In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form x∈⋃i=1mPi\documentclass[12pt]{minimal}
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\begin{document}$$x \in \bigcup _{i=1}^m P_i$$\end{document}, where the Pi\documentclass[12pt]{minimal}
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\begin{document}$$P_i$$\end{document} are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, Q:=conv(⋃i=1mPi×{ϵi})\documentclass[12pt]{minimal}
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\begin{document}$$Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})$$\end{document}, where ϵi\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon ^i$$\end{document} is the ith unit vector in Rm\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^m$$\end{document}. 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⊂Rd\documentclass[12pt]{minimal}
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\begin{document}$$D \subset {\mathbb {R}}^d$$\end{document} 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.
