Balas formulation for the union of polytopes is optimal

TL;DR

This paper proves that Balas's linear mixed-integer formulation for the union of two polytopes is optimal in terms of the number of inequalities and variables, establishing fundamental limits on such formulations.

Contribution

It demonstrates that any polynomial-size formulation for the convex hull of two polytopes must have a number of additional variables linear in the dimension, confirming the optimality of Balas's approach.

Findings

Balas's formulation is proven to be optimal in inequality count.

Any efficient formulation requires a linear number of extra variables.

Results extend to approximation and lift-and-project settings.

Abstract

A celebrated theorem of Balas gives a linear mixed-integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union. The number of inequalities in Balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled. In this paper we show that this is best possible: in every dimension there exist two nonempty polytopes such that if a formulation for the convex hull of their union has a number of inequalities that is polynomial in the number of inequalities that describe the two polytopes, then the number of additional variables is at least linear in the dimension of the polytopes. We then show that this result essentially carries over if one wants to approximate the convex hull of the union of two polytopes and also in the more restrictive setting of lift-and-project.