An adaptive search algorithm for numerical optimization
type: Item
instance of: scholarly article
title: An adaptive search algorithm for numerical optimization
zbMATH Open document ID: 0641.65055
DOI: 10.1016/0096-3003(87)90060-9
author: B. George
published in: Applied Mathematics and Computation
publication date: 1987-01-01T00:00:00Z
review text: This paper takes a new look at the popular simplex method of \textit{A. J. Nelder} and \textit{R. Mead} [(*) A simplex method for function minimization, Comput. J. 308 (1964)], with which we compute minima of functions of several variables without computing any first or second derivatives. A section of the paper carefully and clearly reviews the simplex method (*); it then presents two improvements to the basic algorithm. Finally, the results of numerical experiments appear. The algorithm (*) involves reflection, expansion, and contraction steps. The authors replace the expansion and reflection by a special line search step. The line search itself is interesting, since it involves using Fibonacci ratios both to first expand the original interval to bracket the minimum and then to contract the bracketing interval. The line search makes the algorithm (*) more efficient and less sensitive to scaling. Viewed another way, it relieves the user of some of the burden of ``tuning'' the algorithm. The second proposed modification is use of a stochastic algorithm to determine a starting simplex for the algorithm (*), in order to converge to a global minimum instead of just to local one. The numerical tests include Rosenbrock's functions and variants, and a special multimodal function. Besides being clearly written, the paper is well illustrated.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4045041
zbMATH Keywords: nonlinear optimization
zbMATH Keywords: derivative-free optimization
zbMATH Keywords: simplex method
zbMATH Keywords: numerical experiments
zbMATH Keywords: reflection
zbMATH Keywords: expansion
zbMATH Keywords: contraction
zbMATH Keywords: line search
zbMATH Keywords: scaling
zbMATH Keywords: stochastic algorithm
zbMATH Keywords: global minimum
zbMATH Keywords: numerical tests
zbMATH Keywords: Rosenbrock's functions
reviewed by: Ralph Baker Kearfott
MaRDI profile type: Publication
full work available at URL: 0096-3003(87)90060-9
OpenAlex ID: W1990584978
cites work: Q5652460
cites work: A Simplex Method for Function Minimization
description: scientific article; zbMATH DE number 4045041