On the integrality gap of convex mixed-integer programs

TL;DR

This paper investigates the integrality gap in convex mixed-integer programs, identifying classes with finite gaps and exploring approximation methods, highlighting limitations of polyhedral approximations for nonlinear sets.

Contribution

It characterizes classes of convex sets with finite integrality gaps and analyzes the effectiveness of polyhedral approximations for estimating these gaps.

Findings

Finite integrality gaps for Dirichlet convex sets and sets with recession cones.

Estimates for the integrality gap in certain convex set classes.

Polyhedral approximations may provide arbitrarily poor bounds for nonlinear convex sets.

Abstract

We study the integrality gap of convex mixed-integer programs, that is, the difference between the optimal value of such a problem and the optimal value of its continuous relaxation. We study classes of convex sets whose associated optimization problem have finite integrality gap: Dirichlet convex sets, sets with full-dimensional recession cones and sets that can be approximated by polyhedral sets. In the latter two cases, we provide estimates for the value of the integrality gap. Finally, we study the possibility of estimating the integrality gap of nonlinear convex mixed-integer programs via rational polyhedral approximations of their feasible regions and argue that, in general, such an approach may yield arbitrarily worse bounds compared to integrality gap estimations specifically derived by studying the associated nonlinear set.