Interior point methods in the year 2025

Abstract

Interior point methods (IPMs) have hugely influenced the field of optimization. Their fast development has been triggered by the seminal paper of Narendra Karmarkar published in 1984 which delivered a polynomial algorithm for linear programming and suggested that it might be implemented into a very efficient method in practice. Indeed, this has been demonstrated within a few years after 1984 and has gained IPMs a status of exceptionally powerful optimization tool. Linear Programming (LP) is at the centre of many operational research techniques including mixed-integer programming, network optimization and various decomposition techniques. Therefore, any progress in LP has far-reaching consequences. IPMs certainly did not disappoint in this context: they have become a heavily used methodology in modern optimization and operational research. Their accuracy, efficiency and reliability have been particularly appreciated when IPMs are applied to truly large scale problems which challenge any alternative approaches.

In this survey we will discuss several issues related to interior point methods. We will recall techniques which provide the building blocks of IPMs, and observe that actually at least some of them have been developed before 1984. We will briefly comment on the worst-case complexity results for different variants of IPMs and then focus on key aspects of their implementation. We will also address some of the most spectacular features of IPMs and discuss their potential advantages when applied in decomposition algorithms, cutting planes scheme and column generation technique.