An adaptive search algorithm for numerical optimization
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.
scientific article; zbMATH DE number 4045041