Comments on “Fitting Parameters to Complex Models by Direct Search”
R. R. Meyer, Michael H. Rothkopf · Journal of Marketing Research · 1971
purpose of this note is to present a different view of the state of the art of nonlinear regression methods than that presented in Van Wormer and Weiss' recent JMR article, Fitting Parameters to Complex Models by Direct Search, and to call attention to certain mathematical assertions that we feel are misleading. Van Wormer and Weiss assert, The power of direct search rests (1) in its built-in strategy for deciding where on the response surface to search next for improvements and (2) the speed of the data processing equipment [4, p. 504]. They fail to point out that Levenberg's original procedure [1] and its subsequent variants [2, 3] not only have built-in (and rather more sophisticated) strategies for locating new trial parameter values, but are as suitable for implementation on high-speed computers as any other numerical algorithm. In fact, rather than merely being classical techniques, nonlinear regression methods are commonly used for the numerical solution of nonlinear least squares problems, since these methods exploit the special structure of these problems and, as a consequence, usually converge at a rapid rate. This is in sharp contrast to the poor convergence properties of direct search acknowledged by Van Wormer and Weiss and illustrated in their numerical example. It should also be recognized that the sum of squares function for a nonlinear model will, in general, not be unimodal even when the model is well behaved, as Van Wormer and Weiss claim. operation of squaring the differences between the observed values and the model tends to introduce valleys not present in the model itself.