函数的次微分性质

本文给出了函数的Fenchel次微分、Frechet次微分,Hadamard次微分,Gateaux次微分的一些重要性质,并对函数的性质尤其是凸性给出其次微分刻画。     

ε-次微分与边际函数的解

借助ε-次微分讨论一类对偶边际函数的次微分,并得到此类函数解集的特征．     

函数的次微分连续性与邻近正则性

研究了函数和集在某点的邻近正则性与次微分连续性,给出邻近正则函数的次微分在某种意义下的单调性及函数的邻近正则性与其上图的邻近正则性的关系．     

DICR函数的刻画及其在优化问题中的应用

在拓扑向量空间中研究DICR函数.引入该函数关于支撑集、次微分的概念,研究该函数支撑集、次微分之间的关系.也研究了与严格DICR函数相关的集合的最大元,得到严格DICR函数差的全局最小值的充要条件.     

扰动多目标规划的次微分稳定性

本文利用共轭对偶算子定义了次微分,在一般拓扑向量空间中系统地讨论了多目标规划次微分稳定性.在目标函数为锥严格凸,约束函数为拟凸以及锥半连续的条件下,得到扰动多目标规划问题的整体稳定性.另外,通过引进点集,映射在一点凸的定义,得到问题的局部稳定性.我们将所得到的结论应用于有限维欧氏空间中控制结构为正锥的情形,还得到一些特殊结果.     

基于前向差分的一阶数值微分公式验证与实践

根据多项式插值理论,可以通过构造相应的插值多项式来逼近未知的目标函数,再进一步求一阶导数,从而得到该目标函数的一阶数值微分公式.对于此数值微分公式,探讨基于前向差分的未知目标函数的多点一阶微分近似公式;即,等间距情况下的二至十六个数据点的前向差分公式.计算机数值实验进一步验证与表明,该用于未知目标函数一阶数值微分的多点公式可以取得较高的计算精度.     

广义次微分及其应用

通过应用广义次微分来研究不可微规划的最优解，得到了适当函数类在强意义下的最优性条件，并给出了广义次微分在稳定性理论和极小化方法中的应用     

Markov调制的随机微分延迟方程的函数稳定性

随机微分延迟方程的指数稳定性被人们广泛研究,但讨论带Markov调制的随机微分延迟方程的函数稳定性的不多.本文主要研究了两种类型的函数稳定性.我们采用了一例特定的Lyapunov函数,来研究带Markov调制的随机微分延迟方程的p阶矩ψα-函数稳定性,并对其几乎必然ψβ/p-函数稳定性也进行了探讨.     

多目标决策中的ε—共轭对偶理论

本文首先讨论了ε—有效解的性质,证明了ε—有效解集的连通性。第二,在通常的Pareto有效解的意义下,利用ε—次微分和ε—共轭映射,讨论了Pareto有效解的共轭对偶定理、拉格朗日对偶定理和鞍点定理。还证明了ε—次微分的存在性定理。§1 ε—有效解和连通性近年来,对多目标最优化的共轭对偶理论已有了许多讨论。Tanino,T.[1]在Pareto有效解的意义下利用向量值函数的次微分给出了多目标最     

一类三次多项式微分系统的周期解

根据Mironenko的反射函数理论,给出一种利用多项式方程探讨三次多项式微分系统周期解的几何性质的新方法.该文首先研究一类系统具有满足特定关系式的反射函数的结构,由此建立三次多项式微分系统与多项式方程之间的解的对应关系,然后利用此对应关系探讨三次多项式微分系统的周期解的几何性质.     

Subdifferentials with Autoconjugate Fitzpatrick Function

In the framework of real Banach spaces, the present paper provides a necessary and sufficient condition for the Fitzpatrick function of the subdifferential of a proper lower semicontinuous convex function to be autoconjugate. This enables us to: obtain a new proof of the fact that subdifferentials of indicator and sublinear functions have autoconjugate Fitzpatrick functions; characterize those classes of functions whose subdifferentials fulfill the condition under study in the same special way as indicator and sublinear functions do; prove that, in the one-dimensional case, the functions of these classes are the only ones whose subdifferentials have autoconjugate Fitzpatrick functions, while this is not true in higher dimensions.     

The Compatibility with Order of Some Subdifferentials

It is well known that elementary subdifferentials which are the simplest and the most precise among known subdifferentials do not enjoy good calculus rules, whereas more elaborated subdifferentials do have calculus rules but are not as precise and, in particular, do not preserve order. This paper explores an order preservation property for the subdifferentials of the second category. This property concerns the case in which a distance function is involved. It emphasizes the crucial role played by such functions in nonsmooth analysis. The result enables one to get in a simple, unified way the passage from the properties of subdifferentials for Lipschitzian functions to the same properties for the case of lower semicontinuous functions.     

Calculus of directional subdifferentials and coderivatives in Banach spaces

In this work we study the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We establish some characterizations and basic properties of these constructions, and then develop calculus including sum rules and chain rules involving smooth functions. As an application, we also explore the upper estimates of the directional Mordukhovich subdifferentials and singular subdifferentials of marginal functions.     

Relationships Between Convexificators and Greenberg–Pierskalla Subdifferentials for Quasiconvex Functions

In this paper, the relationship between convexificators and Greenberg–Pierskalla-based (GP-based) subdifferentials for quasiconvex functions is proved. The established results lead to a mean value theorem, a chain rule, and the closedness property for GP-based subdifferentials. Furthermore, the connection between Clarke generalized gradient and Mordukhovich subdifferential with GP-based subdifferentials is highlighted.     

Subdifferential representation of homogeneous functions and extension of smoothness in Banach spaces

In this paper, we mainly consider proximal subdifferentials of lower semicontinuous functions defined on real Hilbert space and Clarke＇s subdifferentials of locally Lipschitzian functions defined on Banach space respectively, and obtain the generalized Euler identity of homogenous functions. Then, by introducing a multifunction F, we extend the smoothness of sphere and differentiability of norm function in Banach space.     

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel-Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.     

Subdifferentiation of Regularized Functions

We study the Moreau regularization process for functions satisfying a general growth condition on general Banach spaces. We give differentiability criteria and we study the relationships between the subdifferentials of the function and the subdifferentials of its approximations. We also consider the Lasry-Lions process.     

Generalized Euler identity for subdifferentials of homogeneous functions and applications

In this paper, we mainly consider subdifferentials and basic subdifferentials of homogeneous functions defined on real Banach space and Asplund space respectively, and obtain the generalized Euler identity. As applications, we consider constrained optimization problems and several geometric properties of Banach space.     

Banach空间的p— Asplund 伴随空间

我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质（FDP）,如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ－子集上Frechet可微。本文主要证明了：对任何Banach空间E,均存在一个局部凸相容拓扑p使得1）（E,p)是Hausdorff局部凸空间;2） E上的每个范数连续具有FDP的凸函数均是p－连续的;3）每个p-连续的凸函数均具有FDP ;4）p等价某个范数拓扑当且仅不E是Asplund空间。     

Generalized Monotonicity of Subdifferentials and Generalized Convexity

Characterizations of convexity and quasiconvexity of lower semicontinuous functions on a Banach space X are presented in terms of the contingent and Fréchet subdifferentials. They rely on a general mean-value theorem for such subdifferentials, which is valid in a class of spaces which contains the class of Asplund spaces.