On Finding the Closest Zonotope to a Polytope in Hausdorff Distance
<|reference_start|>On Finding the Closest Zonotope to a Polytope in Hausdorff Distance: 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.<|reference_end|>
@article{torres2024on,
    title={On Finding the Closest Zonotope to a Polytope in Hausdorff Distance},
    author={George D. Torres},
    journal={arXiv preprint arXiv:2407.13125},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.13125},
    primaryClass={math.OC cs.CG}
}