Sequential formulae for the normal cone to sublevel sets
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- by A. Cabot and L. Thibault PDF
- Trans. Amer. Math. Soc. 366 (2014), 6591-6628 Request permission
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 $S$: \begin{equation*} N_S(\overline {x}) = \limsup _{x\to \overline {x}} \mathbb {R}_+ \partial \Phi (x) \;\; \text {if} \;\; \Phi (\overline {x}) = \lambda , \tag {$\star $} \end{equation*} 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
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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
- 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
- Received by editor(s): February 5, 2013
- Received by editor(s) in revised form: April 11, 2013
- Published electronically: May 2, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 6591-6628
- MSC (2010): Primary 90C25, 52A41, 49J52; Secondary 34A60
- DOI: https://doi.org/10.1090/S0002-9947-2014-06151-5
- MathSciNet review: 3267020