Mixed-Integer Convex Representability

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 nonrepresentability results for various nonconvex 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.

Shapes and recession cones in mixed-integer convex representability

“…If S is MICP-R, but not binary MICP-R we call it general-integer MICP representable (general-integer MICP-R) to emphasize the need for unbounded integer variables to model it (note that any MICP formulation with only bounded integer variables can be converted to one with only binary variables through a standard affine transformation [9,Footnote 4])…”

“…It is an easy corollary of Definition 1.1 (see e.g. [9,Theorem 4.1]) that if M ⊆ R n+p+d induces an MICP formulation of S ⊆ R n , then…”

“…Hence, a binary MICP-R set is a finite union of convex sets, and a general-integer MICP-R set is a countably infinite union of convex sets (in both cases the convex sets are projections of closed convex sets, but in principle may not be closed themselves). The work in [8,9] focuses on understanding the specific structure imposed on these unions of convex sets by an MICP formulation. With regards to the finite unions for the binary MICP-R case, MICP formulations proposed prior to [8,9] required the convex sets in the union (1.2) to have the same recession cones (e.g.…”

“…The work in [8,9] focuses on understanding the specific structure imposed on these unions of convex sets by an MICP formulation. With regards to the finite unions for the binary MICP-R case, MICP formulations proposed prior to [8,9] required the convex sets in the union (1.2) to have the same recession cones (e.g. [1,2,6,5,11]).…”

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Mixed-integer convex representable (MICP-R) sets are those sets that can be represented exactly through a mixed-integer convex programming formulation. Following up on recent work by Lubin et al (2017Lubin et al ( , 2020 we investigate structural geometric properties of MICP-R sets, which strongly differentiate them from the class of mixed-integer linear representable sets (MILP-R). First, we provide an example of an MICP-R set which is the countably infinite union of convex sets with countably infinitely many different recession cones.

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