  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.
