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Transactions of the American Mathematical Society

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Sequential formulae for the normal cone to sublevel sets

Authors: A. Cabot and L. Thibault
Journal: Trans. Amer. Math. Soc. 366 (2014), 6591-6628
MSC (2010): Primary 90C25, 52A41, 49J52; Secondary 34A60
Published electronically: May 2, 2014
MathSciNet review: 3267020
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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.

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Additional Information

A. Cabot
Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

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

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

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