Nelder-Mead Function Minimization Method

An implementation of the Nelder-Mead algorithm for derivative-free optimization / function minimization.

nelder_mead(fn, x0, ..., adapt = TRUE,
            tol = 1e-08, maxfeval = 5000, 
      step = rep(1.0, length(x0)))

Arguments

fn

nonlinear function to be minimized.

x0

starting point for the iteration.

...

additional arguments to be passed to the function.

adapt

logical; adapt to parameter dimension.

tol

terminating limit for the variance of function values; can be made *very* small, like tol=1e-50.

maxfeval

maximum number of function evaluations.

step

size and shape of initial simplex; relative magnitudes of its elements should reflect the units of the variables.

Details

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.

The simplex function minimisation procedure due to Nelder and Mead (1965), as implemented by O'Neill (1971), with subsequent comments by Chambers and Ertel 1974, Benyon 1976, and Hill 1978. For another elaborate implementation of Nelder-Mead in R based on Matlab code by Kelley see package `dfoptim'.

nelder_mead can be used up to 20 dimensions (then `tol' and `maxfeval' need to be increased). With adapt=TRUE it applies adaptive coefficients for the simplicial search, depending on the problem dimension – see Fuchang and Lixing (2012). This approach especially reduces the number of function calls.

Value

List with following components:

xmin

minimum solution found.

fmin

value of f at minimum.

count

number of iterations performed.

info

list with solver name and no. of restarts.

References

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.

Note

Original FORTRAN77 version by R O'Neill; MATLAB version by John Burkardt under LGPL license. Re-implemented in R by Hans W. Borchers.

See also

hooke_jeeves

Examples