A Fast Polynomial-time Primal-Dual Projection Algorithm for Linear Programming

TL;DR

This paper introduces a new polynomial-time primal-dual projection algorithm for linear programming that balances performance on feasible and infeasible instances, outperforming previous methods.

Contribution

The paper develops the PPDP algorithm, explicitly incorporating duality to improve efficiency and balance performance across different problem types.

Findings

PPDP achieves balanced performance on feasible and infeasible instances.

PPDP outperforms previous algorithms in numerical experiments.

Theoretical bounds explain faster performance on infeasible instances.

Abstract

Traditionally, there are several polynomial algorithms for linear programming including the ellipsoid method, the interior point method and other variants. Recently, Chubanov [Chubanov, 2015] proposed a projection and rescaling algorithm, which has become a potentially \emph{practical} class of polynomial algorithms for linear feasibility problems and also for the general linear programming. However, the Chubanov-type algorithms usually perform much better on the infeasible instances than on the feasible instances in practice. To explain this phenomenon, we derive a new theoretical complexity bound for the infeasible instances based on the condition number, which shows that algorithms can indeed run much faster on infeasible instances in certain situations. In order to speed up the feasible instances, we propose a \emph{Polynomial-time Primal-Dual Projection} algorithm (called PPDP) by explicitly developing the dual algorithm. The numerical results validate that our PPDP algorithm achieves a quite balanced performance between feasible and infeasible instances, and its performance is remarkably better than previous algorithms.