换和近似逼近三个步骤的连续化解法:(1)借助阶跃函数把0-1离散变量转化为[0,1]区间上的连续变量;(2)对目标函数采用逼近折中阶跃函数近光滑打磨函数,约束条件采用线性打磨函数逼近折中阶跃函数,把0-1规划问题由离散问题转化为连续优化模型;(3)利用高阶光滑的解法求解优化模型.该方法打破了特定求解方法仅适用于特定类型0-1规划问题惯例,使求解0-1规划问题的方法更加一般化.在具体求解时,采用正弦型光滑打磨函数来逼近折中阶跃函数,计算效果很好.     

基于新光滑函数的P0映射非线性互补问题的光滑牛顿法(英文)

非线性互补问题(NCP)可以重新表述为一个非光滑方程组的解.通过引入一个新的光滑函数,将问题近似为参数化光滑方程组.基于这个光滑函数,我们提出了一个求解P₀映射和R₀映射非线性互补问题的光滑牛顿法.该算法每次迭代只求解一个线性方程和一次线搜索.在适当的条件下,证明了该方法是全局和局部二次收敛的.数值结果表明,该算法是有效的.     

P0函数非线性互补问题的非内部连续化算法

提出了一种新的光滑函数,它具有现存的一些光滑函数不具备的性质.基于此光滑函数,讨论了求解P₀函数非线性互补问题的光滑路径的存在性和连续性.在非线性互补问题的解集非空有界的假设下,利用新光滑函数的特性,研究了求解P₀函数非线性互补问题的非内部连续化算法得到的迭代序列的有界性.解集非空有界的条件弱于一些现存的求解非线性互补问题的连续化算法所要求的假设条件.     

鲁棒稀疏重构问题的凝聚同伦算法

鲁棒稀疏重构问题是信号处理领域的重要问题,该问题的数学本质是一个NP难的数学优化问题.同伦算法是一类典型的路径跟踪算法,该算法是解非线性问题的一类成熟算法,具有全局收敛性,且易于并行实现.本文考虑同伦算法在鲁棒稀疏重构问题中的数值求解.基于l_∞范数及罚函数策略,我们首先将原始的基于l_0范数的最优化模型,转化为含参数的无约束极大极小值问题,进而构造凝聚函数光滑化模型中的极大值函数,并构造凝聚同伦算法数值求解.数值仿真实验验证了新方法的有效性,为大规模鲁棒重构问题的并行化数值求解奠定基础.     

带有P0函数的非线性互补问题的一个新的非内点连续算法

研究带有P₀函数的非线性互补问题. 基于一个新的光滑函数, 把问题近似成参数化的光滑方程组, 并且给出一个新的非内点连续算法. 所给算法在每步迭代只需要求解一个线性方程组和执行一次Armijo类型的线搜索. 在不需要严格互补条件的情况下, 证明了算法是全局收敛和超线性收敛的. 并且, 在一个较弱的条件下该算法具有局部二阶收敛性. 数值实验证实了算法的可行性和有效性.     

基于SCN函数共轭梯度方向的稀疏向量特征提取算法

稀疏向量特征提取是指在优化时利用各种范数对解进行约束,从而获得带有稀疏特征的最优解,其广泛应用于复杂系统中的机器学习、深度学习和大数据分析等领域的特征提取问题.大量的研究表明各种范数如L₀范数、L₁范数和L₂范数的方法都存在各自的缺点,主要表现在越容易求解的范数越不精准稀疏,越精准稀疏的范数越难求解.文章提出了一种基于SCN函数共轭梯度方向的稀疏向量特征发现算法(CGDL),稀疏向量特征发现可以用一个稀疏特征提取优化模型建立,其目标函数是一个SCN函数,对其中的L₀范数进行转换,形成一个具有特殊结构优化问题,这个问题等价于双层规划的凸-凹极小极大化问题,这类问题可以解决稀疏回归、图像特征和压缩感知等问题.文章给出了上述模型的稀疏特征提取算法的详细计算步骤和收敛性分析证明,并且对给定的实际数据集和高维模拟数据集对算法的有效性、复杂性和收敛速度进行了数值对比实验,表明了该算法在精准度和稀疏性上显著优于其他对比方法,并且具有较好的收敛速度.     

最小化三个凸函数之和的一个简单原始-对偶算法

提出一个简单的原始-对偶算法求解三个凸函数之和的最小化问题, 其中目标函数包含有梯度李普希兹连续的光滑函数, 非光滑函数和含有复合算子的非光滑函数. 在新方法中, 对偶变量迭代使用预估-矫正的方案. 分析了算法的收敛性和收敛速率. 最后, 数值实验说明了算法的有效性.     

一个求解P0函数非线性互补问题的非内部连续化算法

基于黄正海等2001年提出的光滑函数,本文给出一个求解P0函数非线性互补问题的非内部连续化算法.所给算法拥有一些好的特性.在较弱的条件下,证明了所给算法或者是全局线性收敛,或者是全局和局部超线性收敛.给出了所给算法求解两个标准测试问题的数值试验结果.     

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.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

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.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

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.     

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