Acta Numerica

Research Article

Direct search algorithms for optimization calculations

M. J. D. Powella1

a1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CBS 9EW, England. E-mail: mjdp@damtp.cam.ac.uk

Abstract

Many different procedures have been proposed for optimization calculations when first derivatives are not available. Further, several researchers have contributed to the subject, including some who wish to prove convergence theorems, and some who wish to make any reduction in the least calculated value of the objective function. There is not even a key idea that can be used as a foundation of a review, except for the problem itself, which is the adjustment of variables so that a function becomes least, where each value of the function is returned by a subroutine for each trial vector of variables. Therefore the paper is a collection of essays on particular strategies and algorithms, in order to consider the advantages, limitations and theory of several techniques. The subjects addressed are line search methods, the restriction of vectors of variables to discrete grids, the use of geometric simplices, conjugate direction procedures, trust region algorithms that form linear or quadratic approximations to the objective function, and simulated annealing. We study the main features of the methods themselves, instead of providing a catalogue of references to published work, because an understanding of these features may be very helpful to future research.