Normal cone ($N_C(z)$)

Normal cones describe generalized supporting directions to sets. For a convex set, the normal condition is $⟨w,z^{\prime }-z⟩\le 0$ for every $z^{\prime }$ in the set. For nonconvex sets, the Fréchet normal cone and the limiting normal cone distinguish local regular support from limits of such supports.

The limiting normal cone consists of limits $w_k\to w$ of vectors from the Fréchet normal cone $w_k\in {\hat{N}}_C(z_k)$ with $z_k\to z$ in $C$. It need not be convex. Including normals from nearby graph pieces is essential to the Mordukhovich criterion.

A vector $w$ is a regular normal at $z\in C$ when ${\mathrm{limsup}}_{z^{\prime }\to z,z^{\prime }\in C\setminus \{z\}}⟨w,z^{\prime }-z⟩/\Vert z^{\prime }-z\Vert \le 0$. At an isolated point all vectors satisfy this condition. It is the local first-order supporting cone.