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Global convergence and the Powell singular function

Trond Steihaug (Trond.Steihaug***at***ii.uib.no)
Sara Suleiman (Sara.Suleiman***at***ii.uib.no)

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.

Keywords: System of nonlinear equations - global optimization - higher order methods - Newton's method

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Unconstrained Optimization )

Category 3: Nonlinear Optimization (Nonlinear Systems and Least-Squares )

Citation: Trond Steihaug and Sara Suleiman, Journal of Global Optimization, 2012, DOI:10.1007/s10898-012-9898-z

Download: [PDF]

Entry Submitted: 03/06/2012
Entry Accepted: 03/06/2012
Entry Last Modified: 04/24/2012

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