Polynomial zonotopes, a non-convex set representation, have a wide range of applications from real-time motion planning and control in robotics, to reachability analysis of nonlinear systems and safety shielding in reinforcement learning. Despite this widespread use, a frequently overlooked difficulty associated with polynomial zonotopes is intersection checking. Determining whether the reachable set, represented as a polynomial zonotope, intersects an unsafe set is not straightforward. In fact, we show that this fundamental operation is NP-hard, even for a simple class of polynomial zonotopes. The standard method for intersection checking with polynomial zonotopes is a two-part algorithm that overapproximates a polynomial zonotope with a regular zonotope and then, if the overapproximation error is deemed too large, splits the set and recursively tries again. Beyond the possible need for a large number of splits, we identify two sources of concern related to this algorithm: (1) overapproximating a polynomial zonotope with a zonotope has unbounded error, and (2) after splitting a polynomial zonotope, the overapproximation error can actually increase. Taken together, this implies there may be a possibility that the algorithm does not always terminate.We perform a rigorous analysis of the method and detail necessary conditions for the union of overapproximations to provably converge to the original polynomial zonotope.

翻译：多项式zonotopes作为一种非凸集合表示方法，广泛应用于机器人实时运动规划与控制、非线性系统可达性分析以及强化学习安全屏蔽等领域。尽管应用广泛，但多项式zonotopes交集检验中一个常被忽视的难题值得关注。判断以多项式zonotope表示的可达集是否与非安全集相交并非易事。事实上，我们证明即使对于简单类别的多项式zonotopes，这一基本操作也是NP-hard问题。目前交集检验的标准方法采用两阶段算法：先通过正则zonotope对多项式zonotope进行过近似，若过近似误差过大则分裂集合并递归重试。除可能需要大量分裂操作外，我们发现该算法存在两个关键问题：（1）用zonotope过近似多项式zonotope的误差无界，（2）分裂多项式zonotope后过近似误差可能反而增大。综合来看，这意味着算法存在不收敛的可能性。我们对该方法进行严格分析，并详细阐述了保证过近似并集可验证收敛到原始多项式zonotope的必要条件。