Nelder, Mead, and the Other Simplex Method

In the mid-1960s, two English statisticians working at the National Vegetable Research Station invented the Nelder–Mead “simplex” direct search method. The method emerged at a propitious time, when there was great and growing interest in computer solution of complex nonlinear real-world optimization problems. Because obtaining first derivatives of the function f to be optimized was frequently impossible, the strong preference of most practitioners was for a “direct search” method that required only the values of f ; the new Nelder– Mead method fit the bill perfectly. Since then, the Nelder–Mead method has consistently been one of the most used and cited methods for unconstrained optimization. We are fortunate indeed that the late John Nelder has left us a detailed picture of the method’s inspiration and development [11, 14]. For Nelder, the starting point was a 1963 conference talk by William Spendley of Imperial Chemical Industries about a “simplex” method recently proposed by Spendley, Hext, and Himsworth for response surface exploration [15]. Despite its name, this method is not related to George Dantzig’s simplex method for linear programming, which dates from 1947. Nonetheless, the name is entirely appropriate because the Spendley, Hext, and Himsworth method is defined by a simplex; the method constructs a pattern of n + 1 points in dimension n, which moves across the surface to be explored, sometimes changing size, but always retaining the same shape. Inspired by Spendley’s talk, Nelder had what he describes as “one useful new idea”: while defining each iteration via a simplex, add the crucial ingredient that the shape of the simplex should “adapt itself to the local landscape” [12]. During a sequence of lively discussions with his colleague Roger Mead, where “each of us [was] able to try out the ideas of the previous evening on the other the following morning”, they developed a method in which the simplex could “elongate itself to move down long gentle slopes”, or “contract itself on to the final minimum” [11]. And, as they say, the rest is history.