  We provide a local theory for the optimization of the Hausdorff distance
between a polytope and a zonotope. To do this, we compute explicit local
formulae for the Hausdorff function $d(P, -) : Z_n \to \mathbb{R}$, where $P$
is a fixed polytope and $Z_n$ is the space of rank $n$ zonotopes. This local
theory is then used to provide an optimization algorithm based on subgradient
descent that converges to critical points of $d(P, -)$. We also express the
condition of being at a local minimum as a polyhedral feasibility condition.
