Wintner定理的推广与全局光滑线性化

微分方程光滑线性化的研究,目前仅限于原点近傍的小邻域,本文给出了一个全局光滑线性化的结论.全局拓扑线性化或全局光滑线性化的先决条件是任一解的存在区间为全实轴.因此,本文也讨论了Wintner定理的推广问题.     

全局拓扑线性化理论的推广

微分方程dx/dt=Ax+f(x)(其中A的特征根实部异于零)拓扑线性化的经典结论是由Hartman与Grobman给出的,但是他们的结论都是局部拓扑线性化,即要求同胚函数限制在原点的小邻域内.如果要延伸到全局上的话,必须f(x)有界.本文研究了系统(1.3),证明当此系统满足适当的条件时可全局线性化.     

等式约束优化的信赖域法

本文研究了约束优化信赖域法中的线性化约束条件在信赖域内无解的问题.利用一种基于增广Lagrange函数的方法.获得了一个改进的约束优化的信赖域法.该法的线性化约束条件在信赖内有解,并且具有全局收敛性和超线性收敛性.     

全局线性化及同胚函数的■连续性

本文证明了微分方程全局线性化中的同胚函数及其逆同胚函数都具有连续性，同时给出了具体的估计式．     

全局拓扑线性化理论的推广

微分方程dx/dt=Ax f(x)(其中A的特征根实部异于零)拓扑线性化的经典结论是由Hartman与 Grobman给出的,但是他们的结论都是局部拓扑线性化,即要求同胚函数限制在原点的小邻域内．如 果要延伸到全局上的话,必须f(x)有界．本文研究了系统(1．3),证明当此系统满足适当的条件时可全 局线性化．     

Kac方程的解及其性质

研究Kac方程的初值问题．证明了该类方程存在唯一的全局分布解．并且使用一种新的线性化方法证明了该类方程的解具有相应的多项式衰减性．     

具扩散的生物控制模型的渐进性

讨论了具有空间扩散的生物控制系统在齐次Neumann边界条件下解的整体性态.通过线性化方法给出了局部渐近稳定条件,通过构造Lyapunov函数给出了全局稳定性的证明.     

三个可分离算子凸优化的线性化方法

本文研究了三个可分离算子不含交叉变量的线性约束凸优化问题.利用定制的邻近点算法,对其变分不等式子问题进行线性化处理,并增加一邻近点项,使其子问题成为易于运算的单调线性变分不等式,得到了线性化定制的邻近点算法,并证明了全局收敛性,推广了文献中的研究结果.     

临界情形下的全局拓扑线性化

非线性系ｘ＇＝Ａｘ＝ｆ（ｘ）线性化的基本条件是Ａ的特征根实部异于零．经典线性化的结论只限于原点小邻域［１］．后改进为全局性的［２,３］,但要求ｆ（ｘ）有界．在文［４］中我们去掉了ｆ（ｘ）有界的限制．本文将进一步去掉Ａ的特征根实部异于零的限制,证明了,只要ｆ（ｘ）有适当结构,全局线性化仍是可能的．     

一类具有Allee影响的捕食者-食饵扩散系统研究

讨论一类具有Allee影响的捕食者-食饵扩散模型解的整体性态.通过线性化方法和Lyapunov泛函方法分别证明了该模型正平衡点的局部渐近稳定性和全局渐近稳定性.     

Backward error of polynomial eigenvalue problems solved by linearization

It is commonplace in many application domains to utilize polynomial eigenvalue problems to model the behaviour of physical systems. Many techniques exist to compute solutions of these polynomial eigenvalue problems. One of the most frequently used techniques is linearization, in which the polynomial eigenvalue problem is turned into an equivalent linear eigenvalue problem with the same eigenvalues, and with easily recoverable eigenvectors. The eigenvalues and eigenvectors of the linearization are usually computed using a backward stable solver such as the QZ algorithm. Such backward stable algorithms ensure that the computed eigenvalues and eigenvectors of the linearization are exactly those of a nearby linear pencil, where the perturbations are bounded in terms of the machine precision and the norms of the matrices defining the linearization. Although we have solved a nearby linear eigenvalue problem, we are not certain that our computed solution is in fact the exact solution of a nearby polynomial eigenvalue problem. Here, we perform a backward error analysis for the solution of a specific linearization for polynomials expressed in the monomial basis. We use a suitable one-sided factorization of the linearization that allows us to map generic perturbations of the linearization onto structured perturbations of the polynomial coefficients. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computable Linearizations of Well-partial-orderings

We analyze results on well-partial-orderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every well-partial-order has a linearization of maximal order type. We show that such a linearization can be found computably. We also show that the process of finding such a linearization is not computably uniform, not even hyperarithmetically.     

A linearization framework for unconstrained quadratic (0-1) problems

In this paper, we are interested in linearization techniques for the exact solution of the Unconstrained Quadratic (0-1) Problem. Our purpose is to propose “economical” linear formulations. We first extend current techniques in a general linearization framework containing many other schemes and propose a new linear formulation. Numerical results comparing classical, Glover’s and the new linearization are reported.     

Construction of upper and lower solutions for flow past a non-uniformly heated plate

The Oseen linearization and the modified Oseen linearization are often used in studying fluid mechanical problems, but whether the linearized solution is accurate is usually difficult to assess. For the sample problem of uniform flow past a plate, we use a comparison theorem to show that the Oseen linearization, used in two ways, gives both an upper and a lower solution. Further, we make use of the comparison theorem and the modified Oseen linearization to construct a sharper upper solution valid in the boundary layer. We then go on to consider the case when the plate temperature increases along the plate according to a power law. Upper and lower solutions for the temperature equation are constructed, and bounds on the temperature gradient at the plate are obtained. With a minor modification, similar results for the case when the logarithmic derivative of the plate temperature lies between two power law curves are obtained.     

“Miniaturized” Linearizations for Quadratic 0/1 Problems

In order to solve a quadratic 0/1 problem, some techniques, consisting in deriving a linear integer formulation, are used. Those techniques, called “linearization”, usually involve a huge number of additional variables. As a consequence, the exact resolution of the linear model is, in general, very difficult. Our aim, in this paper, is to propose “economical” linear models. Starting from an existing linearization (typically the so-called “classical linearization”), we find a new linearization with fewer variables. The resulting model is called “Miniaturized” linearization. Based on this approach, we propose a new linearization scheme for which numerical tests have been performed.     

在满足广义指数型二分性条件下的线性化

本文将Ｈａｒｔｍａｎ线性化定理和Ｐａｌｍｅｒ线性化定理推广到具有广义指数型二分性的系统,证明了等价函数的强一致连续性;作为应用,在例子中给出了一类系 统的稳定性的判别方法．     

非线性广义系统平衡点的定性分析

本文研究非线性广义系统平衡点的定性行为．利用系统的正则性，证明了可以通过其线性系统的定性行为研究非线性系统的定性性态，为定性研究非线性系统提供了一种方法．     

Improvement of Hartman''''s linearization theorem

Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) is bounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphism of Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x = Ax. In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we prove the result of global topological linearization without any special limitation and adding any condition. Thus, Hartman's linearization theorem is improved essentially.     

Stochastic graph partitioning: quadratic versus SOCP formulations

We consider a variant of the graph partitioning problem involving knapsack constraints with Gaussian random coefficients. In this new variant, under this assumption of probability distribution, the problem can be traditionally formulated as a binary SOCP for which the continuous relaxation is convex. In this paper, we reformulate the problem as a binary quadratic constrained program for which the continuous relaxation is not necessarily convex. We propose several linearization techniques for latter: the classical linearization proposed by Fortet (Trabajos de Estadistica 11(2):111–118, 1960) and the linearization proposed by Sherali and Smith (Optim Lett 1(1):33–47, 2007). In addition to the basic implementation of the latter, we propose an improvement which includes, in the computation, constraints coming from the SOCP formulation. Numerical results show that an improvement of Sherali–Smith’s linearization outperforms largely the binary SOCP program and the classical linearization when investigated in a branch-and-bound approach.     

对带有盒约束的二次整数规划的一种线性化方法

本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合0-1整数规划问题,然后利用Ilog-cplex或Excel软件中的规划求解工具进行求解,从而解决原二次整数规划.