Mixed-integer convex representability
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.
Cite
@article{arxiv.1611.07491,
title = {Mixed-integer convex representability},
author = {Miles Lubin and Ilias Zadik and Juan Pablo Vielma},
journal= {arXiv preprint arXiv:1611.07491},
year = {2017}
}