  This work develops a class of relaxations in between the big-M and convex
hull formulations of disjunctions, drawing advantages from both. The proposed
"P-split" formulations split convex additively separable constraints into P
partitions and form the convex hull of the partitioned disjuncts. Parameter P
represents the trade-off of model size vs. relaxation strength. We examine the
novel formulations and prove that, under certain assumptions, the relaxations
form a hierarchy starting from a big-M equivalent and converging to the convex
hull. We computationally compare the proposed formulations to big-M and convex
hull formulations on a test set including: K-means clustering, P_ball problems,
and ReLU neural networks. The computational results show that the intermediate
P-split formulations can form strong outer approximations of the convex hull
with fewer variables and constraints than the extended convex hull
formulations, giving significant computational advantages over both the big-M
and convex hull.
