Log-barrier interior point methods are not strongly polynomial

TL;DR

This paper demonstrates that primal-dual log-barrier interior point methods are not strongly polynomial by constructing specific linear programs with exponential iteration complexity and curvature, using tropical geometry techniques.

Contribution

The paper introduces a novel tropicalization approach to prove exponential lower bounds on iteration count and curvature for interior point methods, disproving strong polynomiality.

Findings

Constructed linear programs with exponential iteration complexity

Proved exponential total curvature of the central path

Disproved a continuous analogue of the Hirsch conjecture

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.