Journal of Optimization Theory and Applications

, Volume 93, Issue 2, pp 395–415

New Version of the Newton Method for Nonsmooth Equations


  • H. Xu
    • Department of MathematicsNingbo University
    • Department of Mathematics and Computer SciencesUniversity of Dundee
  • B. M. Glover
    • School of Information Technology and Mathematical SciencesUniversity of Ballarat

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 mappingsweak Jacobianssemismooth functionsfinite-difference approximationsinexact Newton methodsglobal convergence

Copyright information

© Plenum Publishing Corporation 1997