/usr/share/octave/packages/optim-1.4.0/de_min.m is in octave-optim 1.4.0-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 | ## Copyright (C) 1996, 1997 R. Storn
## Copyright (C) 2009-2010 Christian Fischer <cfischer@itm.uni-stuttgart.de>
##
## 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/>.
## de_min: global optimisation using differential evolution
##
## Usage: [x, obj_value, nfeval, convergence] = de_min(fcn, control)
##
## minimization of a user-supplied function with respect to x(1:D),
## using the differential evolution (DE) method based on an algorithm
## by Rainer Storn (http://www.icsi.berkeley.edu/~storn/code.html)
## See: http://www.softcomputing.net/tevc2009_1.pdf
##
##
## Arguments:
## ---------------
## fcn string : Name of function. Must return a real value
## control vector : (Optional) Control variables, described below
## or struct
##
## Returned values:
## ----------------
## x vector : parameter vector of best solution
## obj_value scalar : objective function value of best solution
## nfeval scalar : number of function evaluations
## convergence : 1 = best below value to reach (VTR)
## 0 = population has reached defined quality (tol)
## -1 = some values are close to constraints/boundaries
## -2 = max number of iterations reached (maxiter)
## -3 = max number of functions evaluations reached (maxnfe)
##
## Control variable: (optional) may be named arguments (i.e. "name",value
## ---------------- pairs), a struct, or a vector, where
## NaN's are ignored.
##
## XVmin : vector of lower bounds of initial population
## *** note: by default these are no constraints ***
## XVmax : vector of upper bounds of initial population
## constr : 1 -> enforce the bounds not just for the initial population
## const : data vector (remains fixed during the minimization)
## NP : number of population members
## F : difference factor from interval [0, 2]
## CR : crossover probability constant from interval [0, 1]
## strategy : 1 --> DE/best/1/exp 7 --> DE/best/1/bin
## 2 --> DE/rand/1/exp 8 --> DE/rand/1/bin
## 3 --> DE/target-to-best/1/exp 9 --> DE/target-to-best/1/bin
## 4 --> DE/best/2/exp 10--> DE/best/2/bin
## 5 --> DE/rand/2/exp 11--> DE/rand/2/bin
## 6 --> DEGL/SAW/exp else DEGL/SAW/bin
## refresh : intermediate output will be produced after "refresh"
## iterations. No intermediate output will be produced
## if refresh is < 1
## VTR : Stopping criterion: "Value To Reach"
## de_min will stop when obj_value <= VTR.
## Use this if you know which value you expect.
## tol : Stopping criterion: "tolerance"
## stops if (best-worst)/max(1,worst) < tol
## This stops basically if the whole population is "good".
## maxnfe : maximum number of function evaluations
## maxiter : maximum number of iterations (generations)
##
## The algorithm seems to work well only if [XVmin,XVmax] covers the
## region where the global minimum is expected.
## DE is also somewhat sensitive to the choice of the
## difference factor F. A good initial guess is to choose F from
## interval [0.5, 1], e.g. 0.8.
## CR, the crossover probability constant from interval [0, 1]
## helps to maintain the diversity of the population and is
## rather uncritical but affects strongly the convergence speed.
## If the parameters are correlated, high values of CR work better.
## The reverse is true for no correlation.
## Experiments suggest that /bin likes to have a slightly
## larger CR than /exp.
## The number of population members NP is also not very critical. A
## good initial guess is 10*D. Depending on the difficulty of the
## problem NP can be lower than 10*D or must be higher than 10*D
## to achieve convergence.
##
## Default Values:
## ---------------
## XVmin = [-2];
## XVmax = [ 2];
## constr= 0;
## const = [];
## NP = 10 *D
## F = 0.8;
## CR = 0.9;
## strategy = 12;
## refresh = 0;
## VTR = -Inf;
## tol = 1.e-3;
## maxnfe = 1e6;
## maxiter = 1000;
##
##
## Example to find the minimum of the Rosenbrock saddle:
## ----------------------------------------------------
## Define f as:
## function result = f(x);
## result = 100 * (x(2) - x(1)^2)^2 + (1 - x(1))^2;
## end
## Then type:
##
## ctl.XVmin = [-2 -2];
## ctl.XVmax = [ 2 2];
## [x, obj_value, nfeval, convergence] = de_min (@f, ctl);
##
## Keywords: global-optimisation optimisation minimisation
function [bestmem, bestval, nfeval, convergence] = de_min(fcn, varargin)
## Default options
XVmin = [-2 ];
XVmax = [ 2 ];
constr= 0;
const = [];
NP = 0; # NP will be set later
F = 0.8;
CR = 0.9;
strategy = 12;
refresh = 0;
VTR = -Inf;
tol = 1.e-3;
maxnfe = 1e6;
maxiter = 1000;
## ------------ Check input variables (ctl) --------------------------------
if nargin >= 2, # Read control arguments
va_arg_cnt = 1;
if nargin > 2,
ctl = struct (varargin{:});
else
ctl = varargin{va_arg_cnt++};
end
if isnumeric (ctl)
if length (ctl)>=1 && !isnan (ctl(1)), XVmin = ctl(1); end
if length (ctl)>=2 && !isnan (ctl(2)), XVmax = ctl(2); end
if length (ctl)>=3 && !isnan (ctl(3)), constr = ctl(3); end
if length (ctl)>=4 && !isnan (ctl(4)), const = ctl(4); end
if length (ctl)>=5 && !isnan (ctl(5)), NP = ctl(5); end
if length (ctl)>=6 && !isnan (ctl(6)), F = ctl(6); end
if length (ctl)>=7 && !isnan (ctl(7)), CR = ctl(7); end
if length (ctl)>=8 && !isnan (ctl(8)), strategy = ctl(8); end
if length (ctl)>=9 && !isnan (ctl(9)), refresh = ctl(9); end
if length (ctl)>=10&& !isnan (ctl(10)), VTR = ctl(10); end
if length (ctl)>=11&& !isnan (ctl(11)), tol = ctl(11); end
if length (ctl)>=12&& !isnan (ctl(12)), maxnfe = ctl(12); end
if length (ctl)>=13&& !isnan (ctl(13)), maxiter = ctl(13); end
else
if isfield (ctl,"XVmin") && !isnan (ctl.XVmin), XVmin=ctl.XVmin; end
if isfield (ctl,"XVmax") && !isnan (ctl.XVmax), XVmax=ctl.XVmax; end
if isfield (ctl,"constr")&& !isnan (ctl.constr), constr=ctl.constr; end
if isfield (ctl,"const") && !isnan (ctl.const), const=ctl.const; end
if isfield (ctl, "NP" ) && ! isnan (ctl.NP ), NP = ctl.NP ; end
if isfield (ctl, "F" ) && ! isnan (ctl.F ), F = ctl.F ; end
if isfield (ctl, "CR" ) && ! isnan (ctl.CR ), CR = ctl.CR ; end
if isfield (ctl, "strategy") && ! isnan (ctl.strategy),
strategy = ctl.strategy ; end
if isfield (ctl, "refresh") && ! isnan (ctl.refresh),
refresh = ctl.refresh ; end
if isfield (ctl, "VTR") && ! isnan (ctl.VTR ), VTR = ctl.VTR ; end
if isfield (ctl, "tol") && ! isnan (ctl.tol ), tol = ctl.tol ; end
if isfield (ctl, "maxnfe") && ! isnan (ctl.maxnfe)
maxnfe = ctl.maxnfe;
end
if isfield (ctl, "maxiter") && ! isnan (ctl.maxiter)
maxiter = ctl.maxiter;
end
end
end
## set dimension D and population size NP
D = length (XVmin);
if (NP == 0);
NP = 10 * D;
end
## -------- do a few sanity checks --------------------------------
if (length (XVmin) != length (XVmax))
error("Length of upper and lower bounds does not match.")
end
if (NP < 5)
error("Population size NP must be bigger than 5.")
end
if ((F <= 0) || (F > 2))
error("Difference Factor F out of range (0,2].")
end
if ((CR < 0) || (CR > 1))
error("CR value out of range [0,1].")
end
if (maxiter <= 0)
error("maxiter must be positive.")
end
if (maxnfe <= 0)
error("maxnfe must be positive.")
end
refresh = floor(abs(refresh));
## ----- Initialize population and some arrays --------------------------
pop = zeros(NP,D); # initialize pop
## pop is a matrix of size NPxD. It will be initialized with
## random values between the min and max values of the parameters
for i = 1:NP
pop(i,:) = XVmin + rand (1,D) .* (XVmax - XVmin);
end
## initialise the weighting factors between 0.0 and 1.0
w = rand (NP,1);
wi = w;
popold = zeros (size (pop)); # toggle population
val = zeros (1, NP); # create and reset the "cost array"
bestmem = zeros (1, D); # best population member ever
bestmemit = zeros (1 ,D); # best population member in iteration
nfeval = 0; # number of function evaluations
## ------ Evaluate the best member after initialization ------------------
ibest = 1; # start with first population member
val(1) = feval (fcn, [pop(ibest,:) const]);
bestval = val(1); # best objective function value so far
bestw = w(1); # weighting of best design so far
for i = 2:NP # check the remaining members
val(i) = feval (fcn, [pop(i,:) const]);
if (val(i) < bestval) # if member is better
ibest = i; # save its location
bestval = val(i);
bestw = w(i);
end
end
nfeval = nfeval + NP;
bestmemit = pop(ibest,:); # best member of current iteration
bestvalit = bestval; # best value of current iteration
bestmem = bestmemit; # best member ever
## ------ DE - Minimization ---------------------------------------
## popold is the population which has to compete. It is static
## through one iteration. pop is the newly emerging population.
bm_n= zeros (NP, D); # initialize bestmember matrix in neighbourh.
lpm1= zeros (NP, D); # initialize local population matrix 1
lpm1= zeros (NP, D); # initialize local population matrix 2
rot = 0:1:NP-1; # rotating index array (size NP)
rotd= 0:1:D-1; # rotating index array (size D)
iter = 1;
while ((iter < maxiter) && (nfeval < maxnfe) && (bestval > VTR) && ...
((abs (max (val) - bestval) / max (1, abs (max (val))) > tol)))
popold = pop; # save the old population
wold = w; # save the old weighting factors
ind = randperm (4); # index pointer array
a1 = randperm (NP); # shuffle locations of vectors
rt = rem (rot + ind(1), NP); # rotate indices by ind(1) positions
a2 = a1(rt+1); # rotate vector locations
rt = rem (rot + ind(2), NP);
a3 = a2(rt+1);
rt = rem (rot +ind(3), NP);
a4 = a3(rt+1);
rt = rem (rot + ind(4), NP);
a5 = a4(rt+1);
pm1 = popold(a1,:); # shuffled population 1
pm2 = popold(a2,:); # shuffled population 2
pm3 = popold(a3,:); # shuffled population 3
w1 = wold(a4); # shuffled weightings 1
w2 = wold(a5); # shuffled weightings 2
bm = repmat (bestmemit, NP, 1); # population filled with the best member
# of the last iteration
bw = repmat (bestw, NP, 1); # the same for the weighting of the best
mui = rand (NP, D) < CR; # mask for intermediate population
# all random numbers < CR are 1, 0 otherwise
if (strategy > 6)
st = strategy - 6; # binomial crossover
else
st = strategy; # exponential crossover
mui = sort (mui'); # transpose, collect 1's in each column
for i = 1:NP
n = floor (rand * D);
if (n > 0)
rtd = rem (rotd + n, D);
mui(:,i) = mui(rtd+1,i); #rotate column i by n
endif
endfor
mui = mui'; # transpose back
endif
mpo = mui < 0.5; # inverse mask to mui
if (st == 1) # DE/best/1
ui = bm + F*(pm1 - pm2); # differential variation
elseif (st == 2) # DE/rand/1
ui = pm3 + F*(pm1 - pm2); # differential variation
elseif (st == 3) # DE/target-to-best/1
ui = popold + F*(bm-popold) + F*(pm1 - pm2);
elseif (st == 4) # DE/best/2
pm4 = popold(a4,:); # shuffled population 4
pm5 = popold(a5,:); # shuffled population 5
ui = bm + F*(pm1 - pm2 + pm3 - pm4); # differential variation
elseif (st == 5) # DE/rand/2
pm4 = popold(a4,:); # shuffled population 4
pm5 = popold(a5,:); # shuffled population 5
ui = pm5 + F*(pm1 - pm2 + pm3 - pm4); # differential variation
else # DEGL/SAW
## The DEGL/SAW method is more complicated.
## We have to generate a neighbourhood first.
## The neighbourhood size is 10% of NP and at least 1.
nr = max (1, ceil ((0.1*NP -1)/2)); # neighbourhood radius
## FIXME: I don't know how to vectorise this. - if possible
for i = 1:NP
neigh_ind = i-nr:i+nr; # index range of neighbourhood
neigh_ind = neigh_ind + ((neigh_ind <= 0)-(neigh_ind > NP))*NP;
# do wrap around
[x, ix] = min (val(neigh_ind)); # find the local best and its index
bm_n(i,:) = popold(neigh_ind(ix),:); # copy the data from the local best
neigh_ind(nr+1) = []; # remove "i"
pq = neigh_ind(randperm (length (neigh_ind)));
# permutation of the remaining ind.
lpm1(i,:) = popold(pq(1),:); # create the local pop member matrix
lpm2(i,:) = popold(pq(2),:); # for the random point p,q
endfor
## calculate the new weihting factors
wi = wold + F*(bw - wold) + F*(w1 - w2); # use DE/target-to-best/1/nocross
# for optimisation of weightings
## fix bounds for weightings
o = ones (NP, 1);
wi = sort ([0.05*o, wi, 0.95*o],2)(:,2); # sort and take the second column
## fill weighting matrix
wm = repmat (wi, 1, D);
li = popold + F*(bm_n- popold) + F*(lpm1 - lpm2);
gi = popold + F*(bm - popold) + F*(pm1 - pm2);
ui = wm.*gi + (1-wm).*li; # combine global and local part
endif
## crossover
ui = popold.*mpo + ui.*mui;
## enforce initial bounds/constraints if specified
if (constr == 1)
for i = 1:NP
ui(i,:) = max (ui(i,:), XVmin);
ui(i,:) = min (ui(i,:), XVmax);
end
end
## ----- Select which vectors are allowed to enter the new population ------
for i = 1:NP
tempval = feval (fcn, [ui(i,:) const]); # check cost of competitor
if (tempval <= val(i)) # if competitor is better
pop(i,:) = ui(i,:); # replace old vector with new one
val(i) = tempval; # save value in "cost array"
w(i) = wi(i); # save the weighting factor
## we update bestval only in case of success to save time
if (tempval <= bestval) # if competitor better than the best one ever
bestval = tempval; # new best value
bestmem = ui(i,:); # new best parameter vector ever
bestw = wi(i); # save best weighting
end
end
endfor #---end for i = 1:NP
nfeval = nfeval + NP; # increase number of function evaluations
bestmemit = bestmem; # freeze the best member of this iteration for the
# coming iteration. This is needed for some of the
# strategies.
## ---- Output section ----------------------------------------------------
if (refresh > 0)
if (rem (iter, refresh) == 0)
printf ('Iteration: %d, Best: %8.4e, Worst: %8.4e\n', ...
iter, bestval, max(val));
for n = 1:D
printf ('x(%d) = %e\n', n, bestmem(n));
end
end
end
iter = iter + 1;
endwhile #---end while ((iter < maxiter) ...
## check that all variables are well within bounds/constraints
boundsOK = 1;
for i = 1:NP
range = XVmax - XVmin;
if (ui(i,:) < XVmin + 0.01*range)
boundsOK = 0;
end
if (ui(i,:) > XVmax - 0.01*range)
boundsOK = 0;
end
end
## create the convergence result
if (bestval <= VTR)
convergence = 1;
elseif (abs (max (val) - bestval) / max (1, abs (max (val))) <= tol)
convergence = 0;
elseif (boundsOK == 0)
convergence = -1;
elseif (iter >= maxiter)
convergence = -2;
elseif (nfeval >= maxnfe)
convergence = -3;
end
endfunction
%!function result = f(x);
%! result = 100 * (x(2) - x(1)^2)^2 + (1 - x(1))^2;
%!test
%! tol = 1.0e-4;
%! ctl.tol = 0.0;
%! ctl.VTR = 1.0e-6;
%! ctl.XVmin = [-2 -2];
%! ctl.XVmax = [ 2 2];
%! rand("state", 11)
%! [x, obj_value, nfeval, convergence] = de_min (@f, ctl);
%! assert (convergence == 1);
%! assert (f(x) == obj_value);
%! assert (obj_value < ctl.VTR);
%!demo
%! ## define a simple example function
%! f = @(x) peaks(x(1), x(2));
%! ## plot the function to see where the minimum might be
%! peaks()
%! ## first we set the region where we expect the minimum
%! ctl.XVmin = [-3 -3];
%! ctl.XVmax = [ 3 3];
%! ## and solve it with de_min
%! [x, obj_value, nfeval, convergence] = de_min (f, ctl)
|