Nonsmooth Newton Methods for Solving the Best Approximation Problem; with Applications to Linear Programming

Abstract

In this thesis, we study the effects of applying a modified Levenberg-Marquardt regularization to a nonsmooth Newton method. We expand this application to exact and inexact nonsmooth Newton methods and apply it to the best approximation constrained to a polyhedral set problem. We also demonstrate that linear programs can be represented as a best approximation problem, extending the application of nonsmooth Newton methods to linear programming. This application provides us with insight into an external path following algorithm that, like the simplex method, takes a finite number of steps on the boundary of the polyhedral set. However, unlike the simplex method, these steps do not use basic feasible solutions.