Nelder-Mead Minimization Method

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Description

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

Usage

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nelmin(fn, x0, tol = 1e-10, ..., 
        maxfeval = 1e4, step = rep(1.0, length(x0)))

nelminb(fn, x0, lower, upper, tol = 1e-10, ..., 
        maxfeval = 10000, step = rep(1, length(x0)))

Arguments

fn

nonlinear function to be minimized.

x0

starting point for the iteration.

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.

...

additional arguments to be passed to the function.

lower, upper

lower and upper bounds.

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’.

nelminb uses transfinite to define the function on all of R^n and to retransform the solution to the bounded domain. The starting value is not allowed to lie on the boundary.

Value

List with following components:

xmin

minimum solution found.

fmin

value of f at minimum.

fcount

number of iterations performed.

restarts

number of restarts.

errmess

error message

Note

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

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.

See Also

dfoptim::nmk

Examples

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##  Classical tests as in the article by Nelder and Mead
# Rosenbrock's parabolic valley
rpv <- function(x) 100*(x[2] - x[1]^2)^2 + (1 - x[1])^2
x0 <- c(-2, 1)
nelmin(rpv, x0)                         #  1 1

# Fletcher and Powell's helic valley
fphv <- function(x)
    100*(x[3] - 10*atan2(x[2], x[1])/(2*pi))^2 + 
        (sqrt(x[1]^2 + x[2]^2) - 1)^2 +x[3]^2
x0 <- c(-1, 0, 0)
nelmin(fphv, x0)                        #  1 0 0

# Powell's Singular Function (PSF)
psf <- function(x)  (x[1] + 10*x[2])^2 + 5*(x[3] - x[4])^2 + 
                    (x[2] - 2*x[3])^4 + 10*(x[1] - x[4])^4
x0 <- c(3, -1, 0, 1)
nelmin(psf, x0)         #  0 0 0 0, needs maximum number of function calls

# Bounded version of Nelder-Mead
lower <- c(-Inf, 0,   0)
upper <- c( Inf, 0.5, 1)
x0 <- c(0, 0.1, 0.1)
nelminb(fnRosenbrock, c(0, 0.1, 0.1), lower, upper)
# $xmin = c(0.7085595, 0.5000000, 0.2500000)
# $fmin = 0.3353605

## Not run: 
# Can run Rosenbrock's function in 30 dimensions in one and a half minutes:
nelmin(fnRosenbrock, rep(0, 30), tol=1e-20, maxfeval=10^7)
# $xmin
#  [1]  0.9999998 1.0000004 1.0000000 1.0000001 1.0000000 1.0000001
#  [7]  1.0000002 1.0000001 0.9999997 0.9999999 0.9999997 1.0000000
# [13]  0.9999999 0.9999994 0.9999998 0.9999999 0.9999999 0.9999999
# [19]  0.9999999 1.0000001 0.9999998 1.0000000 1.0000003 0.9999999
# [25]  1.0000000 0.9999996 0.9999995 0.9999990 0.9999973 0.9999947
# $fmin
# [1] 5.617352e-10
# $fcount
# [1] 1426085
# elapsed time is 96.008000 seconds 
## End(Not run)