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A Generalized Univariate Newton Method Motivated by Proximal Regularization

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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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Acknowledgements

Authors would like to thank Gábor Kassay for motivating discussions on antiresolvents of proximal-projection methods during his visit to the School of Mathematics and Statistics at the University of South Australia. The iteration formula (15) in Lemma 4.1 is an interpretation of Roberto Cominetti after seeing our generalized Newton iteration formula in (3). The authors are indebted to him for his observation, which consequently made the proof of Theorem 4.1 neater. They also thank the referees, whose comments and suggestions improved the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of South Australia, Mawson Lakes, 5095, Australia

    Regina S. Burachik & C. Yalçın Kaya

  2. Department of Mathematics, The Technion—Israel Institute of Technology, 32000, Haifa, Israel

    Shoham Sabach

Authors
  1. Regina S. Burachik
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  2. C. Yalçın Kaya
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  3. Shoham Sabach
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Correspondence to C. Yalçın Kaya.

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Communicated by Liqun Qi.

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Burachik, R.S., Kaya, C.Y. & Sabach, S. A Generalized Univariate Newton Method Motivated by Proximal Regularization. J Optim Theory Appl 155, 923–940 (2012). https://doi.org/10.1007/s10957-012-0095-5

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  • DOI: https://doi.org/10.1007/s10957-012-0095-5

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