任意矩阵的行列式与应用

推广行列式的概念,利用格兰姆行列式给出任意一个矩阵的行列式的定义,讨论该行列式的性质,说明其几何意义,并应用于求点到子空间和点到线性流形的距离.     

基于均值的Toeplitz矩阵填充的子空间算法

本文提出一种基于均值的Toeplitz矩阵填充的子空间算法.通过在左奇异向量空间中对已知元素的最小二乘逼近,形成了新的可行矩阵;并利用对角线上的均值化使得迭代后的矩阵保持Toeplitz结构,从而减少了奇异向量空间的分解时间.理论上,证明了在一定条件下该算法收敛于一个低秩的Toeplitz矩阵.通过不同已知率的矩阵填充数值实验展示了Toeplitz矩阵填充的新算法比阈值增广Lagrange乘子算法在时间上和精度上更有效.     

一类线性约束矩阵不等式及其最小二乘问题

本文主要考虑一类线性矩阵不等式及其最小二乘问题,它等价于相应的矩阵不等式最小非负偏差问题.之前相关文献提出了求解该类最小非负偏差问题的迭代方法,但该方法在每步迭代过程中需要精确求解一个约束最小二乘子问题,因此对规模较大的问题,整个迭代过程需要耗费巨大的计算量.为了提高计算效率,本文在现有算法的基础上,提出了一类修正迭代方法.该方法在每步迭代过程中利用有限步的矩阵型LSQR方法求解一个低维矩阵Krylov子空间上的约束最小二乘子问题,降低了整个迭代所需的计算量.进一步运用投影定理以及相关的矩阵分析方法证明了该修正算法的收敛性,最后通过数值例子验证了本文的理论结果以及算法的有效性.     

FC-空间中的参数型KKM定理及其应用

首先在Fc-空间中证明了一个特征性质,然后通过引进线性序空间,利用集合的连通性,在非紧的FC-空间中证明了一个参数型KKM定理.应用参数型KKM定理得到非紧的极大极小不等式、鞍点定理和截口定理.这些结果改进和推广了这个领域中的一些相关结果.     

新的滤子方法

本文定义了一种新的滤子方法,并提出了求解光滑不等式约束最优化问题的滤子QP-free非可行域方法.通过乘子和分片线性非线性互补函数,构造一个等价于原约束问题一阶KKT条件的非光滑方程组.在此基础上,通过牛顿-拟牛顿迭代得到满足KKT最优条件的解,在迭代中采用了滤子线搜索方法,证明了该算法是可实现,并具有全局收敛性.另外,在较弱条件下可以证明该方法具有超线性收敛性.     

新的滤子方法(英)

本文定义了一种新的滤子方法,并提出了求解光滑不等式约束最优化问题的滤子QP-free非可行域方法. 通过乘子和分片线性非线性互补函数,构造一个等价于原约束问题一阶KKT条件的非光滑方程组.在此基础上, 通过牛顿-拟牛顿迭代得到满足KKT最优条件的解,在迭代中采用了滤子线搜索方法,证明了该算法是可实现,并具有全局收敛性. 另外,在较弱条件下可以证明该方法具有超线性收敛性.     

Choi不等式的一些研究和应用

1974年,Man-DuenChoi结合正定线性算子定义,给出了矩阵分析学中的一个重要不等式,这个不等式是Fiedler不等式的推广,利用一些正定矩阵的性质,给出Choi不等式的一个新的推到方法,并且用该不等式给出了Fiedler不等式的一个新证明.     

利用辛空间上子空间构作(d,r;z]-disjunct矩阵

一个(d,r;z]-disjunct矩阵在许多领域有着极为广泛的应用.利用辛空间上m维(m,s)型子空间的性质构作了(d,r;z]-disjunct矩阵,并利用子空间的计数定理计算了它的参数.     

解决大规模信赖域子问题的一种新算法

信赖域方法是解决无约束优化问题的一类有效的方法,而求解信赖域子问题又是信赖域方法的一个重要的组成部分。在本文中,我们首先介绍Hager[4]的序列子空间方法,并分析了对于不同的子空间序列,该算法所具有的性质。随后我们在以上分析的启发下,给出SSM算法的一种改进算法,改进后的算法不仅是全局收敛的,而且进一步减少了矩阵运算量。最后我们给出一些初步的数值试验报告。     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

基于“直觉模糊集的距离测度”的联想

直觉模糊集不同于模糊集的最重要指标是犹豫度.在直觉模糊集的距离定义和公式中应突出体现犹豫度;在直觉模糊集的距离定义中也应包含经典距离的性质—三角不等式.基于论文"直觉模糊集的距离测度",提出直觉模糊集的新距离定义和新公式.应用实例表明,新提出的直觉模糊集的距离公式是实用的.     

The matrix valued triangle inequality:quaternion version

The matrix valued triangle inequality is shown to hold for matrices over the quaternions. The history of the inequality and open questions connected with it are described.     

Notes on the Norm Estimates for the Sum of Two Matrices

This is a lecture note of my joint work with Chi-Kwong Li concerning various results on the norm structure of n x n matrices (as Hilbert-space operators). The main result says that the triangle inequality serves as the ultimate norm estimate for the upper bounds of summation of two matrices.In the case of summation of two normal matrices, the result turns out to be a norm estimate in terms of the spectral variation for normal matrices.     

Traveling salesman games

In this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited. We introduce two types of cooperative games—fixed-route traveling salesman games and traveling salesman games—as a tool to attack this problem. Under very mild conditions we prove that fixed-route traveling salesman games have non-empty cores if the fixed route is a solution of the classical traveling salesman problem. Core elements provide us with fair cost allocations. A traveling salesman game may have an empty core, even if the cost matrix satisfies the triangle inequality. In this paper we introduce a class of matrices defining TS-games with non-empty cores.     

On the distance function of a connected graph

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.     

The generalized main inequality of Reich-Strebel and its applications

In this paper,a new version of the general form of the main inequality of Reich-Strebel is given.As applications,we improve the strong triangle inequality and generalize the Delta inequality in certain sense.     

Refinements of generalized triangle inequalities

In this paper, we discuss refinements of the well-known triangle inequality and it is reverse inequality for strongly integrable functions with values in a Banach space X. We also discuss refinement of a generalized triangle inequality of the second kind for Lp functions. For both cases, the attainability of the equality is also investigated.     

Fuzzy partitions and relations; an axiomatic basis for clustering

In this paper some connections between fuzzy partitions and similarity relations are explored. A new definition of transitivity for fuzzy relations yields a relation-theoretic characterization of the class of all psuedo-metrics on a fixed (finite) data set into the closed unit interval. This notion of transitivity also links the triangle inequality to convex decompositions of fuzzy similarity relations in a manner which may generate new techniques for fuzzy clustering. Finally, we show that every fuzzy c-partition of a finite data set induces a psuedo-metric of the type described above on the data.     

A rigidity theorem in Alexandrov spaces with lower curvature bound

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.     

Minimax Theorems with Bregman Distance in Banach Spaces

Since Bregman distance is a generalized distance function which does not satisfy the triangle inequality nor symmetry, it is important to study minimax theorems in reflexive Banach spaces. Our minimax theorems differ from those in previous literature.