A Generalized Univariate Newton Method Motivated by Proximal Regularization

Regina S. Burachik (), C. Yalçın Kaya () and Shoham Sabach ()
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Regina S. Burachik: University of South Australia
C. Yalçın Kaya: University of South Australia
Shoham Sabach: The Technion—Israel Institute of Technology

Journal of Optimization Theory and Applications, 2012, vol. 155, issue 3, No 10, 923-940

Abstract: 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.

Keywords: Newton–Raphson method; Generalized Newton methods; Numerical analysis; Nonlinear equations; Bregman distances; Antiresolvent (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10957-012-0095-5

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