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IMA Journal of Numerical Analysis Advance Access originally published online on February 27, 2008
IMA Journal of Numerical Analysis 2008 28(4):649-664; doi:10.1093/imanum/drm047
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Developments of NEWUOA for minimization without derivatives

M. J. D. Powell{dagger}

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

1 Email: mjdp{at}cam.ac.uk

Received on 8 June 2007. Revised on 21 November 2007.


  Abstract

The NEWUOA software is described briefly, with some numerical results that show good efficiency and accuracy in the unconstrained minimization without derivatives of functions of up to 320 variables. Some preliminary work on an extension of NEWUOA that allows simple bounds on the variables is also described. It suggests a variation of a technique in NEWUOA for maintaining adequate linear independence in the interpolation conditions that are used, which leads to five versions of the technique including the original one. Numerical experiments suggest that the new versions have some merit, but the details of the calculations are influenced strongly by computer rounding errors. The dependence of the number of iterations on the number of interpolation conditions is also investigated numerically. A surprising case with n = 160 is found, n being the number of variables, where the number of iterations is reduced when the number of conditions is decreased from 2n + 1 to n + 6. The given conclusions may assist the development of some new software for unconstrained optimization.

Dedicated to Ya-Xiang Yuan, with much gratitude for my large family of research students in China, and for wonderful conferences that celebrated my 60th and 70th birthdays in Beijing.


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