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

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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.

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Authors and Affiliations

  1. Assistant Professor, Department of Mathematics, University of Texas-Pan American, Edinburg, Texas

    M. C. Villalobos

  2. Professor, Department of Computational and Applied Mathematics, Rice University, Houston, Texas

    R. A. Tapia & Y. Zhang

Authors
  1. M. C. Villalobos
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  2. R. A. Tapia
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  3. Y. Zhang
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Villalobos, M.C., Tapia, R.A. & Zhang, Y. Sphere of Convergence of Newton's Method on Two Equivalent Systems from Nonlinear Programming. Journal of Optimization Theory and Applications 121, 489–514 (2004). https://doi.org/10.1023/B:JOTA.0000037601.54325.3d

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  • DOI: https://doi.org/10.1023/B:JOTA.0000037601.54325.3d

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