No self-concordant barrier interior point method is strongly polynomial

TL;DR

This paper proves that no self-concordant barrier interior point method can be strongly polynomial for linear programming, by showing the degeneracy of the central path and providing an exponential lower bound example.

Contribution

It establishes that all self-concordant barrier interior point methods are inherently not strongly polynomial, resolving an open question in the theory.

Findings

Central path degenerates to a piecewise linear curve

Explicit LP example with exponential iteration complexity

Negative result for the existence of strongly polynomial methods

Abstract

It is an open question to determine if the theory of self-concordant barriers can provide an interior point method with strongly polynomial complexity in linear programming. In the special case of the logarithmic barrier, it was shown in [Allamigeon, Benchimol, Gaubert and Joswig, SIAM J. on Applied Algebra and Geometry, 2018] that the answer is negative. In this paper, we show that none of the self-concordant barrier interior point methods is strongly polynomial. This result is obtained by establishing that, on parametric families of convex optimization problems, the log-limit of the central path degenerates to a piecewise linear curve, independently of the choice of the barrier function. We provide an explicit linear program that falls in the same class as the Klee-Minty counterexample, i.e., in dimension $n$ with $2n$ constraints, in which the number of iterations is $\Omega(2^n)$.