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Approximate Newton Methods for Nonsmooth Equations

Author

Listed:
  • H. Xu

    (Ningbo University
    University of Dundee)

  • X. W. Chang

    (Ningbo University)

Abstract

We develop general approximate Newton methods for solving Lipschitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian, thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: the classic difference approximation method and the ε-generalized Jacobian method. The former can be applied to problems with specific structures, while the latter is expected to work well for general problems. Numerical tests show that the two methods are efficient. Finally, a norm-reducing technique for the global convergence of the generalized Newton method is briefly discussed.

Suggested Citation

  • H. Xu & X. W. Chang, 1997. "Approximate Newton Methods for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 373-394, May.
    • Handle: RePEc:spr:joptap:v:93:y:1997:i:2:d:10.1023_a:1022606224224
    DOI: 10.1023/A:1022606224224
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    References listed on IDEAS

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    1. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    2. Jong-Shi Pang, 1990. "Newton's Method for B-Differentiable Equations," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 311-341, May.
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    Cited by:

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    2. Long, Qiang & Wu, Changzhi & Wang, Xiangyu, 2015. "A system of nonsmooth equations solver based upon subgradient method," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 284-299.
    3. H. Xu, 2001. "Adaptive Smoothing Method, Deterministically Computable Generalized Jacobians, and the Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 215-224, April.

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