基于全牛顿求解P*(κ)阵水平线性互补问题的内点算法

提出一种求解P_*(k)阵水平线性互补问题的全牛顿内点算法,全牛顿算法的优势在于每次迭代中不需要线性搜寻.当给定适当的中心路径邻域的阈值和更新势垒参数,证明算法中心邻域的全牛顿是局部二次收敛的,最后给出算法迭代复杂性O(√n)log(n+1+k)/εμ₀.     

定比分点向量公式、向量基本定理的理解

<正>一、定比分点向量公式如图1,设P₁（x1.y1）、P₂（x2,y2）为直线l上的两点,点P是l上不同于P₁、P₂的任一点,则存在一个实数λ,使■=λ■,λ叫做点P分有向线段■所成的比,则■=     

双会议服务器选址问题研究

中位选址问题一直是管理学科的研究热点,本文考虑平面点集选址问题中的双会议服务器选址问题,该问题可以看成是2中位问题的衍生问题。令P为平面上包含n个点的点集,双会议服务器选址问题即为寻找由该点集构成的一棵二星树,使得这棵树上所有叶子之间的距离和最小。本文给出求解该问题的关键几何结构和最优解算法设计,并证明所给算法时间复杂性为O(n³logn)。     

给定度序列图的覆盖成本和反向覆盖成本的研究

具有n个顶点且度序列为(m,2,…,2,1,…,1)(1的重数为m)的连通图不止一个(这些图均为树),而每个树对应唯一一个段序列(l₁,l₂,…,l_m).通过对任意一树移动最长段的悬挂点到最短段悬挂点的方式得到另一树,比较前后两树的覆盖成本和反向覆盖成本,给出了具有最小覆盖成本和反向覆盖成本的极树,并且进一步给出了取得最小覆盖成本和反向覆盖成本的顶点.     

关于非均匀圆柱型正交各向异性图板材料性质的研究

在我们已发表的一文中,根据梁的弯曲理论得到了非均匀圆柱型正交各向异性圆板的折算刚度随半径按指数变化的规律.本文的目的在于进一步证明半径对材料性质的影响.按照平面应力一应交关系,我们得到材料性质的数值E_γ,E_θ,ν_θγ是半径γ的函数.我们所得到的解析结果与实验结果是一致的.     

数学问题解答

2021年1月号问题解答(解答由问题提供人给出)2581如图1,半径为r、R的⊙B、⊙C外切于点A(R>r),两圆的一条外公切线与⊙B相切于点D,与⊙C相切于点E,点H₁、H₂在BC上,且BH₁=CH₂.过点A作DE的垂线,与过点H₁垂直于BC的直线相交于点F₁与过点H₂垂直.于BC的直线相交于点F₂.求证.     

极大平面图的算术——几何指标（英文）

给定一个具有点集V(G)={v₁,v₂,…,v_n}和边集E(G)的图G,其算术—几何指标AG定义为■其中d_i表记G中点v_i的度数.最近,该拓扑指标在化学图论中被引入并得到广泛研究.本文分别刻画了直径为2且最小度为4的n阶极大平面图中具有最大和最小AG-指标的极大平面图.     

随机环境下两个上临界分支过程的参数比较

设(Z_1,n)_n≥0和(Z_2,n)_n≥0是两个在独立同分布随机环境下的上临界分支过程,并且其关键参数分别为μ₁和μ₂.容易知道,在适当条件下,■和■分别依概率收敛到μ₁和μ₂.该文旨在讨论两个上临界分支过程的关键参数之差μ₁-μ₂的估计问题,它可以被看作是一类双样本U统计量问题.我们得到了■的中心极限定理,非一致性Berry-Esseen估计和Cramér型中偏差.最后,作为应用部分,指出了以上的结果可用于关键参数置信区间的构造.     

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.     

具有学习效应的两台机上最优混合流水作业算法

研究相同工件在两台机器（分别称为机器M₁和M₂）上的混合流水作业问题,每个给定工件有两个任务,分别称之为任务A和任务B,任务B只能在任务A完工后才能开始加工,每个工件有两种加工模式供选择：模式1是将两个任务都安排在机器M₂上加工;模式2是将任务A和B分别安排在机器M₁和M₂上加工.假设在加工工件时,机器具有学习效应,即工件的实际加工时间与工件的加工位置有关.目标函数是最小化最大完工时间.分别讨论了具有无缓冲区与无限缓冲区两种加工环境情况,两种情况下都得到了最优算法.     

Steinhaus问题及其证明

本文利用Pythagoras数组的性质,导出了与此问题等价的相关量的表述,证明了可以按某种方式把平面上的点划分为不相交的四类点集,而在每一类点集中都不存在整距点.     

Best proximity point results in non-Archimedean fuzzy metric spaces

We consider the problem of finding a best proximity point which achieves the minimum distance between two nonempty sets in a non-Archimedean fuzzy metric space. First we prove the existence and uniqueness of the best proximity point by using different contractive conditions, then we present some examples to support our best proximity point theorems.     

The gradual covering decay location problem on a network

In covering problems it is assumed that there is a critical distance within which the demand point is fully covered, while beyond this distance it is not covered at all. In this paper we define two distances. Within the lower distance a demand point is fully covered and beyond the larger distance it is not covered at all. For a distance between these two values we assume a gradual coverage decreasing from full coverage at the lower distance to no coverage at the larger distance.     

重大突发事件应急设施多重覆盖选址模型及算法

为了解决应对重大突发事件过程中应急需求的多点同时需求和多次需求问题,本文研究了应对重大突发事件的应急服务设施布局中的覆盖问题:针对重大突发事件应急响应的特点,引入最大临界距离和最小临界距离的概念,在阶梯型覆盖质量水平的基础上,建立了多重数量和质量覆盖模型。模型的优化目标是满足需求点的多次覆盖需求和多需求点同时需求的要求条件下,覆盖的人口期望最大,并用改进的遗传算法进行求解;最后给出的算例证明了模型和算法的有效性,从而应急设施的多重覆盖选址模型能够为有效应对重大突发事件的应急设施选址决策提供参考依据。     

The directional p-median problem: Definition,complexity, and algorithms

An instance of a p-median problem gives n demand points. The objective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. p-Median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this paper, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the assigned supply point for a given demand point to lie above and to the right of it. In a previous work, we showed that the directional p-median problem is polynomially solvable in one dimension; we give here an improved solution through reformulating the problem as a special case of the constrained shortest path problem. We have previously proven that the problem is NP-complete in two or more dimensions; we present here an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for real-world applications with thousands of demand points.     

求解点覆盖问题的拟物转换及算法

本文首先基于CNF－SAT问题已有的拟物型的求解思想,将点覆盖问题转换为连续情形的目标优化问题,进而提出一种新的竞争式下降梯度方法,从而获得了求解点覆盖问题的高效实用的快速算法．     

Distance transforms for real-valued functions

A set in a metric space gives rise to its distance function that associates with every point its distance to the nearest point in the set. This function is called the distance transform of the original set. In the same vein, given a real-valued function f we consider the expected distances from any point to a level set of f taken at a random height. This produces another function called a distance transform of f. Such transforms are called grey-scale distance transforms to signpost their differences from the binary case when sets (or their indicators) give rise to conventional distance functions. Basic properties of the introduced grey-scale distance transform are discussed. The most important issue is the uniqueness problem whether two different functions may share the same distance transform. We answer this problem in a generality completely sufficient for all practical applications in imaging sciences, the full-scale problem remains open.     

Using the analytic center in the feasibility pump

The feasibility pump (FP) has proved to be a successful heuristic for finding feasible solutions of mixed integer linear problems. Briefly, FP alternates between two sequences of points: one of feasible solutions for the relaxed problem, and another of integer points. This short paper extends FP, such that the integer point is obtained by rounding a point on the (feasible) segment between the computed feasible point and the analytic center for the relaxed linear problem.     

Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever μ≤0.945 (μ∈(0,1) by definition). In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold.     

带惩罚的相同容量k-均值问题的局部搜索算法

经典$k$-均值问题是一类应用广泛的聚类问题,它是指给定$\mathbb{R}^d$中观测点集合$D$和整数$k$,目的是在空间中寻找$k$个点作为中心集合$S$,使得集合$D$中的每个观测点到$S$中离它最近的中心的距离平方求和最小。这是个NP-难问题。经典$k$-均值问题有很多推广,本文研究的带惩罚的相同容量$k$-均值问题就是其中之一。与经典$k$-均值问题相比,惩罚性质是指每个观测点都给定惩罚费用,当某个观测点到最近中心的距离大于惩罚费用时,其对目标函数的贡献就用该观测点的惩罚费用来代替最近的距离的平方,相同容量约束要求每个中心至多连接$U$个观测点。针对这种问题,我们设计了局部搜索算法,该算法在至多选取$（3+\alpha）k$个中心的情况下,可以达到$\beta$-近似,其中,参数$\alpha>34$,$\beta>\frac{\alpha+34}{\alpha-34}$。