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Global convergence and the Powell singular function

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  • Trond Steihaug
  • Sara Suleiman

Abstract

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton’s method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton’s method and methods in the Halley class can be extended to this class. Newton’s method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton’s method. We illustrate some of these properties with numerical experiments. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Trond Steihaug & Sara Suleiman, 2013. "Global convergence and the Powell singular function," Journal of Global Optimization, Springer, vol. 56(3), pages 845-853, July.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:3:p:845-853
    DOI: 10.1007/s10898-012-9898-z
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    References listed on IDEAS

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    1. Jiaqiao Hu & Michael C. Fu & Steven I. Marcus, 2007. "A Model Reference Adaptive Search Method for Global Optimization," Operations Research, INFORMS, vol. 55(3), pages 549-568, June.
    2. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, September.
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