When Does Primal Interior Point Method Beat Primal-dual in Linear Optimization?

TL;DR

This paper shows that a stabilized primal interior point method can outperform primal-dual methods in linear optimization, especially near convergence, by leveraging stability and preconditioning techniques.

Contribution

It introduces a stabilized primal IPM that can be faster than primal-dual methods through normal equation acceleration and stability analysis.

Findings

Primal IPM can outperform primal-dual methods near convergence.

Stability of the primal scaling matrix enables effective preconditioning.

Experimental results demonstrate the efficiency of the proposed primal IPM.

Abstract

The primal-dual interior point method (IPM) is widely regarded as the most efficient IPM variant for linear optimization. In this paper, we demonstrate that the improved stability of the pure primal IPM can allow speedups relative to a primal-dual solver, particularly as the IPM approaches convergence. The stability of the primal scaling matrix makes it possible to accelerate each primal IPM step using fast preconditioned iterative solvers for the normal equations. Crucially, we identify properties of the central path that make it possible to stabilize the normal equations. Experiments on benchmark datasets demonstrate the efficiency of primal IPM and showcase its potential for practical applications in linear optimization and beyond.