Sequential formulae for the normal cone to sublevel sets

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.