Mathematical Programming

, Volume 153, Issue 1, pp 67–74 | Cite as

Tropical lower bounds for extended formulations

Full Length Paper Series B

Abstract

The tropical arithmetic operations on \(\mathbb {R}\), defined as \(\oplus :(a,b)\rightarrow \min \{a,b\}\) and \(\otimes :(a,b)\rightarrow a+b\), 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.

Keywords

Convex polytope Extended formulation Tropical mathematics 

Mathematics Subject Classification

15A23 15B48 52B05 

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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