Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sequential formulae for the normal cone to sublevel sets


Authors: A. Cabot and L. Thibault
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 90C25, 52A41, 49J52; Secondary 34A60
Published electronically: May 2, 2014
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a reflexive Banach space and let $ \Phi $ be an extended real-valued lower semicontinuous convex function on $ X$. Given a real $ \lambda $ and the sublevel set $ S=[\Phi \leq \lambda ]$, we establish at $ {\overline {x}}\in S$ the following formula for the normal cone to $ S$:

$\displaystyle N_S({\overline {x}})=\limsup _{x\to {\overline {x}}}\, \mathbb{R}_+\, \partial \Phi (x)$$\displaystyle \quad \mbox {if} \quad \Phi ({\overline {x}})=\lambda ,$ ($ \star $)

without any qualification condition. The case $ \Phi ({\overline {x}})<\lambda $ is also studied. Here $ \mathbb{R}_+:=[0,+\infty [$ and $ \partial \Phi $ stands for the subdifferential of $ \Phi $ in the sense of convex analysis. The proof is based on the sequential convex subdifferential calculus developed previously by the second author. Formula $ (\star )$ is extended to nonreflexive Banach spaces via the use of nets. The normal cone to the intersection of finitely many sublevel sets is also examined, thus leading to new formulae without a qualification condition. Our study goes beyond the convex framework: when $ \dim X<+\infty $, we show that the inclusion of the left member of $ (\star )$ into the right one still holds true for a locally Lipschitz continuous function. Finally, an application of formula $ (\star )$ is given to the study of the asymptotic behavior of some gradient dynamical system.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 90C25, 52A41, 49J52, 34A60

Retrieve articles in all journals with MSC (2010): 90C25, 52A41, 49J52, 34A60


Additional Information

A. Cabot
Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
Email: acabot@math.univ-montp2.fr

L. Thibault
Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
Address at time of publication: Centro de Modelamiento Matemático (CMM), Universidad de Chile, Santiago, Chile
Email: thibault@math.univ-montp2.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06151-5
PII: S 0002-9947(2014)06151-5
Keywords: Convex function, subdifferential, sequential subdifferential calculus, sublevel set, normal cone, nonsmooth analysis
Received by editor(s): February 5, 2013
Received by editor(s) in revised form: April 11, 2013
Published electronically: May 2, 2014
Article copyright: © Copyright 2014 American Mathematical Society