Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations

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Abstract

We study local convergence of smoothing quasi-Newton methods for solving a system of nonsmooth (nondifferentiable) equations in ℝn. The feature of smoothing quasi-Newton methods is to use a smooth function to approximate the nonsmooth mapping and update the quasi-Newton matrix at each step. Convergence results are given under directional derivative consistence property. Without differentiability we establish a Dennis-Moré-type superlinear convergence theorem for smoothing quasi-Newton methods and we prove linear convergence of the smoothing Broyden method. Furthermore, we propose a superlinear convergent smoothing Newton-Broyden method without using the generalized Jacobian and the semismooth assumption. We illustrate the smoothing approach on box constrained variational inequalities.
Original languageEnglish
Pages (from-to)105-126
Number of pages22
JournalJournal of Computational and Applied Mathematics
Volume80
Issue number1
DOIs
Publication statusPublished - 1 Jan 1997
Externally publishedYes

Keywords

  • Nonsmooth equations
  • Quasi-Newton method
  • Smooth approximation
  • Superlinear convergence
  • Variational inequalities

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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