The strength of multi-row models

We develop a method for computing facet-defining valid inequalities for any mixed-integer set $$P_J$$PJ. While our practical implementation does not return only facet-defining inequalities, it is able to find a separating cut whenever one exists. The separator is not comparable in speed with the specific cutting-plane generators used in branch-and-cut solvers, but it is general-purpose. We can thus use it to compute cuts derived from any reasonably small relaxation $$P_J$$PJ of a general mixed-integer problem, even when there exists no specific implementation for computing cuts with $$P_J$$PJ. Exploiting this, we evaluate, from a computational perspective, the usefulness of cuts derived from several types of multi-row relaxations. In particular, we present results with four different strengthenings of the two-row intersection cut model, and multi-row models with up to fifteen rows. We conclude that only fully-strengthened two-row cuts seem to offer a significant advantage over two-row intersection cuts. Our results also indicate that the improvement obtained by going from models with very few rows to models with up to fifteen rows may not be worth the increased computing cost.