P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints
This work addresses the challenge of efficient and accurate constraint formulation for optimization practitioners, offering a novel intermediate approach that balances relaxation strength and computational burden, though it is incremental in building upon existing big-M and convex hull methods.
The paper tackles the problem of formulating disjunctive constraints in mixed-integer programming by introducing P-split formulations, which are intermediate between big-M and convex hull methods, aiming for tight relaxations with low computational cost. The result shows that on 344 test instances, including K-means clustering and neural network optimization, P-split formulations reduce solution time by an order of magnitude compared to convex hull while matching its node exploration and outperforming big-M in both time and nodes.
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.