Sequential formulae for the normal cone to sublevel sets

Abstract

Let X X be a reflexive Banach space and let Φ \Phi be an extended real-valued lower semicontinuous convex function on X X . Given a real λ \lambda and the sublevel set S = [ Φ ≤ λ ] S=[\Phi \leq \lambda ] , we establish at x ¯ ∈ S {\overline {x}}\in S the following formula for the normal cone to S S : ( ⋆ ) N S ( x ¯ ) = lim sup x → x ¯ R + ∂ Φ ( x ) if Φ ( x ¯ ) = λ , \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 Φ ( x ¯ ) > λ \Phi ({\overline {x}})>\lambda is also studied. Here R + := [ 0 , + ∞ [ \mathbb {R}_+:=[0,+\infty [ and ∂ Φ \partial \Phi stands for the subdifferential of Φ \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 > + ∞ \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.