For a given set of valid inequalities $\cal V$
$$
\left\{\sum_{i}^n w_k x_i + c_k \le 0\right\}_k
$$
we can obtain a polyhedron $P$ in $n$-dimensional space. It's known that the polyhedron $P$ can be either represented by its vertices, which is named as the V-representation, or represented by its facet inequalities and equalities, which is called the H-representation. To switch from those two representations, we can utilize different algorithms and packages like lrs
, cdd
.
Definitely we can calculate the V-representation from the set of valid inequalities $\cal V$, and then switch to the H-representation to obtain all the facet inequalities. Well, this indirect approach is extremely time-consuming sometimes.
So the question is, is there a better way to get all the facet inequalities from a set of valid inequalities?