Dear Olaf,
Here is a new version that addresses your remarks.
To avoid eliciting a warning, I think optimset would need a new
option, which I've called SearchDirections, to hold the initial set of
direction vectors.
Best,
Nir
On Sat, Nov 26, 2011 at 4:19 PM, Olaf Till <olaf.t...@uni-jena.de> wrote:
> But a few remarks to some coding aspects.
>
> - It is now obsolete (and should
> not be done in new code) to have an argument ('args' in your case) with
> extra arguments for the user function. These extra arguments can be passed
> using anonymous functions, e.g.
>
> optimizer (@ (x) user_function (x, extra_argument, ...), ...)
>
> and need not be cared for by the optimizer. Only the argument which is
> optimized is to be cared for
>
> - For the optimization functions of core Octave, 'optimset' is now used
> for handling optimization options. In the optim package, some functions
> (which were programmed before Octaves 'optimset' was available)
> have different mechanisms of option handling. But this makes things more
> complicated. In new code as yours, we should stick to some standard of
> option handling. And as things stand now, the standard is Octaves 'optimset'.
> So I think you should re-consider the handling of the 'control' argument.
> Note that if there is a _good_ reason (which possibly should be discussed
> on the list) new optimset options can be defined.
>
> Olaf
## Copyright (C) 2011 Nir Krakauer
##
## This program is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} [@var{p}, @var{obj_value}, @var{convergence}, @var{iters}, @var{nevs}] = powell (@var{f}, @var{p0}, @var{control})
##powell: implements a direction-set (Powell's) method for multidimensional minimization of a function without calculation of the gradient [1, 2]
##
## @subheading Arguments
##
## @itemize @bullet
## @item
## @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)
##
## @item
## @var{p0}: An initial value of the function argument to minimize
##
## @item
## @var{options}: an optional structure, which can be generated by optimset, with some or all of the following fields:
## @itemize @minus
## @item
## MaxIter: maximum iterations (positive integer, or -1 or Inf for unlimited (default))
## @item
## TolFun: minimum amount by which function value must decrease in each iteration to continue (default is 1E-8)
## @item
## MaxFunEvals: maximum function evaluations (positive integer, or -1 or Inf for unlimited (default))
## @item
## 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)
## @end itemize
## @end itemize
##
## @subheading Examples
##
## @example
## @group
## y = @@(x, s) x(1) .^ 2 + x(2) .^ 2 + s;
## o = optimset('MaxIter', 100, 'TolFun', 1E-10);
## s = 1;
## [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
## %should return something like x_optim = [4E-14 3E-14], y_min = 1, conv = 1, iters = 2, nevs = 24
## @end group
##
## @end example
##
## @subheading Returns:
##
## @itemize @bullet
## @item
## @var{p}: the minimizing value of the function argument
## @item
## @var{obj_value}: the value of @var{f}() at @var{p}
## @item
## @var{convergence}: 1 if normal convergence, 0 if not
## @item
## @var{iters}: number of iterations performed
## @item
## @var{nevs}: number of function evaluations
## @end itemize
##
## @subheading References
##
## @enumerate
## @item
## 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
##
## @item
## 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)
## @end enumerate
## @end deftypefn
## Author: Nir Krakauer <nkraka...@ccny.cuny.edu>
## Description: Multidimensional minimization (direction-set method)
function [p, obj_value, convergence, iters, nevs] = powell(f, p0, options);
# check number of arguments
if ((nargin < 2) || (nargin > 3))
usage('powell: you must supply 2 or 3 arguments');
endif
%default or input values
if nargin < 3
options = struct ();
end
xi_set = 0;
xi = optimget(options, 'SearchDirections');
if ~isempty(xi)
if isvector(xi) %assume that xi is is n*1 or 1*n
xi = diag(x);
end
xi_set = 1;
end
MaxIter = optimget(options, 'MaxIter', Inf);
if MaxIter < 0, MaxIter = Inf; end
MaxFunEvals = optimget(options, 'MaxFunEvals', Inf);
TolFun = optimget(options, 'TolFun', 1E-8);
nevs = 0;
iters = 0;
convergence = 0;
p = p0; %initial value of the argument being minimized
try
obj_value = f(p);
catch
error("function does not exist or cannot be evaluated");
end
nevs++;
n = numel(p); %number of dimensions to minimize over
xit = zeros(n, 1);
if ~xi_set
xi = eye(n);
end
%do an iteration
while (iters <= MaxIter) && (nevs <= MaxFunEvals) && (~convergence)
iters++;
pt = p; %best point as iteration begins
fp = obj_value; %value of the objective function as iteration begins
ibig = 0; %will hold direction along which the objective function decreased the most in this iteration
dlt = 0; %will hold decrease in objective function value in this iteration
for i = 1:n
xit = reshape(xi(:, i), size(p));
fptt = obj_value;
[a, obj_value, nev] = line_min(f, xit, {p});
nevs = nevs + nev;
p = p + a*xit;
change = fptt - obj_value;
if (change > dlt)
dlt = change;
ibig = i;
end
end
if ( 2*abs(fp-obj_value) <= TolFun*(abs(fp) + abs(obj_value)) )
convergence = 1;
return
end
if iters == MaxIter
disp('iteration maximum exceeded')
return
end
%attempt parabolic extrapolation
ptt = 2*p - pt;
xit = p - pt;
fptt = f(ptt);
nevs++;
if fptt < fp %check whether the extrapolation actually makes the objective function smaller
t = 2 * (fp - 2*obj_value + fptt) * (fp-obj_value-dlt)^2 - dlt * (fp-fptt)^2;
if t < 0
p = ptt;
[a, obj_value, nev] = line_min(f, xit, {p});
nevs = nevs + nev;
p = p + a*xit;
%add the net direction from this iteration to the direction set
xi(:, ibig) = xi(:, n);
xi(:, n) = xit(:);
end
end
end
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