New Version of the Newton Method for Nonsmooth Equations

  • H. Xu
  • B. M. Glover


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 


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  1. 1.
    Qi, L., and Jiang, H., Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving These Equations, Report 5, Applied Mathematics Department, University of New South Wales, 1994.Google Scholar
  2. 2.
    Harker, P. T., and Xiao, B., Newton's Method for Linear Complementarity Problem: A B-Differentiable Equation Approach, Mathematical Programming, Vol. 48, pp. 339–357, 1990.Google Scholar
  3. 3.
    Pang, J. S., A B-Differentiable Equation-Based, Globally, and Locally Quadratically Convergent Algorithm for Nonlinear Programs, Complementarity, and Variational Inequality Problems, Mathematical Programming, Vol. 51, pp. 101–131, 1991.Google Scholar
  4. 4.
    Pang, J. S., and Qi, L., Nonsmooth Equations: Motivation and Algorithms, SIAM Journal on Optimization, Vol. 3, pp. 443–465, 1993.Google Scholar
  5. 5.
    Qi, L., Superlinearly Approximate Newton Methods for LC1-Optimization Problems, Mathematical Programming, Vol. 64, pp. 277–294, 1994.Google Scholar
  6. 6.
    Sun, J., and Qi, L., An Interior-Point Algorithm of O(\(\sqrt m \)|log ε|) Iterations for C1-Convex Programming, Mathematical Programming, Vol. 57, pp. 239–257, 1992.Google Scholar
  7. 7.
    Xiao, B., and Harher, P. T., A Nonsmooth Newton Method for Variational Inequalities, Part 1: Theory, Mathematical Programming, Vol. 65, pp. 151–194, 1994.Google Scholar
  8. 8.
    Ip, C. M., and Kyparisis, J., Local Convergence of Quasi-Newton Methods for B-Differentiable Equations, Mathematical Programming, Vol. 58, pp. 71–89, 1992.Google Scholar
  9. 9.
    Chen, X., and Qi, L., A Parametrized Newton Method and a Quasi-Newton Method for Nonsmooth Equations, Computational Optimization and Applications, Vol. 3, pp. 157–179, 1994.Google Scholar
  10. 10.
    Kojima, M., and Shindo, S., Extensions of Newton and Quasi-Newton Methods to Systems of PC1-Equations, Journal of the Operation Research Society of Japan, Vol. 29, pp. 352–374, 1986.Google Scholar
  11. 11.
    Kummer, B., Newton's Method for Nondifferentiable Functions, Advances in Mathematical Optimization, Edited by J. Guddat et al. Akademie Verlag, Berlin, Germany, pp. 114–125, 1988.Google Scholar
  12. 12.
    Pang, J. S., Newton's Method for B-Differentiable Equations, Mathematics of Operation Research, Vol. 15, pp. 311–341, 1990.Google Scholar
  13. 13.
    Qi, L., Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operation Research, Vol. 18, pp. 227–244, 1993.Google Scholar
  14. 14.
    Qi, L., and Sun, J., A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353–367, 1993.Google Scholar
  15. 15.
    Clarke, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983.Google Scholar
  16. 16.
    Xu, H., Approximate Newton Methods for Nonsmooth Equations, Report 158, Mathematics and Computer Sciences Department, Dundee University, Dundee, Scotland, 1994.Google Scholar
  17. 17.
    Dembo, R. S., Eisentat, S. C., and Steihaug, T., Inexact Newton Methods, SIAM Journal on Numerical Analysis, Vol. 19, pp. 400–408, 1982.Google Scholar
  18. 18.
    Eisenstat, S. C., and Walker, H. F., Globally Convergence Inexact Newton Methods, SIAM Journal on Optimization, Vol. 4, pp. 393–422, 1994.Google Scholar
  19. 19.
    Martinez, J. M., and Qi, L., Inexact Newton Methods for Solving Nonsmooth Equations, Journal of Computational and Applied Mathematics, Vol. 60, pp. 127–145, 1995.Google Scholar
  20. 20.
    Dennis, J. E., and More, J. J., Quasi-Newton Methods: Motivation and Theory, SIAM Review, Vol. 19, pp. 46–89, 1977.Google Scholar
  21. 21.
    Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.Google Scholar

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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