A factorisation-based regularised interior point method using the augmented system

TL;DR

This paper introduces a new interior point solver for linear programming in HiGHS that employs a factorisation-based approach with augmented systems, achieving improved performance and stability.

Contribution

It presents a novel implementation of an interior point method using direct factorisation and augmented systems, with detailed strategies for stability and performance optimization.

Findings

Enhanced solver performance over previous HiGHS methods

Effective handling of ill-conditioned matrices through pivoting and regularisation

Demonstrated improvements on multiple problem collections

Abstract

This paper describes the implementation of a new interior point solver for linear programming for the open-source optimization library HiGHS. The solver uses a direct factorisation to solve the Newton systems, choosing the best approach between the normal equations and augmented system. Details of the multifrontal factorisation routine are given, with attention to the features that allow to achieve high performance, like storage formats, use of efficient dense linear algebra subroutines and parallelism. The paper also describes the use of pivoting and regularisation strategies to ensure that a stable factorisation is obtained, despite the ill-conditioning of the matrices. Results on three different collections of problems are presented which highlight the improved performance of the solver compared to the existing HiGHS interior point method.