P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints
Jan Kronqvist, Ruth Misener, Calvin Tsay
TL;DR
This work introduces P-split formulations, a lifted partitioning approach for disjunctive constraints that interpolates between the classical big-M and convex-hull formulations. By partitioning constraint indices into $P$ blocks and introducing auxiliary variables $\alpha_s^l$, the disjunction is reformulated in a smaller, linearized space while preserving feasibility in the original $\boldsymbol{x}$-space; the continuous relaxations form a hierarchy from big-M ($P=1$) to convex hull ($P=n$) under additive bounds. Theoretical results establish that increasing $P$ yields equally tight or tighter relaxations and, in favorable cases, can recover the convex hull; the framework generalizes to multiple constraints per disjunct and nonconvex within disjuncts via convex envelopes and linking constraints. Computational experiments on 344 instances across P_ball problems, OSIF ReLU-NN optimization, and clustering tasks show that intermediate $P$-split formulations typically reduce solve time and node counts relative to big-M, while offering competitive performance to convex hull without the same degree of complexity, especially when problem structure aligns with the assumptions. Overall, P-splits provide a tunable, scalable option for strong yet tractable disjunctive encodings with practical guidance on partitioning and strengthening strategies.
Abstract
We develop a class of mixed-integer formulations for disjunctive constraints intermediate to the big-M and convex hull formulations in terms of relaxation strength. The main idea is to capture the best of both the big-M and convex hull formulations: a computationally light formulation with a tight relaxation. The "P-split" formulations are based on a lifted transformation that splits convex additively separable constraints into P partitions and forms the convex hull of the linearized and partitioned disjunction. The "P-split" formulations are derived for disjunctive constraints with convex constraints within each disjunct, and we generalize the results for the case with nonconvex constraints within the disjuncts. We analyze the continuous relaxation of the P-split formulations and show that, under certain assumptions, the formulations form a hierarchy starting from a big-M equivalent and converging to the convex hull. We computationally compare the P-split formulations against big-M and convex hull formulations on 344 test instances. The test problems include K-means clustering, semi-supervised clustering, P_ball problems, and optimization over trained ReLU neural networks. The computational results show promising potential of the P-split formulations. For many of the test problems, P-split formulations are solved with a similar number of explored nodes as the convex hull formulation, while reducing the solution time by an order of magnitude and outperforming big-M both in time and number of explored nodes.