Regularized Nonsmooth Newton Algorithms for Best Approximation

TL;DR

This paper introduces a regularized nonsmooth Newton method for best approximation problems, demonstrating superior empirical performance over classical projection methods in large-scale linear programming applications.

Contribution

The paper develops a novel regularized nonsmooth Newton algorithm for polyhedral set approximation, with empirical performance improvements and applications to large-scale linear programming.

Findings

The regularized nonsmooth method outperforms classical projection methods empirically.

The method is not guaranteed to converge due to singular Jacobian issues.

Applications include large-scale linear programs and generalized constrained least squares.

Abstract

We consider the problem of finding the best approximation point from a polyhedral set, and its applications, in particular to solving large-scale linear programs. The classical projection problem has many various and many applications. We study a regularized nonsmooth Newton type solution method where the Jacobian is singular; and we compare the computational performance to that of the classical projection method of Halperin-Lions-Wittmann-Bauschke (HLWB). We observe empirically that the regularized nonsmooth method significantly outperforms the HLWB method. However, the HLWB has a convergence guarantee while the nonsmooth method is not monotonic and does not guarantee convergence due in part to singularity of the generalized Jacobian. Our application to solving large-scale linear programs uses a parametrized projection problem. This leads to a \emph{stepping stone external path following} algorithm. Other applications are finding triangles from branch and bound methods, and generalized constrained linear least squares. We include scaling methods that improve the efficiency and robustness.