On Finding the Closest Zonotope to a Polytope in Hausdorff Distance

TL;DR

This paper develops a local optimization framework for approximating a polytope with a zonotope by minimizing the Hausdorff distance, providing explicit formulas and a subgradient descent algorithm.

Contribution

It introduces explicit local formulas for the Hausdorff distance between a polytope and a zonotope and proposes a subgradient-based optimization method for finding the closest zonotope.

Findings

Derived explicit local formulas for the Hausdorff distance.

Developed a subgradient descent algorithm for optimization.

Characterized local minima as polyhedral feasibility conditions.

Abstract

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.