Limiting normal cone ($N_C(z)={\mathrm{limsup}}_{z^{\prime }\to z}{\hat{N}}_C(z^{\prime })$)

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.