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 *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.

## Citation

December/2022

## Article

View Regularized Nonsmooth Newton Algorithms for Best Approximation