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.

在本文中，我们重新考虑了一种已知的技术，该技术可为\（x \ in \ bigcup _ {i = 1} ^ m P_i \）的形式的析构约束构建强MIP公式，其中\（P_i \）是多面体。该公式基于多面体并集的Cayley嵌入，即\（Q：= \ mathrm {conv}（\ bigcup _ {i = 1} ^ m P_i \ times \ {\ epsilon ^ i \}）\ ），其中\（\ epsilon ^ i \）是\（{\ mathbb {R}} ^ m \）中的第i个单位向量。我们的主要贡献是对Q方面的全面描述，只要它具有一定的网络表示形式。在本文的后半部分，我们从文献中计算出许多应用，例如类型2的特殊有序集，逻辑约束，表示多位基数的基数，简单性的并集等，以及最近出现的更为复杂的情况。例子。此外，我们使用D中网格点数的对数辅助变量，描述了在矩形区域\（D \ subset {\ mathbb {R}} ^ d \）的网格三角剖分中定义的分段线性函数的新公式对于任何固定的d。该系列应用程序证明了可以应用我们的方法的一类析取约束的丰富性。