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