Embedding Formulations and Complexity for Unions of Polyhedra

TL;DR

This paper introduces embedding formulations and complexity as a systematic way to construct minimal size MIP formulations for unions of polyhedra, improving efficiency in solving disjunctive constraints.

Contribution

It presents a new formulation paradigm that guarantees minimal size ideal formulations for disjunctive constraints, including SOS2 and certain piecewise linear functions.

Findings

Optimal formulations for SOS2 constraints characterized.

Significant computational advantages demonstrated for piecewise linear functions.

Framework ensures smallest possible ideal formulations using 0-1 variables.

Abstract

It is well known that selecting a good Mixed Integer Programming (MIP) formulation is crucial for an effective solution with state-of-the art solvers. While best practices and guidelines for constructing good formulations abound, there is rarely a systematic construction leading to the best possible formulation. We introduce embedding formulations and complexity as a new MIP formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size. More specifically, they yield the smallest possible ideal formulation (i.e. one whose LP relaxation has integral extreme points) among all formulations that only use 0-1 auxiliary variables. We use the paradigm to characterize optimal formulations for SOS2 constraints and certain piecewise linear functions of two variables. We also show that the resulting formulations can provide a significant computational advantage over all known formulations for piecewise linear functions.