Mixed-Integer Convex Representability

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 MICP problems. We establish several results this direction, including first complete characterization mixed-binary case a simple necessary condition general case. use latter to derive non-representability various non-convex such set rank-1 matrices prime numbers. Finally, correspondence with seminal work on linear representability Jeroslow Lowe, under rationality assumptions. Under these assumptions, that representable obey strong regularity properties periodicity, provide subsets natural numbers compact sets. Interestingly, numbers, our clear separation between mathematical modeling power optimization. In sets, imply using unbounded integer variables is only