A Generalized Univariate Newton Method Motivated by Proximal Regularization

Regina S. Burachik, C. Yalçin Kaya, Shoham Sabach

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)923-940
Number of pages18
JournalJournal of Optimization Theory and Applications
Issue number3
StatePublished - Dec 2012


  • Antiresolvent
  • Bregman distances
  • Generalized Newton methods
  • Newton-Raphson method
  • Nonlinear equations
  • Numerical analysis

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics


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