Exploring the Numerics of Branch-and-Cut for Mixed Integer Linear Optimization

TL;DR

This paper examines how the numerical properties of LP relaxations change during branch-and-cut algorithms in mixed integer linear optimization, aiming to predict and improve solver performance based on numerical conditioning.

Contribution

It provides an analysis of the numerical behavior of LP relaxations in branch-and-cut solvers, laying groundwork for predictive models to enhance algorithmic decisions.

Findings

Numerical properties evolve during the solution process.

Intuitive understanding of numerical behavior is partially confirmed.

Potential for predicting solver performance based on numerical conditioning.

Abstract

We investigate how the numerical properties of the LP relaxations evolve throughout the solution procedure in a solver employing the branch-and-cut algorithm. The long-term goal of this work is to determine whether the effect on the numerical conditioning of the LP relaxations resulting from the branching and cutting operations can be effectively predicted and whether such predictions can be used to make better algorithmic choices. In a first step towards this goal, we discuss here the numerical behavior of an existing solver in order to determine whether our intuitive understanding of this behavior is correct.