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Local Behavior of the Newton Method on Two Equivalent Systems from Linear Programming

Author

Listed:
  • C. Villalobos

    (University of Texas at El Paso)

  • R. Tapia

    (Rice University)

  • Y. Zhang

    (Rice University)

Abstract

Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems, i.e., problems with the same solution but different mathematical formulations. In this paper, we investigate differences in the local behavior of Newton's method when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation and the equations in the so-called perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more effectively on the latter system.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:112:y:2002:i:2:d:10.1023_a:1013665605315
    DOI: 10.1023/A:1013665605315
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    References listed on IDEAS

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    1. 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. D.C. Jamrog & R.A. Tapia & Y. Zhang, 2002. "Comparison of Two Sets of First-Order Conditions as Bases of Interior-Point Newton Methods for Optimization with Simple Bounds," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 21-40, April.
    2. 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.
    3. 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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