Minimization of a multidimensional function using some of the algorithms described in:

http://www.gnu.org/software/gsl/manual/html_node/Multidimensional-Minimization.html

The example in the GSL manual:

f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30

main = do
    let (s,p) = minimize NMSimplex2 1E-2 30 [1,1] f [5,7]
    print s
    print p

> main
[0.9920430849306288,1.9969168063253182]
 0.000  512.500  1.130  6.500  5.000
 1.000  290.625  1.409  5.250  4.000
 2.000  290.625  1.409  5.250  4.000
 3.000  252.500  1.409  5.500  1.000
 ...
22.000   30.001  0.013  0.992  1.997
23.000   30.001  0.008  0.992  1.997

The path to the solution can be graphically shown by means of:

Graphics.Plot.mplot $ drop 3 (toColumns p)

Taken from the GSL manual:

The vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.

The bfgs2 version of this minimizer is the most efficient version available, and is a faithful implementation of the line minimization scheme described in Fletcher's Practical Methods of Optimization, Algorithms 2.6.2 and 2.6.4. It supercedes the original bfgs routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance tol corresponds to the parameter sigma used by Fletcher. A value of 0.1 is recommended for typical use (larger values correspond to less accurate line searches).

The nmsimplex2 version is a new O(N) implementation of the earlier O(N^2) nmsimplex minimiser. It calculates the size of simplex as the rms distance of each vertex from the center rather than the mean distance, which has the advantage of allowing a linear update.