A Generalized Univariate Newton Method Motivated by Proximal Regularization

A Generalized Univariate Newton Method Motivated by Proximal Regularization

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Article ID: iaor20128242
Volume: 155
Issue: 3
Start Page Number: 923
End Page Number: 940
Publication Date: Dec 2012
Journal: Journal of Optimization Theory and Applications
Authors: , ,
Keywords: heuristics, programming: nonlinear
Abstract:

We devise a new generalized univariate Newton method for solving nonlinear equations, motivated by Bregman distances and proximal regularization of optimization problems. We prove quadratic convergence of the new method, a special instance of which is the classical Newton method. We illustrate the possible benefits of the new method over the classical Newton method by means of test problems involving the Lambert W function, Kullback–Leibler distance, and a polynomial. These test problems provide insight as to which instance of the generalized method could be chosen for a given nonlinear equation. Finally, we derive a closed‐form expression for the asymptotic error constant of the generalized method and make further comparisons involving this constant.

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