--- /dev/null
+## Copyright (C) 2002-2008 Etienne Grossmann <etienne@egdn.net>
+##
+## 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/>.
+
+## [x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization
+##
+## Minimize 'f' using the Nelder-Mead algorithm. This function is inspired
+## from the that found in the book "Numerical Recipes".
+##
+## ARGUMENTS
+## ---------
+## f : string : Name of function. Must return a real value
+## args : list : Arguments passed to f.
+## or matrix : f's only argument
+## ctl : vector : (Optional) Control variables, described below
+## or struct
+##
+## RETURNED VALUES
+## ---------------
+## x0 : matrix : Local minimum of f
+## v : real : Value of f in x0
+## nev : number : Number of function evaluations
+##
+## CONTROL VARIABLE : (optional) may be named arguments (i.e. "name",value
+## ------------------ pairs), a struct, or a vector of length <= 6, where
+## NaN's are ignored. Default values are written <value>.
+## OPT. VECTOR
+## NAME POS
+## ftol,f N/A : Stopping criterion : stop search when values at simplex
+## vertices are all alike, as tested by
+##
+## f > (max_i (f_i) - min_i (f_i)) /max(max(|f_i|),1)
+##
+## where f_i are the values of f at the vertices. <10*eps>
+##
+## rtol,r N/A : Stop search when biggest radius of simplex, using
+## infinity-norm, is small, as tested by :
+##
+## ctl(2) > Radius <10*eps>
+##
+## vtol,v N/A : Stop search when volume of simplex is small, tested by
+##
+## ctl(2) > Vol
+##
+## crit,c ctl(1) : Set one stopping criterion, 'ftol' (c=1), 'rtol' (c=2)
+## or 'vtol' (c=3) to the value of the 'tol' option. <1>
+##
+## tol, t ctl(2) : Threshold in termination test chosen by 'crit' <10*eps>
+##
+## narg ctl(3) : Position of the minimized argument in args <1>
+## maxev ctl(4) : Maximum number of function evaluations. This number <inf>
+## may be slightly exceeded.
+## isz ctl(5) : Size of initial simplex, which is : <1>
+##
+## { x + e_i | i in 0..N }
+##
+## Where x == args{narg} is the initial value
+## e_0 == zeros (size (x)),
+## e_i(j) == 0 if j != i and e_i(i) == ctl(5)
+## e_i has same size as x
+##
+## Set ctl(5) to the distance you expect between the starting
+## point and the minimum.
+##
+## rst ctl(6) : When a minimum is found the algorithm restarts next to
+## it until the minimum does not improve anymore. ctl(6) is
+## the maximum number of restarts. Set ctl(6) to zero if
+## you know the function is well-behaved or if you don't
+## mind not getting a true minimum. <0>
+##
+## verbose, v Be more or less verbose (quiet=0) <0>
+
+function [x,v,nev] = nelder_mead_min (f, args, varargin)
+
+verbose = 0;
+
+ # Default control variables
+ftol = rtol = 10*eps; # Stop either by likeness of values or
+vtol = nan; # radius, but don't care about volume.
+crit = 0; # Stopping criterion ctl(1)
+tol = 10*eps; # Stopping test's threshold ctl(2)
+narg = 1; # Position of minimized arg ctl(3)
+maxev = inf; # Max num of func evaluations ctl(4)
+isz = 1; # Initial size ctl(5)
+rst = 0; # Max # of restarts
+
+
+if nargin >= 3, # Read control arguments
+ va_arg_cnt = 1;
+ if nargin > 3,
+ ctl = struct (varargin{:});
+ else
+ ctl = varargin{va_arg_cnt++};
+ end
+ if isnumeric (ctl)
+ if length (ctl)>=1 && !isnan (ctl(1)), crit = ctl(1); end
+ if length (ctl)>=2 && !isnan (ctl(2)), tol = ctl(2); end
+ if length (ctl)>=3 && !isnan (ctl(3)), narg = ctl(3); end
+ if length (ctl)>=4 && !isnan (ctl(4)), maxev = ctl(4); end
+ if length (ctl)>=5 && !isnan (ctl(5)), isz = ctl(5); end
+ if length (ctl)>=6 && !isnan (ctl(6)), rst = ctl(6); end
+ else
+ if isfield (ctl, "crit") && ! isnan (ctl.crit ), crit = ctl.crit ; end
+ if isfield (ctl, "tol") && ! isnan (ctl.tol ), tol = ctl.tol ; end
+ if isfield (ctl, "ftol") && ! isnan (ctl.ftol ), ftol = ctl.ftol ; end
+ if isfield (ctl, "rtol") && ! isnan (ctl.rtol ), rtol = ctl.rtol ; end
+ if isfield (ctl, "vtol") && ! isnan (ctl.vtol ), vtol = ctl.vtol ; end
+ if isfield (ctl, "narg") && ! isnan (ctl.narg ), narg = ctl.narg ; end
+ if isfield (ctl,"maxev") && ! isnan (ctl.maxev), maxev = ctl.maxev; end
+ if isfield (ctl, "isz") && ! isnan (ctl.isz ), isz = ctl.isz ; end
+ if isfield (ctl, "rst") && ! isnan (ctl.rst ), rst = ctl.rst ; end
+ if isfield(ctl,"verbose")&& !isnan(ctl.verbose),verbose=ctl.verbose;end
+ end
+end
+
+
+if crit == 1, ftol = tol;
+elseif crit == 2, rtol = tol;
+elseif crit == 3, vtol = tol;
+elseif crit, error ("crit is %i. Should be 1,2 or 3.\n");
+end
+
+if iscell (args)
+ x = args{1};
+else # Single argument
+ x = args;
+ args = {args};
+endif
+
+if narg > length (args) # Check
+ error ("nelder_mead_min : narg==%i, length (args)==%i\n",
+ narg, length (args));
+end
+
+[R,C] = size (x);
+N = R*C; # Size of argument
+x = x(:);
+ # Initial simplex
+u = isz * eye (N+1,N) + ones(N+1,1)*x';
+
+y = zeros (N+1,1);
+for i = 1:N+1,
+ y(i) = feval (f, args{1:narg-1},reshape(u(i,:),R,C),args{narg+1:end});
+end ;
+nev = N+1;
+
+[ymin,imin] = min(y);
+ymin0 = ymin;
+## y
+nextprint = 0 ;
+v = nan;
+while nev <= maxev,
+
+ ## ymin, ymax, ymx2 : lowest, highest and 2nd highest function values
+ ## imin, imax, imx2 : indices of vertices with these values
+ [ymin,imin] = min(y); [ymax,imax] = max(y) ;
+ y(imax) = ymin ;
+ [ymx2,imx2] = max(y) ;
+ y(imax) = ymax ;
+
+ ## ymin may be > ymin0 after restarting
+ ## if ymin > ymin0 ,
+ ## "nelder-mead : Whoa 'downsimplex' Should be renamed 'upsimplex'"
+ ## keyboard
+ ## end
+
+ # Compute stopping criterion
+ done = 0;
+ if ! isnan (ftol),
+ done |= ((max(y)-min(y)) / max(1,max(abs(y))) < ftol);
+ end
+ if ! isnan (rtol),
+ done |= (2*max (max (u) - min (u)) < rtol);
+ end
+ if ! isnan (vtol)
+ done |= (abs (det (u(1:N,:)-ones(N,1)*u(N+1,:)))/factorial(N) < vtol);
+ end
+ ## [ 2*max (max (u) - min (u)), abs (det (u(1:N,:)-ones(N,1)*u(N+1,:)))/factorial(N);\
+ ## rtol, vtol]
+
+ # Eventually print some info
+ if verbose && nev > nextprint && ! done
+
+ printf("nev=%-5d imin=%-3d ymin=%-8.3g done=%i\n",\
+ nev,imin,ymin,done) ;
+
+ nextprint = nextprint + 100 ;
+ end
+
+ if done # Termination test
+ if (rst > 0) && (isnan (v) || v > ymin)
+ rst--;
+ if verbose
+ if isnan (v),
+ printf ("Restarting next to minimum %10.3e\n",ymin);
+ else
+ printf ("Restarting next to minimum %10.3e\n",ymin-v);
+ end
+ end
+ # Keep best minimum
+ x = reshape (u(imin,:), R, C) ;
+ v = ymin ;
+
+ jumplen = 10 * max (max (u) - min (u));
+
+ u += jumplen * randn (size (u));
+ for i = 1:N+1, y(i) = \
+ feval (f, args{1:narg-1},reshape(u(i,:),R,C),args{narg+1:length(args)});
+ end
+ nev += N+1;
+ [ymin,imin] = min(y); [ymax,imax] = max(y);
+ y(imax) = ymin;
+ [ymx2,imx2] = max(y);
+ y(imax) = ymax ;
+ else
+ if isnan (v),
+ x = reshape (u(imin,:), R, C) ;
+ v = ymin ;
+ end
+ if verbose,
+ printf("nev=%-5d imin=%-3d ymin=%-8.3g done=%i. Done\n",\
+ nev,imin,ymin,done) ;
+ end
+ return
+ end
+
+ end
+ ## [ y' u ]
+
+ tra = 0 ; # 'trace' debug var contains flags
+ if verbose > 1, str = sprintf (" %i : %10.3e --",done,ymin); end
+
+ # Look for a new point
+ xsum = sum(u) ; # Consider reflection of worst vertice
+ # around centroid.
+ ## f1 = (1-(-1))/N = 2/N;
+ ## f2 = f1 - (-1) = 2/N + 1 = (N+2)/N
+ xnew = (2*xsum - (N+2)*u(imax,:)) / N;
+ ## xnew = (2*xsum - N*u(imax,:)) / N;
+ ynew = feval (f, args{1:narg-1},reshape(xnew,R,C),args{narg+1:length(args)});
+ nev++;
+
+ if ynew <= ymin , # Reflection is good
+
+ tra += 1 ;
+ if verbose > 1
+ str = [str,sprintf(" %3i : %10.3e good refl >>",nev,ynew-ymin)];
+ end
+ y(imax) = ynew; u(imax,:) = xnew ;
+ ## ymin = ynew;
+ ## imin = imax;
+ xsum = sum(u) ;
+
+ ## f1 = (1-2)/N = -1/N
+ ## f2 = f1 - 2 = -1/N - 2 = -(2*N+1)/N
+ xnew = ( -xsum + (2*N+1)*u(imax,:) ) / N;
+ ynew = feval (f, args{1:narg-1},reshape(xnew,R,C),args{narg+1:length(args)});
+ nev++;
+
+ if ynew <= ymin , # expansion improves
+ tra += 2 ;
+ ## 'expanded reflection'
+ y(imax) = ynew ; u(imax,:) = xnew ;
+ xsum = sum(u) ;
+ if verbose > 1
+ str = [str,sprintf(" %3i : %10.3e expd refl",nev,ynew-ymin)];
+ end
+ else
+ tra += 4 ;
+ ## 'plain reflection'
+ ## Updating of y and u has already been done
+ if verbose > 1
+ str = [str,sprintf(" %3i : %10.3e plain ref",nev,ynew-ymin)];
+ end
+ end
+ # Reflexion is really bad
+ elseif ynew >= ymax ,
+
+ tra += 8 ;
+ if verbose > 1
+ str = [str,sprintf(" %3i : %10.3e intermedt >>",nev,ynew-ymin)];
+ end
+ ## look for intermediate point
+ # Bring worst point closer to centroid
+ ## f1 = (1-0.5)/N = 0.5/N
+ ## f2 = f1 - 0.5 = 0.5*(1 - N)/N
+ xnew = 0.5*(xsum + (N-1)*u(imax,:)) / N;
+ ynew = feval (f, args{1:narg-1},reshape(xnew,R,C),args{narg+1:length(args)});
+ nev++;
+
+ if ynew >= ymax , # New point is even worse. Contract whole
+ # simplex
+
+ nev += N + 1 ;
+ ## u0 = u;
+ u = (u + ones(N+1,1)*u(imin,:)) / 2;
+ ## keyboard
+
+ ## Code that doesn't care about value of empty_list_elements_ok
+ if imin == 1 , ii = 2:N+1;
+ elseif imin == N+1, ii = 1:N;
+ else ii = [1:imin-1,imin+1:N+1]; end
+ for i = ii
+ y(i) = \
+ ynew = feval (f, args{1:narg-1},reshape(u(i,:),R,C),args{narg+1:length(args)});
+ end
+ ## 'contraction'
+ tra += 16 ;
+ if verbose > 1
+ str = [str,sprintf(" %3i contractn",nev)];
+ end
+ else # Replace highest point
+ y(imax) = ynew ; u(imax,:) = xnew ;
+ xsum = sum(u) ;
+ ## 'intermediate'
+ tra += 32 ;
+ if verbose > 1
+ str = [str,sprintf(" %3i : %10.3e intermedt",nev,ynew-ymin)];
+ end
+ end
+
+ else # Reflexion is neither good nor bad
+ y(imax) = ynew ; u(imax,:) = xnew ;
+ xsum = sum(u) ;
+ ## 'plain reflection (2)'
+ tra += 64 ;
+ if verbose > 1
+ str = [str,sprintf(" %3i : %10.3e keep refl",nev,ynew-ymin)];
+ end
+ end
+ if verbose > 1, printf ("%s\n",str); end
+end
+
+if verbose >= 0
+ printf ("nelder_mead : Too many iterations. Returning\n");
+end
+
+if isnan (v) || v > ymin,
+ x = reshape (u(imin,:), R, C) ;
+ v = ymin ;
+end
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