--- /dev/null
+%% Copyright (C) 2002 N.J.Higham
+%% Copyright (C) 2003 Andy Adler <adler@ncf.ca>
+%%
+%% 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/>.
+
+%%MDSMAX Multidirectional search method for direct search optimization.
+%% [x, fmax, nf] = MDSMAX(FUN, x0, STOPIT, SAVIT) attempts to
+%% maximize the function FUN, using the starting vector x0.
+%% The method of multidirectional search is used.
+%% Output arguments:
+%% x = vector yielding largest function value found,
+%% fmax = function value at x,
+%% nf = number of function evaluations.
+%% The iteration is terminated when either
+%% - the relative size of the simplex is <= STOPIT(1)
+%% (default 1e-3),
+%% - STOPIT(2) function evaluations have been performed
+%% (default inf, i.e., no limit), or
+%% - a function value equals or exceeds STOPIT(3)
+%% (default inf, i.e., no test on function values).
+%% The form of the initial simplex is determined by STOPIT(4):
+%% STOPIT(4) = 0: regular simplex (sides of equal length, the default),
+%% STOPIT(4) = 1: right-angled simplex.
+%% Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
+%% If a non-empty fourth parameter string SAVIT is present, then
+%% `SAVE SAVIT x fmax nf' is executed after each inner iteration.
+%% NB: x0 can be a matrix. In the output argument, in SAVIT saves,
+%% and in function calls, x has the same shape as x0.
+%% MDSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
+%% arguments to be passed to fun, via feval(fun,x,P1,P2,...).
+%%
+%% This implementation uses 2n^2 elements of storage (two simplices), where x0
+%% is an n-vector. It is based on the algorithm statement in [2, sec.3],
+%% modified so as to halve the storage (with a slight loss in readability).
+%%
+%% References:
+%% [1] V. J. Torczon, Multi-directional search: A direct search algorithm for
+%% parallel machines, Ph.D. Thesis, Rice University, Houston, Texas, 1989.
+% [2] V. J. Torczon, On the convergence of the multidirectional search
+%% algorithm, SIAM J. Optimization, 1 (1991), pp. 123-145.
+%% [3] N. J. Higham, Optimization by direct search in matrix computations,
+%% SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
+%% [4] N. J. Higham, Accuracy and Stability of Numerical Algorithms,
+%% Second edition, Society for Industrial and Applied Mathematics,
+%% Philadelphia, PA, 2002; sec. 20.5.
+
+% From Matrix Toolbox
+% Copyright (C) 2002 N.J.Higham
+% www.maths.man.ac.uk/~higham/mctoolbox
+% Modifications for octave by A.Adler 2003
+
+function [x, fmax, nf] = mdsmax(fun, x, stopit, savit, varargin)
+
+x0 = x(:); % Work with column vector internally.
+n = length(x0);
+
+mu = 2; % Expansion factor.
+theta = 0.5; % Contraction factor.
+
+% Set up convergence parameters etc.
+if nargin < 3
+ stopit(1) = 1e-3;
+elseif isempty(stopit)
+ stopit(1) = 1e-3;
+endif
+tol = stopit(1); % Tolerance for cgce test based on relative size of simplex.
+if length(stopit) == 1, stopit(2) = inf; end % Max no. of f-evaluations.
+if length(stopit) == 2, stopit(3) = inf; end % Default target for f-values.
+if length(stopit) == 3, stopit(4) = 0; end % Default initial simplex.
+if length(stopit) == 4, stopit(5) = 1; end % Default: show progress.
+trace = stopit(5);
+if length(stopit) == 5, stopit(6) = 1; end % Default: maximize
+dirn= stopit(6);
+if nargin < 4, savit = []; end % File name for snapshots.
+
+V = [zeros(n,1) eye(n)]; T = V;
+f = zeros(n+1,1); ft = f;
+V(:,1) = x0; f(1) = dirn*feval(fun,x,varargin{:});
+fmax_old = f(1);
+
+if trace, fprintf('f(x0) = %9.4e\n', f(1)), end
+
+k = 0; m = 0;
+
+% Set up initial simplex.
+scale = max(norm(x0,inf),1);
+if stopit(4) == 0
+ % Regular simplex - all edges have same length.
+ % Generated from construction given in reference [18, pp. 80-81] of [1].
+ alpha = scale / (n*sqrt(2)) * [ sqrt(n+1)-1+n sqrt(n+1)-1 ];
+ V(:,2:n+1) = (x0 + alpha(2)*ones(n,1)) * ones(1,n);
+ for j=2:n+1
+ V(j-1,j) = x0(j-1) + alpha(1);
+ x(:) = V(:,j); f(j) = dirn*feval(fun,x,varargin{:});
+ end
+else
+ % Right-angled simplex based on co-ordinate axes.
+ alpha = scale*ones(n+1,1);
+ for j=2:n+1
+ V(:,j) = x0 + alpha(j)*V(:,j);
+ x(:) = V(:,j); f(j) = dirn*feval(fun,x,varargin{:});
+ end
+end
+nf = n+1;
+size = 0; % Integer that keeps track of expansions/contractions.
+flag_break = 0; % Flag which becomes true when ready to quit outer loop.
+
+while 1 %%%%%% Outer loop.
+k = k+1;
+
+% Find a new best vertex x and function value fmax = f(x).
+[fmax,j] = max(f);
+V(:,[1 j]) = V(:,[j 1]); v1 = V(:,1);
+if ~isempty(savit), x(:) = v1; eval(['save ' savit ' x fmax nf']), end
+f([1 j]) = f([j 1]);
+if trace
+ fprintf('Iter. %2.0f, inner = %2.0f, size = %2.0f, ', k, m, size)
+ fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ...
+ 100*(fmax-fmax_old)/(abs(fmax_old)+eps))
+end
+fmax_old = fmax;
+
+% Stopping Test 1 - f reached target value?
+if fmax >= stopit(3)
+ msg = ['Exceeded target...quitting\n'];
+ break % Quit.
+end
+
+m = 0;
+while 1 %%% Inner repeat loop.
+ m = m+1;
+
+ % Stopping Test 2 - too many f-evals?
+ if nf >= stopit(2)
+ msg = ['Max no. of function evaluations exceeded...quitting\n'];
+ flag_break = 1; break % Quit.
+ end
+
+ % Stopping Test 3 - converged? This is test (4.3) in [1].
+ size_simplex = norm(V(:,2:n+1)- v1(:,ones(1,n)),1) / max(1, norm(v1,1));
+ if size_simplex <= tol
+ msg = sprintf('Simplex size %9.4e <= %9.4e...quitting\n', ...
+ size_simplex, tol);
+ flag_break = 1; break % Quit.
+ end
+
+ for j=2:n+1 % ---Rotation (reflection) step.
+ T(:,j) = 2*v1 - V(:,j);
+ x(:) = T(:,j); ft(j) = dirn*feval(fun,x,varargin{:});
+ end
+ nf = nf + n;
+
+ replaced = ( max(ft(2:n+1)) > fmax );
+
+ if replaced
+ for j=2:n+1 % ---Expansion step.
+ V(:,j) = (1-mu)*v1 + mu*T(:,j);
+ x(:) = V(:,j); f(j) = dirn*feval(fun,x,varargin{:});
+ end
+ nf = nf + n;
+ % Accept expansion or rotation?
+ if max(ft(2:n+1)) > max(f(2:n+1))
+ V(:,2:n+1) = T(:,2:n+1); f(2:n+1) = ft(2:n+1); % Accept rotation.
+ else
+ size = size + 1; % Accept expansion (f and V already set).
+ end
+ else
+ for j=2:n+1 % ---Contraction step.
+ V(:,j) = (1+theta)*v1 - theta*T(:,j);
+ x(:) = V(:,j); f(j) = dirn*feval(fun,x,varargin{:});
+ end
+ nf = nf + n;
+ replaced = ( max(f(2:n+1)) > fmax );
+ % Accept contraction (f and V already set).
+ size = size - 1;
+ end
+
+ if replaced, break, end
+ if (trace && rem(m, 10) == 0)
+ fprintf(' ...inner = %2.0f...\n', m);
+ end
+ end %%% Of inner repeat loop.
+
+if flag_break, break, end
+end %%%%%% Of outer loop.
+
+% Finished.
+if trace, fprintf(msg), end
+x(:) = v1;
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff