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

Abstract

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_itimes {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.