Smooth Normal Approximations of epi-Lipschitzian subsets of $\mathbbR^n$
Bernard Cornet, Marc-Olivier Czarnecki · SIAM Journal on Control and Optimization · 1999
A sequence (M k ) of closed subsets of $\R^n$ converges normally to $M\subset \R^n$ if (sc) $M=\limsup M_{k}=\liminf M_k$ in the sense of Painlevé--Kuratowski and (nc) $\limsup {\rm G}(N_{M_k})\subset {\rm G}(N_{M})$, where G(N M ) (resp., $ {\rm G}(N_{M_k})$) denotes the graph of N M (resp., $N_{M_k}$), Clarke's normal cone to (resp., $M_k$). This paper studies the normal convergence of subsets of $\R^n$ and mainly shows two results. The first result states that every closed epi-Lipschitzian subset M of $\R^n$, with a compact boundary, can be approximated by a sequence of smooth sets (M k ), which converges normally to M and such that the sets M k and M are lipeomorphic for every k (i.e., the homeomorphism between M and M k and its inverse are both Lipschitzian). The second result shows that, if a sequence (M k ) of closed subsets of $\R^n$ converges normally to an epi-Lipschitzian set M, and if we additionally assume that the boundary of M k remains in a fixed compact set, then, for k large enough, the sets M k and M are lipeomorphic. In Cornet and Czarnecki [ Cahier Eco-Maths 95-55, 1995], direct applications of these results are given to the study (existence, stability, etc.) of the generalized equation $0 \in f(x^*) +N_{M}(x^*)$ when M is a compact epi-Lipschitzian subset of $\R^n$ and $f: M to\R^n$ is a continuous map (or more generally a correspondence).