Adaptive Nelder-Mead Minimization
An implementation of the Nelder-Mead algorithm for derivative-free optimization / function minimization.
anms(fn, x0, ...,
tol = 1e-10, maxfeval = NULL)fn |
nonlinear function to be minimized. |
x0 |
starting vector. |
tol |
relative tolerance, to be used as stopping rule. |
maxfeval |
maximum number of function calls. |
... |
additional arguments to be passed to the function. |
Also called a ‘simplex’ method for finding the local minimum of a function of several variables. The method is a pattern search that compares function values at the vertices of the simplex. The process generates a sequence of simplices with ever reducing sizes.
anms can be used up to 20 or 30 dimensions (then ‘tol’ and ‘maxfeval’
need to be increased). It applies adaptive parameters for simplicial search,
depending on the problem dimension – see Fuchang and Lixing (2012).
With upper and/or lower bounds, anms will apply a transformation of
bounded to unbounded regions before utilizing Nelder-Mead. Of course, if the
optimum is near to the boundary, results will not be as accurate as when the
minimum is in the interior.
List with following components:
xmin |
minimum solution found. |
fmin |
value of |
nfeval |
number of function calls performed. |
Copyright (c) 2012 by F. Gao and L. Han, implemented in Matlab with a permissive license. Implemented in R by Hans W. Borchers. For another elaborate implementation of Nelder-Mead see the package ‘dfoptim’.
Nelder, J., and R. Mead (1965). A simplex method for function minimization. Computer Journal, Volume 7, pp. 308-313.
O'Neill, R. (1971). Algorithm AS 47: Function Minimization Using a Simplex Procedure. Applied Statistics, Volume 20(3), pp. 338-345.
J. C. Lagarias et al. (1998). Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM Journal for Optimization, Vol. 9, No. 1, pp 112-147.
Fuchang Gao and Lixing Han (2012). Implementing the Nelder-Mead simplex algorithm with adaptive parameters. Computational Optimization and Applications, Vol. 51, No. 1, pp. 259-277.
## Rosenbrock function
rosenbrock <- function(x) {
n <- length(x)
x1 <- x[2:n]
x2 <- x[1:(n-1)]
sum(100*(x1-x2^2)^2 + (1-x2)^2)
}
anms(rosenbrock, c(0,0,0,0,0))
# $xmin
# [1] 1 1 1 1 1
# $fmin
# [1] 8.268732e-21
# $nfeval
# [1] 1153Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.