New Version of the Newton Method for Nonsmooth Equations

  • H. Xu
  • B. M. Glover

DOI: 10.1023/A:1022658208295

Cite this article as:
Xu, H. & Glover, B.M. Journal of Optimization Theory and Applications (1997) 93: 395. doi:10.1023/A:1022658208295


In this paper, an inexact Newton scheme is presented which produces a sequence of iterates in which the problem functions are differentiable. It is shown that the use of the inexact Newton scheme does not reduce the convergence rate significantly. To improve the algorithm further, we use a classical finite-difference approximation technique in this context. Locally superlinear convergence results are obtained under reasonable assumptions. To globalize the algorithm, we incorporate features designed to improve convergence from an arbitrary starting point. Convergence results are presented under the condition that the generalized Jacobian of the problem function is nonsingular. Finally, implementations are discussed and numerical results are presented.

Nonsmooth mappings weak Jacobians semismooth functions finite-difference approximations inexact Newton methods global convergence 

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • H. Xu
    • 1
    • 2
  • B. M. Glover
    • 3
  1. 1.Department of MathematicsNingbo UniversityNingbo, ZhejiangP. R. China
  2. 2.Department of Mathematics and Computer SciencesUniversity of DundeeDundeeScotland
  3. 3.School of Information Technology and Mathematical SciencesUniversity of BallaratBallaratAustralia

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