Two computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming

TL;DR

This paper introduces two new polynomial-time infeasible interior-point algorithms for linear programming that are computationally efficient and outperform existing methods in numerical tests.

Contribution

The paper presents two novel arc-search infeasible interior-point algorithms with wider neighborhoods and proven convergence, closing the gap between theory and practice.

Findings

Algorithms are polynomial and converge.

Computational tests show competitiveness with existing methods.

Proposed algorithms outperform some current algorithms in efficiency.

Abstract

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bound are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: "Can we find a theoretically and practically efficient way to reoptimize?" This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version, if it terminates in finite iterations, has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra's predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive comparing to both Mehrotra's predictor-corrector algorithm and the efficient arc-search algorithm.