Between steps: Intermediate relaxations between big-M and convex hull formulations

TL;DR

This paper introduces P-split relaxations that interpolate between big-M and convex hull formulations, offering a flexible trade-off between model size and relaxation strength, with proven hierarchy and computational benefits.

Contribution

The paper proposes a novel P-split formulation that creates a hierarchy of relaxations between big-M and convex hull, improving computational efficiency.

Findings

P-split formulations form a hierarchy converging to the convex hull.

They provide stronger relaxations than big-M with fewer variables.

Computational tests show significant efficiency gains over existing methods.

Abstract

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.