Convergence of the Nelder - Mead Simplex Method Toa Non - Stationary Pointk

This paper analyses the behaviour of the Nelder-Mead simplex method for a family of examples which cause the method to converge to a non-stationary point. All the examples use continuous functions of two variables. The family of functions contains strictly convex functions with up to three continuous derivatives. In all the examples the method repeatedly applies the inside contraction step with the best vertex remaining xed. The simplices tend to a straight line which is orthogonal to the steepest descent direction. It is shown that this behaviour cannot occur for functions with more than three continuous derivatives. The stability of the examples is analysed.