pracma (version 1.9.9)

nelder_mead: Nelder-Mead Minimization

Description

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

Usage

nelder_mead(x0, f, lb = NULL, ub = NULL, tol = 1e-10, maxfeval = 20000, step = rep(1.0, length(x0)), ...)

Arguments

x0

starting vector.

f

nonlinear function to be minimized.

lb, ub

lower and upper of a bounded region.

tol

relative tolerance, to be used as stopping rule.

maxfeval

maximum number of function calls.

step

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

...

additional arguments to be passed to the function.

Value

List with following components:

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.

`nelder_mead()' can be used up to 20 dimensions (then `tol' and `maxfeval' need to be increased). Since version 1.9.8 'nelder_mead()' applies adaptive parameters for the simplicial search, depending on the problem dimension -- see Fuchang and Lixing (2012).

With upper and/or lower bounds, `nelder_mead()' applies 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.

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.

See Also

hooke_jeeves

Examples

##  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)
}

nelder_mead(c(0,0,0,0), rosenbrock)
# $xmin
# [1] 1 1 1 1
# $fmin
# [1] 8.802801e-25
# $nfeval
# [1] 678
# $restarts
# [1] 0

nelder_mead(c(0,0,0,0), rosenbrock, rep(-0.5, 4), rep(0.5, 4))
# $xmin
# [1] 0.50000000 0.26221321 0.07797600 0.00608026
# $fmin
# [1] 1.667875
# $nfeval
# [1] 1501
# $restarts
# [1] 3