  There is often a significant trade-off between formulation strength and size
in mixed integer programming (MIP). When modeling convex disjunctive
constraints (e.g. unions of convex sets), adding auxiliary continuous variables
can sometimes help resolve this trade-off. However, standard formulations that
use such auxiliary continuous variables can have a worse-than-expected
computational effectiveness, which is often attributed precisely to these
auxiliary continuous variables. For this reason, there has been considerable
interest in constructing strong formulations that do not use continuous
auxiliary variables. We introduce a technique to construct formulations without
these detrimental continuous auxiliary variables. To develop this technique we
introduce a natural non-polyhedral generalization of the Cayley embedding of a
family of polytopes and show it inherits many geometric properties of the
original embedding. We then show how the associated formulation technique can
be used to construct small and strong formulation for a wide range of
disjunctive constraints. In particular, we show it can recover and generalize
all known strong formulations without continuous auxiliary variables.
