Solving MIPs via Scaling-based Augmentation

TL;DR

This paper analyzes scaling-based augmentation methods for solving mixed-integer programs, showing their theoretical performance and practical benefits, especially for hard problems, with improved convergence guarantees and empirical results.

Contribution

It extends convergence analysis of scaling methods for MIPs, demonstrating their theoretical performance and practical advantages in solving difficult instances.

Findings

Scaling methods perform similarly in worst-case scenarios.

Geometric scaling can outperform bit scaling in some cases.

Scaling methods often yield high-quality solutions for hard problems.

Abstract

Augmentation methods for mixed-integer (linear) programs are a class of primal solution approaches in which a current iterate is augmented to a better solution or proved optimal. It is well known that the performance of these methods, i.e., number of iterations needed, can theoretically be improved by scaling methods. We extend these results by an improved and extended convergence analysis, which shows that bit scaling and geometric scaling theoretically perform similarly well in the worst case for 0/1 polytopes as well as show that in some cases geometric scaling can outperform bit scaling arbitrarily. We also investigate the performance of implementations of these methods, where the augmentation directions are computed by a MIP solver. It turns out that the number of iterations is usually low. While scaling methods usually do not improve the performance for easier problems, in the case of hard mixed-integer optimization problems they allow to compute solutions of very good quality and are often superior.