This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. We show that this class leads to mixed-integer formulations for trained ReLU neural networks. The approach balances model size and tightness by partitioning node inputs into a number of groups and forming the convex hull over the partitions via disjunctive programming. At one extreme, one partition per input recovers the convex hull of a node, i.e., the tightest possible formulation for each node. For fewer partitions, we develop smaller relaxations that approximate the convex hull, and show that they outperform existing formulations. Specifically, we propose strategies for partitioning variables based on theoretical motivations and validate these strategies using extensive computational experiments. Furthermore, the proposed scheme complements known algorithmic approaches, e.g., optimization-based bound tightening captures dependencies within a partition.

This joint work with Calvin Tsay, Jan Kronqvist, Alexander Thebelt is based on two papers (https://arxiv.org/abs/2102.04373, https://arxiv.org/abs/2101.12708).