P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints

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. 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. The goal of the P -split formulations is to form a strong approximation of the convex hull through a computationally simpler formulation. We computationally compare the P -split formulations against big-M and convex hull formulations on 320 test instances. The test problems include K-means 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.

DisjunctiveProgramming.jl: Generalized Disjunctive Programming Models and Algorithms for JuMP

“…This will include GDP solvers that implement the other solution techniques described in Section 2.4. Future work also involves extending the list of suported reformulation techniques to include the P-Split [19] and True-False [1] reformulations.…”

Outlier Detection in Time Series via Mixed-Integer Conic Quadratic Optimization

“…, m} are given data and F is the feasible region. Problem (1) includes the least trimmed squares as a special case, which is a focus of this paper and discussed at length in §1.1, but also includes regression trees [17] (where 1 − z i = 1 indicates that a given datapoint is routed to a given leaf), regression problems with mismatched data [38] (where variables z indicate the datapoint/response pairs) and k-means [33] (where variables z represent assignment of datapoints to clusters). We point out that few or no mixed-integer optimization (MIO) approaches exist in the literature for (1), as the problems are notoriously hard to solve to optimality, and heuristics are preferred in practice.…”