Empirical Asymptotic Runtime Analysis of Linear Programming Algorithms

TL;DR

This paper empirically investigates the asymptotic runtime growth of key linear programming algorithms across diverse application areas, revealing significant variability that influences future algorithm selection.

Contribution

It provides an empirical analysis of LP algorithms' asymptotic behavior using large language model-generated instances, highlighting variability across algorithms and model types.

Findings

Simple regression models predict runtimes well within a model class.

Asymptotic behavior varies significantly between algorithms.

Results impact future choices of LP algorithms for large models.

Abstract

This paper takes an empirical look at asymptotic runtime growth rates for the most widely used algorithms for solving linear programming (LP) problems across a set of six optimization application areas that are known to produce large and difficult LP models. On the algorithm side, we consider the simplex method, interior-point methods, and PDHG. On the model side, we use a large language model (LLM) to create families of instances in different application areas, allowing us to study model types and sizes that are simultaneously synthetic and realistic. The results indicate that simple regression models typically predict observed runtimes quite well within a model class, and that asymptotic behavior can vary significantly between the different algorithms. This may have a significant impact on which algorithms will be most effective for solving large LP models in the future.