Primal-dual path-following methods and the trust-region updating strategy for linear programming with noisy data

TL;DR

This paper introduces a primal-dual path-following method combined with a trust-region strategy for linear programming problems, especially effective for rank-deficient and noisy data, demonstrating superior robustness in numerical tests.

Contribution

It develops a new primal-dual path-following approach with a trust-region update for noisy and rank-deficient linear programming problems, including a preprocessing step based on QR decomposition.

Findings

The method converges globally when starting from a feasible point.

It outperforms existing interior-point methods on noisy, rank-deficient problems.

Numerical tests confirm its robustness and efficiency.

Abstract

In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rank-deficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.