Ideal, non-extended formulations for disjunctive constraints admitting a network representation

In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form $$x \in \bigcup _{i=1}^m P_i$$ , where the $$P_i$$ are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, $$Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})$$ , where $$\epsilon ^i$$ is the ith unit vector in $${\mathbb {R}}^m$$ . Our main contribution is a full characterization of the facets of Q, provided it has a certain network representation. In the second half of the paper, we work-out a number of applications from the literature, e.g., special ordered sets of type 2, logical constraints, the cardinality indicating polytope, union of simplicies, etc., along with a more complex recent example. Furthermore, we describe a new formulation for piecewise linear functions defined on a grid triangulation of a rectangular region $$D \subset {\mathbb {R}}^d$$ using a logarithmic number of auxilirary variables in the number of gridpoints in D for any fixed d. The series of applications demonstrates the richness of the class of disjunctive constraints for which our method can be applied.