Tropical lower bounds for extended formulations

Abstract

The tropical arithmetic operations on $$\mathbb {R}$$R, defined as $$\oplus :(a,b)\rightarrow \min \{a,b\}$$?:(a,b)?min{a,b} and $$\otimes :(a,b)\rightarrow a+b$$?:(a,b)?a+b, arise from studying the geometry over non-Archimedean fields. We present an application of tropical methods to the study of extended formulations for convex polytopes. We propose a non-Archimedean generalization of the well known Boolean rank bound for the extension complexity. We show how to construct a real polytope with the same extension complexity and combinatorial type as a given non-Archimedean polytope. Our results allow us to develop a method of constructing real polytopes with large extension complexity.