Journal of Optimization Theory and Applications
, Volume 93, Issue 2, pp 395415
First online:
New Version of the Newton Method for Nonsmooth Equations
 H. XuAffiliated withDepartment of Mathematics, Ningbo UniversityDepartment of Mathematics and Computer Sciences, University of Dundee
 , B. M. GloverAffiliated withSchool of Information Technology and Mathematical Sciences, University of Ballarat
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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 finitedifference 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.
 Title
 New Version of the Newton Method for Nonsmooth Equations
 Journal

Journal of Optimization Theory and Applications
Volume 93, Issue 2 , pp 395415
 Cover Date
 199705
 DOI
 10.1023/A:1022658208295
 Print ISSN
 00223239
 Online ISSN
 15732878
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Nonsmooth mappings
 weak Jacobians
 semismooth functions
 finitedifference approximations
 inexact Newton methods
 global convergence
 Industry Sectors
 Authors

 H. Xu ^{(1)} ^{(2)}
 B. M. Glover ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Ningbo University, Ningbo, Zhejiang, P. R. China
 2. Department of Mathematics and Computer Sciences, University of Dundee, Dundee, Scotland
 3. School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Victoria, Australia