On the Difficulty of Intersection Checking with Polynomial Zonotopes

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 sufficient conditions for the union of overapproximations to provably converge to the original polynomial zonotope.