# Small and Strong Formulations for Unions of Convex Sets from the Cayley Embedding

## TL;DR

This paper introduces a novel geometric technique based on Cayley embeddings to create small, strong mixed-integer programming formulations for unions of convex sets without auxiliary continuous variables, improving computational efficiency.

## Contribution

It develops a new geometric approach that generalizes Cayley embeddings, enabling the construction of strong, compact formulations for convex disjunctive constraints without auxiliary variables.

## Key findings

- The technique recovers all known strong formulations without auxiliary variables.
- It produces smaller and stronger formulations for a wide range of disjunctive constraints.
- The approach inherits geometric properties from the Cayley embedding, ensuring robustness.

## Abstract

There is often a significant trade-off between formulation strength and size in mixed integer programming (MIP). When modeling convex disjunctive constraints (e.g. unions of convex sets), adding auxiliary continuous variables can sometimes help resolve this trade-off. However, standard formulations that use such auxiliary continuous variables can have a worse-than-expected computational effectiveness, which is often attributed precisely to these auxiliary continuous variables. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural non-polyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.