Abstract

We prove that primal-dual log-barrier interior point 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. Our method is to tropicalize the central path in linear programming. The tropical central path 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. continuous analogue of the Hirsch conjecture
  4. tropical geometry

MSC codes

  1. 90C51
  2. 14T05

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Information & Authors

Information

Published In

cover image SIAM Journal on Applied Algebra and Geometry
SIAM Journal on Applied Algebra and Geometry
Pages: 140 - 178
ISSN (online): 2470-6566

History

Submitted: 4 August 2017
Accepted: 25 August 2017
Published online: 8 March 2018

Keywords

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

MSC codes

  1. 90C51
  2. 14T05

Authors

Affiliations

Funding Information

Êcole Polytechnique, Université Paris-Saclay https://doi.org/10.13039/501100007242
Êlectricité de France https://doi.org/10.13039/501100006289
Einstein Stiftung Berlin https://doi.org/10.13039/501100006188
Fondation Mathématique Jacques Hadamard https://doi.org/10.13039/501100007493
Université Pierre et Marie Curie https://doi.org/10.13039/501100005737

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