Nonsmooth Newton Methods for Solving the Best Approximation Problem; with Applications to Linear Programming
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Date
2023-12-19
Authors
Weames, Tyler
Journal Title
Journal ISSN
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Publisher
University of Waterloo
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.
Description
Keywords
nonsmooth Newton method, Gauss-Newton method, Levenberg-Marquardt regularization, exact and inexact methods, best approximation problem, first order methods, second order methods, linear programming, sensitivity analysis, time complexity