Mixed-integer convex representability

TL;DR

This paper characterizes which nonconvex sets can be exactly represented by mixed-integer convex optimization problems, providing a complete understanding for finite cases and insights into the power of convex sets in modeling.

Contribution

It offers the first complete characterization for finite integer assignment cases and develops new conditions and negative results for unbounded integer variables.

Findings

Complete characterization for finite integer assignments

Necessary conditions for unbounded integer variables

Insights into modeling power of convex sets over polyhedral sets

Abstract

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.