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

arXiv: https://arxiv.org/abs/1704.03954 · HTML: https://mlsanity.com/paper/1704.03954/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Apr 2017.

## Built on by (most cited first, 5 of 5)

- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Between steps: Intermediate relaxations between big-M and convex hull formulations](https://mlsanity.com/paper/2101.12708/) (2021)
- [P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints](https://mlsanity.com/paper/2202.05198/) (2022)
- [Neural Network Verification as Piecewise Linear Optimization: Formulations for the Composition of Staircase Functions](https://mlsanity.com/paper/2211.14706/) (2022)
- [Tight Mixed-Integer Optimization Formulations for Prescriptive Trees](https://mlsanity.com/paper/2302.14744/) (2023)

## Abstract (arXiv)

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.