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

M. C. Villalobos, R. A. Tapia and Y. Zhang
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M. C. Villalobos: University of Texas-Pan American
R. A. Tapia: Rice University
Y. Zhang: Rice University

Journal of Optimization Theory and Applications, 2004, vol. 121, issue 3, No 2, 489-514

Abstract: 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.

Keywords: Newton's method; equivalent systems; Newton's interior-point method; sphere of convergence (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1023/B:JOTA.0000037601.54325.3d

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