Local Lipschitz-constant Functions and Maximal Subdifferentials

It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)B _X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its local Lipschitz-constant function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.     

The approximate and the Clarke subdifferentials can be different everywhere

We prove that, for a Lipschitz function on , n2, the approximate and the Clarke subdifferentials can differ everywhere. This completely answers a question by A.D. Ioffe, which was partially answered by G. Katriel.     

广义Taylor中值函数若干分析性质

文[2-8]对微分中值定理及Taylor定理"中间点"的渐近性质进行了研究,本文在此基础上,给出了"广义Taylor中值函数"的定义,对"广义Taylor中值函数"的分析性质进行了系统的讨论,证明了"广义Taylor中值函数"的单调性、可积性、连续性、可微性等分析性质.     

The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix

We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative.     

Representation of the Clarke subdifferential for a regular quasidifferentiable function

A class of Lipschitz quasidifferentiable functions is described for which the exact representation of the Clarke subdifferential in terms of a quasidifferential holds. The sufficient conditions formulated are different from those previously established by Rubinov and Akhundov.     

Dirichlet函数与Riemann函数的性质及应用

探讨Dirichlet函数和Riemann函数的连续性、可导性和可积性,并由此分别构造出在一点连续、多点连续、一点可导、多点可导的函数.     

Constraint Qualifications in Nonsmooth Multiobjective Optimization

For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary conditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem.     

积分型Cauchy中值函数若干分析性质

给出"积分型Cauchy中值函数"的定义,对"积分型Cauchy中值函数"的分析性质进行了系统讨论,证明了"积分型Cauchy中值函数"的单调性、可积性、连续性、可微性等分析性质.作为"积分型Cauchy中值函数"的特例,给出了"第一积分中值函数"的定义及"第一积分中值函数"相应的分析性质.     

Upper Subderivatives and Generalized Gradients of the Marginal Function of a Non-Lipschitzian Program

We obtain an upper bound for the upper subderivative of the marginal function of an abstract parametric optimization problem when the objective function is lower semicontinuous. Moreover, we apply the result to a nonlinear program with right-hand side perturbations. As a result, we obtain an upper bound for the upper subderivative of the marginal function of a nonlinear program with right-hand side perturbations, which is expressed in dual form in terms of appropriate Lagrange multipliers. Finally, we present conditions which imply that the marginal function is locally Lipschitzian.     

关于Taylor中值函数若干分析性质的讨论

给出了"Taylor中值函数"的定义,对Taylor中值函数的分析性质进行了系统的综合讨论,证明了Taylor中值函数的单调性、可积性、连续性、可微性等分析性质.     

ON NONLINEAR ELLIPTIC HEMIVARIATIONAL INEQUALITIES OF SECOND ORDER

In this paper, a nonlinear hemivariational inequality of second order with a forcing term of subcritical growth is studied. Using techniques from multivalued analysis and the theory of nonlinear operators of monotone type, an existence theorem for the Dirichlet boundary value problem is proved.     

Integral of the Clarke subdifferential mapping and a generalized Newton-Leibniz formula

In the theory of Lebesgue integration it has been proved that if f is a real Lipschitz function defined on a segment [a,b]⊂R, then the Newton-Leibniz formula (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f^′(⋅) (which is defined almost everywhere on [a,b] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂_Cf(⋅) and the Aumann (set-valued) integral. Among other things, we show that and the equality is valid if and only if f is strictly Hadamard differentiable almost everywhere on [a,b]. The result is derived from a general representation formula, which we obtain herein for the integral of the Clarke subdifferential mapping of a Lipschitz function defined on a separable Banach space.     

Necessary optimality conditions and saddle points for approximate optimization in banach spaces

In this article we study approximate optimality in the setting of a Banach space. We study various solution concepts existing in the literature and develop very general necessary optimality conditions in terms of limiting subdifferentials. We also study saddle point conditions and relate them to various solution concepts. Part of this research was carried out when the author was a post-doctoral fellow at UAB, Barcelona by the Grant No. SB99-B0771103B of the Spanish Ministry of Education and Culture. The hospitality and the facilities provided at CODE, UAB is gratefully acknowledged.     

Henstock引理,导函数的可积性,积分原函数的可导性

对经典实分析非绝对积分理论中的Henstock引理的本质特征进行了讨论,指出:Henstock引理的本质是刻划了导函数的可积性和积分原函数的可导性问题.     

Semicoercive variational hemivariational inequalities

The aim of this paper is the mathematical study of a general class of semicoercive variational hemivariational inequalities introduced by P.D. Panagiotopoulos in order to formulate problems of mechanics involving nonconvex and nonsmooth energy function. Our approach is based on the asymptotic behavior of the functions which are involved in the variational problems.     

HOLDER CONTINUITY AND DIFFERENTIABILITY ON CONVERGING SUBSEQUENCES

It is shown that an arbitrary function from D ?Rn to Rm will become C0,α-continuous in almost every x ∈ D after restriction to a certain subset with limit point x. For n ≥ m differentiability can be ob...     

关于多元函数可微性的一个注记

研究了多元函数的可微性,给出了多元函数连续、可微的一些条件.