The Nelder–Mead Simplex Algorithm Is Sixty Years Old: New Convergence Results and Open Questions

We investigate and compare two versions of the Nelder–Mead simplex algorithm for function minimization. Two types of convergence are studied: the convergence of function values at the simplex vertices and convergence of the simplex sequence. For the first type of convergence, we generalize the main result of Lagarias, Reeds, Wright and Wright (1998). For the second type of convergence, we also improve recent results which indicate that the Lagarias et al.’s version of the Nelder–Mead algorithm has better convergence properties than the original Nelder–Mead method. This paper concludes with some open questions.