efficient way to find one extreme point of a Zonotope in high dimension

Question

Given one set of generators $G \subset \mathbb{R}^d$, one can easily generate a Zonotope using ConvexHull $Z = CH(\{\sum_{i \in S} i| S \subseteq G\})$.

Now my interest is to find one vertex of $Z$, is there any efficient way to do so?

Answer 1

Choose a generic $c\in\Bbb R^d$ (that is, $\langle x,c\rangle\not=0$ for all $x\in G$) and consider the subset

$$G_c:=\{x\in G\mid \langle x,c\rangle > 0\}\subseteq G.$$

The point $v_c:=\sum_{x\in G_c} x$ is then a vertex of $Z$ (and every vertex is of this form).