/usr/share/octave/packages/optim-1.4.0/nelder_mead_min.m is in octave-optim 1.4.0-1.
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##
## 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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