We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show that the computational complexity of the above-mentioned set operations for constrained polynomial zonotopes is at most polynomial in the representation size. The fact that constrained polynomial zonotopes are generalizations of zonotopes, polytopes, polynomial zonotopes, Taylor models, and ellipsoids further substantiates the relevance of this new set representation. In addition, the conversion from other set representations to constrained polynomial zonotopes is at most polynomial with respect to the dimension, and we present efficient methods for representation size reduction and for enclosing constrained polynomial zonotopes by simpler set representations.

我们介绍了受约束的多项式 zonotope，这是一种在线性映射、闵可夫斯基和、笛卡尔积、凸包、交集、并集和二次以及高阶映射下闭合的新型非凸集表示。我们表明，受约束的多项式环带形体的上述集合运算的计算复杂度至多是表示大小的多项式。约束多项式环带是环带、多胞形、多项式环带、泰勒模型和椭球体的推广这一事实进一步证实了这一新集合表示的相关性。此外，从其他集合表示到约束多项式环带形的转换至多是关于维度的多项式，