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Sphere of Convergence of Newton's Method on Two Equivalent Systems from Nonlinear Programming

Author

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  • M. C. Villalobos

    (University of Texas-Pan American)

  • R. A. Tapia

    (Rice University)

  • Y. Zhang

    (Rice University)

Abstract

We study a local feature of two interior-point methods: a logarithmic barrier function method and a primal-dual method. In particular, we provide an asymptotic analysis on the radius of the sphere of convergence of Newton's method on two equivalent systems associated with the two aforementioned interior-point methods for nondegenerate nonlinear programs. We show that the radii of the spheres of convergence have different asymptotic behavior, as the two methods attempt to follow a solution trajectory {x μ} that, under suitable conditions, converges to a solution as μ → 0. We show that, in the case of the barrier function method, the radius of the sphere of convergence of Newton's method is Θ (μ), while for the primal-dual method the radius is bounded away from zero as μ → 0. This work is an extension of the authors earlier work (Ref. 1) on linear programs.

Suggested Citation

  • M. C. Villalobos & R. A. Tapia & Y. Zhang, 2004. "Sphere of Convergence of Newton's Method on Two Equivalent Systems from Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 489-514, June.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:3:d:10.1023_b:jota.0000037601.54325.3d
    DOI: 10.1023/B:JOTA.0000037601.54325.3d
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    References listed on IDEAS

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    1. C. Villalobos & R. Tapia & Y. Zhang, 2002. "Local Behavior of the Newton Method on Two Equivalent Systems from Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 239-263, February.
    2. McLINDEN, L., 1980. "An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem," LIDAM Reprints CORE 443, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. L. A. Melara & A. J. Kearsley & R. A. Tapia, 2007. "Augmented Lagrangian Homotopy Method for the Regularization of Total Variation Denoising Problems," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 15-25, July.

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