--- /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/>.
+
+%%NMSMAX Nelder-Mead simplex method for direct search optimization.
+%% [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to
+%% maximize the function FUN, using the starting vector x0.
+%% The Nelder-Mead direct search method 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).
+%% STOPIT(6) indicates the direction (ie. minimization or
+%% maximization.) Default is 1, maximization.
+%% set STOPIT(6)=-1 for minimization
+%% 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.
+%% NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
+%% arguments to be passed to fun, via feval(fun,x,P1,P2,...).
+%% References:
+%% N. J. Higham, Optimization by direct search in matrix computations,
+%% SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
+%% C. T. Kelley, Iterative Methods for Optimization, Society for Industrial
+%% and Applied Mathematics, Philadelphia, PA, 1999.
+
+% 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] = nmsmax(fun, x, stopit, savit, varargin)
+
+x0 = x(:); % Work with column vector internally.
+n = length(x0);
+
+% Set up convergence parameters etc.
+if (nargin < 3 || isempty(stopit))
+ stopit(1) = 1e-3;
+end
+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)];
+f = zeros(n+1,1);
+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;
+how = 'initial ';
+
+[temp,j] = sort(f);
+j = j(n+1:-1:1);
+f = f(j); V = V(:,j);
+
+alpha = 1; beta = 1/2; gamma = 2;
+
+while 1 %%%%%% Outer (and only) loop.
+k = k+1;
+
+ fmax = f(1);
+ if fmax > fmax_old
+ if ~isempty(savit)
+ x(:) = V(:,1); eval(['save ' savit ' x fmax nf'])
+ end
+ end
+ if trace
+ fprintf('Iter. %2.0f,', k)
+ fprintf([' how = ' how ' ']);
+ fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ...
+ 100*(fmax-fmax_old)/(abs(fmax_old)+eps))
+ end
+ fmax_old = fmax;
+
+ %%% Three stopping tests from MDSMAX.M
+
+ % Stopping Test 1 - f reached target value?
+ if fmax >= stopit(3)
+ msg = ['Exceeded target...quitting\n'];
+ break % Quit.
+ end
+
+ % Stopping Test 2 - too many f-evals?
+ if nf >= stopit(2)
+ msg = ['Max no. of function evaluations exceeded...quitting\n'];
+ break % Quit.
+ end
+
+ % Stopping Test 3 - converged? This is test (4.3) in [1].
+ v1 = V(:,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);
+ break % Quit.
+ end
+
+ % One step of the Nelder-Mead simplex algorithm
+ % NJH: Altered function calls and changed CNT to NF.
+ % Changed each `fr < f(1)' type test to `>' for maximization
+ % and re-ordered function values after sort.
+
+ vbar = (sum(V(:,1:n)')/n)'; % Mean value
+ vr = (1 + alpha)*vbar - alpha*V(:,n+1);
+ x(:) = vr;
+ fr = dirn*feval(fun,x,varargin{:});
+ nf = nf + 1;
+ vk = vr; fk = fr; how = 'reflect, ';
+ if fr > f(n)
+ if fr > f(1)
+ ve = gamma*vr + (1-gamma)*vbar;
+ x(:) = ve;
+ fe = dirn*feval(fun,x,varargin{:});
+ nf = nf + 1;
+ if fe > f(1)
+ vk = ve; fk = fe;
+ how = 'expand, ';
+ end
+ end
+ else
+ vt = V(:,n+1); ft = f(n+1);
+ if fr > ft
+ vt = vr; ft = fr;
+ end
+ vc = beta*vt + (1-beta)*vbar;
+ x(:) = vc;
+ fc = dirn*feval(fun,x,varargin{:});
+ nf = nf + 1;
+ if fc > f(n)
+ vk = vc; fk = fc;
+ how = 'contract,';
+ else
+ for j = 2:n
+ V(:,j) = (V(:,1) + V(:,j))/2;
+ x(:) = V(:,j);
+ f(j) = dirn*feval(fun,x,varargin{:});
+ end
+ nf = nf + n-1;
+ vk = (V(:,1) + V(:,n+1))/2;
+ x(:) = vk;
+ fk = dirn*feval(fun,x,varargin{:});
+ nf = nf + 1;
+ how = 'shrink, ';
+ end
+ end
+ V(:,n+1) = vk;
+ f(n+1) = fk;
+ [temp,j] = sort(f);
+ j = j(n+1:-1:1);
+ f = f(j); V = V(:,j);
+
+end %%%%%% End of outer (and only) loop.
+
+% Finished.
+if trace, fprintf(msg), end
+x(:) = V(:,1);
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