Convergence properties of the Nelder-Mead simplex method in low dimensions

The Nelder-Mead simplex algorithm, first published in 1965, is an enormousl y popular direct search method for multidimensional unconstrained minimizat ion. Despite its widespread use, essentially no theoretical results have be en proved explicitly for the Nelder-Mead algorithm. This paper presents con vergence properties of the Nelder-Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for d imension 1, and various limited convergence results for dimension 2. A coun terexample of McKinnon gives a family of strictly convex functions in two d imensions and a set of initial conditions for which the Nelder-Mead algorit hm converges to a nonminimizer. It is not yet known whether the Nelder-Mead method can be proved to converge to a minimizer for a more specialized cla ss of convex functions in two dimensions.