I have a series of lower semi continuous, eventually level bounded and proper functions $ f^\nu(p)$ that epi-converges to $f(p)$.
In this context, it is known from e.g., [7.33, Variational analyis, Rockafellar and Wets] that the sequence of global minimizers $f^\nu(p)$ converge to a global minimizaer of $f(p)$.
The question is the following. Is it also true that a sequence of local minimizers of $f^\nu(p)$ converge to a local minimizer of $f(p)$?
Additionally, if I have a set sequence $C^\nu$ that converges to $C$ (in the Painlevé-Kuratowski sense), under which circunstances the normal cones converge $N_{C^\nu} \to N_C$?
By the way, neither the functions nor the sets are convex.
Thank you!