anms: Adaptive Nelder-Mead Minimization

In pracma: Practical Numerical Math Functions

Adaptive Nelder-Mead Minimization

Description

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

Usage

anms(fn, x0, ...,
     tol = 1e-10, maxfeval = NULL)

Arguments

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.

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.

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.

Value

List with following components:

xmin	minimum solution found.
fmin	value of f at minimum.
nfeval	number of function calls performed.

Note

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’.

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

optim

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)
}

anms(rosenbrock, c(0,0,0,0,0))
# $xmin
# [1] 1 1 1 1 1
# $fmin
# [1] 8.268732e-21
# $nfeval
# [1] 1153

# To add constraints to the optimization problem, use a slightly 
# modified objective function. Equality constraints not possible.
# Warning: Avoid a starting value too near to the boundary !

## Not run: 
# Example: 0.0 <= x <= 0.5
fun <- function(x) {
    if (any(x < 0) || any(x > 0.5)) 100
    else rosenbrock(x)
}
x0 <- rep(0.1, 5)

anms(fun, x0)
## $xmin
## [1] 0.500000000 0.263051265 0.079972922 0.016228138 0.000267922
## End(Not run)

pracma documentation built on March 19, 2024, 3:05 a.m.