Short-step Methods Are Not Strongly Polynomial-Time

TL;DR

This paper demonstrates that under mild assumptions, short-step interior-point methods for convex optimization are not strongly polynomial-time, highlighting limitations in their efficiency.

Contribution

It provides a theoretical proof that short-step interior-point methods cannot achieve strong polynomiality under mild conditions.

Findings

Short-step methods are not strongly polynomial-time.

The proof relies on mild assumptions about the barrier and neighborhood width.

Highlights limitations of certain interior-point algorithms.

Abstract

Short-step methods are an important class of algorithms for solving convex constrained optimization problems. In this short paper, we show that under very mild assumptions on the self-concordant barrier and the width of the $\ell_2$-neighbourhood, any short-step interior-point method is not strongly polynomial-time.