Representation of Polytopes as Polynomial Zonotopes

TL;DR

This paper introduces a novel polynomial zonotope (Z-representation) for bounded polytopes, offering a potentially more compact and computationally efficient alternative to traditional vertex and halfspace representations.

Contribution

The authors prove that every bounded polytope can be represented as a polynomial zonotope, enabling efficient computations like linear maps and Minkowski addition.

Findings

Z-representation can be more compact than V- and H-representations

Allows polynomial-time computation of linear maps and Minkowski addition

Demonstrated usefulness in range bounding within polytopes

Abstract

We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace representation (H-representation). Depending on the polytope, the Z-representation can be more compact than the V-representation and the H-representation. In addition, the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with a computational complexity that is polynomial in the representation size. The usefulness of the new representation is demonstrated by range bounding within polytopes.