# Mixed-integer convex representability

## TL;DR

This paper investigates which sets can be exactly represented as feasible regions of mixed-integer convex problems, providing characterizations, non-representability results, and insights into the modeling capabilities under rationality assumptions.

## Contribution

It offers the first complete characterization for mixed-binary cases, necessary conditions for general cases, and explores the impact of rationality assumptions on representability.

## Key findings

- Complete characterization for mixed-binary representability.
- Non-representability results for rank-1 matrices and prime numbers.
- Representation of subsets of natural numbers and compact sets under rationality.

## Abstract

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first non-representability results for various non-convex sets such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.