Abstract

Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $\Omega(2^r)$. The total curvature of the central path of these linear programs is also exponential in $r$, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through “logarithmic glasses.” This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature in a general setting.

Keywords

  1. linear programming
  2. central path
  3. strongly polynomial complexity
  4. continuous analogue of the Hirsch conjecture
  5. tropical geometry

MSC codes

  1. 90C51
  2. 14T05

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Information & Authors

Information

Published In

cover image SIAM Review
SIAM Review
Pages: 123 - 164
ISSN (online): 1095-7200

History

Published online: 4 February 2021

Keywords

  1. linear programming
  2. central path
  3. strongly polynomial complexity
  4. continuous analogue of the Hirsch conjecture
  5. tropical geometry

MSC codes

  1. 90C51
  2. 14T05

Authors

Affiliations

Funding Information

Ecole Polytechnique
CMAP
Êlectricité de France https://doi.org/10.13039/501100006289
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : SPP 1489, SFB/TRR 109, SFB/TRR 195
Einstein Stiftung Berlin https://doi.org/10.13039/501100006188
Fondation Mathématique Jacques Hadamard https://doi.org/10.13039/501100007493
Direction Générale de l'Armement https://doi.org/10.13039/501100006021
National Science Foundation https://doi.org/10.13039/100000001 : 1440140
Université Pierre et Marie Curie https://doi.org/10.13039/501100005737

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