The Morse-Sard theorem for Clarke critical values

被引：6

作者：

Barbet, Luc ^{[1
]}

Dambrine, Marc ^{[1
]}

Daniilidis, Aris ^{[2
]}

机构：

[1] Univ Pau & Pays Adour, UMR CNRS 5142, Lab Math & Leurs Applicat, F-64013 Pau, France

[2] Univ Chile, Fac Ciencias, DIM CMM, Santiago, Chile

来源：

ADVANCES IN MATHEMATICS | 2013年 / 242卷

关键词：

Morse-Sard theorem; Lipschitz function; Cantor-Bendixson derivative; Clarke critical value; CRITICAL SET;

D O I：

10.1016/j.aim.2013.03.024

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

The Morse-Sard theorem states that the set of critical values of a C-k smooth function defined on a Euclidean space R-d has Lebesgue measure zero, provided k >= d. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C-k functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null. (C) 2013 Elsevier Inc. All rights reserved.

引用

页码：217 / 227

页数：11

共 18 条

[1]

[Anonymous], 1998, Variational Analysis

[2]

Bolte J., 2005, ANN POL MATH, V87, P13

[3] Clarke subgradients of stratifiable functions [J].

Bolte, Jerome ;

Daniilidis, Aris ;

Lewis, Adrian ;

Shiota, Masahiro .

SIAM JOURNAL ON OPTIMIZATION, 2007, 18 (02) :556-572

[4] Lipschitz functions with maximal Clarke subdifferentials are generic [J].

Borwein, JM ;

Wang, XF .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2000, 128 (11) :3221-3229

[5]

Clarke F.H, 1983, OPTIMIZATION NONSMOO

[6]

Compte G., 2004, LNM, V1834

[7]

COSTE M, 1999, INTRO O MINIMAL GEOM, P81

[8]

Daniilidis A, 2005, J CONVEX ANAL, V12, P315

[9]

Hirsch W., 1976, GRADUATE TEXTS MATH, V33

[10] Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems [J].

Ioffe, A. D. .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2008, 136 (09) :3111-3119

← 1 2 →