1 ## Copyright (C) 2011 Nir Krakauer <nkrakauer@ccny.cuny.edu>
3 ## This program is free software; you can redistribute it and/or modify it under
4 ## the terms of the GNU General Public License as published by the Free Software
5 ## Foundation; either version 3 of the License, or (at your option) any later
8 ## This program is distributed in the hope that it will be useful, but WITHOUT
9 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
10 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
13 ## You should have received a copy of the GNU General Public License along with
14 ## this program; if not, see <http://www.gnu.org/licenses/>.
17 ## @deftypefn {Function File} [@var{p}, @var{obj_value}, @var{convergence}, @var{iters}, @var{nevs}] = powell (@var{f}, @var{p0}, @var{control})
18 ## Multidimensional minimization (direction-set method). Implements a direction-set (Powell's) method for multidimensional minimization of a function without calculation of the gradient [1, 2]
20 ## @subheading Arguments
24 ## @var{f}: name of function to minimize (string or handle), which should accept one input variable (see example for how to pass on additional input arguments)
27 ## @var{p0}: An initial value of the function argument to minimize
30 ## @var{options}: an optional structure, which can be generated by optimset, with some or all of the following fields:
33 ## MaxIter: maximum iterations (positive integer, or -1 or Inf for unlimited (default))
35 ## TolFun: minimum amount by which function value must decrease in each iteration to continue (default is 1E-8)
37 ## MaxFunEvals: maximum function evaluations (positive integer, or -1 or Inf for unlimited (default))
39 ## SearchDirections: an n*n matrix whose columns contain the initial set of (presumably orthogonal) directions to minimize along, where n is the number of elements in the argument to be minimized for; or an n*1 vector of magnitudes for the initial directions (defaults to the set of unit direction vectors)
43 ## @subheading Examples
47 ## y = @@(x, s) x(1) ^ 2 + x(2) ^ 2 + s;
48 ## o = optimset('MaxIter', 100, 'TolFun', 1E-10);
50 ## [x_optim, y_min, conv, iters, nevs] = powell(@@(x) y(x, s), [1 0.5], o); %pass y wrapped in an anonymous function so that all other arguments to y, which are held constant, are set
51 ## %should return something like x_optim = [4E-14 3E-14], y_min = 1, conv = 1, iters = 2, nevs = 24
56 ## @subheading Returns:
60 ## @var{p}: the minimizing value of the function argument
62 ## @var{obj_value}: the value of @var{f}() at @var{p}
64 ## @var{convergence}: 1 if normal convergence, 0 if not
66 ## @var{iters}: number of iterations performed
68 ## @var{nevs}: number of function evaluations
71 ## @subheading References
75 ## Powell MJD (1964), An efficient method for finding the minimum of a function of several variables without calculating derivatives, @cite{Computer Journal}, 7 :155-162
78 ## Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (1992). @cite{Numerical Recipes in Fortran: The Art of Scientific Computing} (2nd Ed.). New York: Cambridge University Press (Section 10.5)
82 ## PKG_ADD: __all_opts__ ("powell");
84 function [p, obj_value, convergence, iters, nevs] = powell (f, p0, options);
86 if (nargin == 1 && ischar (f) && strcmp (f, "defaults"))
87 p = optimset ("MaxIter", Inf, \
90 "SearchDirections", []);
94 ## check number of arguments
95 if ((nargin < 2) || (nargin > 3))
96 usage('powell: you must supply 2 or 3 arguments');
100 ## default or input values
107 xi = optimget (options, 'SearchDirections');
109 if (isvector (xi)) # assume that xi is is n*1 or 1*n
116 MaxIter = optimget (options, 'MaxIter', Inf);
117 if (MaxIter < 0) MaxIter = Inf; endif
118 MaxFunEvals = optimget (options, 'MaxFunEvals', Inf);
119 TolFun = optimget (options, 'TolFun', 1E-8);
125 p = p0; # initial value of the argument being minimized
130 error ("function does not exist or cannot be evaluated");
135 n = numel (p); # number of dimensions to minimize over
145 while (iters <= MaxIter && nevs <= MaxFunEvals && ! convergence)
147 pt = p; # best point as iteration begins
148 fp = obj_value; # value of the objective function as iteration begins
149 ibig = 0; # will hold direction along which the objective function decreased the most in this iteration
150 dlt = 0; # will hold decrease in objective function value in this iteration
152 xit = reshape (xi(:, i), size(p));
154 [a, obj_value, nev] = line_min (f, xit, {p});
157 change = fptt - obj_value;
164 if ( 2*abs(fp-obj_value) <= TolFun*(abs(fp) + abs(obj_value)) )
169 if (iters == MaxIter)
170 disp ("iteration maximum exceeded");
174 ## attempt parabolic extrapolation
179 if (fptt < fp) # check whether the extrapolation actually makes the objective function smaller
180 t = 2 * (fp - 2*obj_value + fptt) * (fp-obj_value-dlt)^2 - dlt * (fp-fptt)^2;
183 [a, obj_value, nev] = line_min (f, xit, {p});
187 ## add the net direction from this iteration to the direction set
188 xi(:, ibig) = xi(:, n);