| devec3(fname,VTR,D,XVmin,XVmax,y,NP,itermax,F,CR,strategy,refresh); |
function [bestmem,bestval,nfeval] = devec3(fname,VTR,D,XVmin,XVmax,y,NP,itermax,F,CR,strategy,refresh);
% minimization of a user-supplied function with respect to x(1:D),
% using the differential evolution (DE) algorithm of Rainer Storn
% (http://www.icsi.berkeley.edu/~storn/code.html)
%
% Special thanks go to Ken Price (kprice@solano.community.net) and
% Arnold Neumaier (http://solon.cma.univie.ac.at/~neum/) for their
% valuable contributions to improve the code.
%
% Strategies with exponential crossover, further input variable
% tests, and arbitrary function name implemented by Jim Van Zandt
% <jrv@vanzandt.mv.com>, 12/97.
%
% Output arguments:
% ----------------
% bestmem parameter vector with best solution
% bestval best objective function value
% nfeval number of function evaluations
%
% Input arguments:
% ---------------
%
% fname string naming a function f(x,y) to minimize
% VTR "Value To Reach". devec3 will stop its minimization
% if either the maximum number of iterations "itermax"
% is reached or the best parameter vector "bestmem"
% has found a value f(bestmem,y) <= VTR.
% D number of parameters of the objective function
% XVmin vector of lower bounds XVmin(1) ... XVmin(D)
% of initial population
% *** note: these are not bound constraints!! ***
% XVmax vector of upper bounds XVmax(1) ... XVmax(D)
% of initial population
% y problem data vector (must remain fixed during the
% minimization)
% NP number of population members
% itermax maximum number of iterations (generations)
% F DE-stepsize F from interval [0, 2]
% CR crossover probability constant from interval [0, 1]
% strategy 1 --> DE/best/1/exp 6 --> DE/best/1/bin
% 2 --> DE/rand/1/exp 7 --> DE/rand/1/bin
% 3 --> DE/rand-to-best/1/exp 8 --> DE/rand-to-best/1/bin
% 4 --> DE/best/2/exp 9 --> DE/best/2/bin
% 5 --> DE/rand/2/exp else DE/rand/2/bin
% Experiments suggest that /bin likes to have a slightly
% larger CR than /exp.
% refresh intermediate output will be produced after "refresh"
% iterations. No intermediate output will be produced
% if refresh is < 1
%
% The first four arguments are essential (though they have
% default values, too). In particular, 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 stepsize 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. 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.
% If the parameters are correlated, high values of CR work better.
% The reverse is true for no correlation.
%
% default values in case of missing input arguments:
% VTR = 1.e-6;
% D = 2;
% XVmin = [-2 -2];
% XVmax = [2 2];
% y=[];
% NP = 10*D;
% itermax = 200;
% F = 0.8;
% CR = 0.5;
% strategy = 7;
% refresh = 10;
%
% Cost function: function result = f(x,y);
% has to be defined by the user and is minimized
% w.r. to x(1:D).
%
% Example to find the minimum of the Rosenbrock saddle:
% ----------------------------------------------------
% Define f.m as:
% function result = f(x,y);
% result = 100*(x(2)-x(1)^2)^2+(1-x(1))^2;
% end
% Then type:
%
% VTR = 1.e-6;
% D = 2;
% XVmin = [-2 -2];
% XVmax = [2 2];
% [bestmem,bestval,nfeval] = devec3("f",VTR,D,XVmin,XVmax);
%
% The same example with a more complete argument list is handled in
% run1.m
%
% About devec3.m
% --------------
% Differential Evolution for MATLAB
% Copyright (C) 1996, 1997 R. Storn
% International Computer Science Institute (ICSI)
% 1947 Center Street, Suite 600
% Berkeley, CA 94704
% E-mail: storn@icsi.berkeley.edu
% WWW: http://http.icsi.berkeley.edu/~storn
%
% devec is a vectorized variant of DE which, however, has a
% propertiy which differs from the original version of DE:
% 1) The random selection of vectors is performed by shuffling the
% population array. Hence a certain vector can't be chosen twice
% in the same term of the perturbation expression.
%
% Due to the vectorized expressions devec3 executes fairly fast
% in MATLAB's interpreter environment.
%
% 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 1, 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. A copy of the GNU
% General Public License can be obtained from the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
%-----Check input variables---------------------------------------------
err=[];
if nargin<1, error('devec3 1st argument must be function name'); else
if exist(fname)<1; err(1,length(err)+1)=1; end; end;
if nargin<2, VTR = 1.e-6; else
if length(VTR)~=1; err(1,length(err)+1)=2; end; end;
if nargin<3, D = 2; else
if length(D)~=1; err(1,length(err)+1)=3; end; end;
if nargin<4, XVmin = [-2 -2];else
if length(XVmin)~=D; err(1,length(err)+1)=4; end; end;
if nargin<5, XVmax = [2 2]; else
if length(XVmax)~=D; err(1,length(err)+1)=5; end; end;
if nargin<6, y=[]; end;
if nargin<7, NP = 10*D; else
if length(NP)~=1; err(1,length(err)+1)=7; end; end;
if nargin<8, itermax = 200; else
if length(itermax)~=1; err(1,length(err)+1)=8; end; end;
if nargin<9, F = 0.8; else
if length(F)~=1; err(1,length(err)+1)=9; end; end;
if nargin<10, CR = 0.5; else
if length(CR)~=1; err(1,length(err)+1)=10; end; end;
if nargin<11, strategy = 7; else
if length(strategy)~=1; err(1,length(err)+1)=11; end; end;
if nargin<12, refresh = 10; else
if length(refresh)~=1; err(1,length(err)+1)=12; end; end;
if length(err)>0
fprintf(stdout,'error in parameter %d\n', err);
usage('devec3 (string,scalar,scalar,vector,vector,any,integer,integer,scalar,scalar,integer,integer)');
end
if (NP < 5)
NP=5;
fprintf(1,' NP increased to minimal value 5\n');
end
if ((CR < 0) | (CR > 1))
CR=0.5;
fprintf(1,'CR should be from interval [0,1]; set to default value 0.5\n');
end
if (itermax <= 0)
itermax = 200;
fprintf(1,'itermax should be > 0; set to default value 200\n');
end
refresh = floor(refresh);
%-----Initialize population and some arrays-------------------------------
pop = zeros(NP,D); %initialize pop to gain speed
%----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
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(fname,pop(ibest,:),y);
bestval = val(1); % best objective function value so far
nfeval = nfeval + 1;
for i=2:NP % check the remaining members
val(i) = feval(fname,pop(i,:),y);
nfeval = nfeval + 1;
if (val(i) < bestval) % if member is better
ibest = i; % save its location
bestval = val(i);
end
end
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.----------------------------------------
pm1 = zeros(NP,D); % initialize population matrix 1
pm2 = zeros(NP,D); % initialize population matrix 2
pm3 = zeros(NP,D); % initialize population matrix 3
pm4 = zeros(NP,D); % initialize population matrix 4
pm5 = zeros(NP,D); % initialize population matrix 5
bm = zeros(NP,D); % initialize bestmember matrix
ui = zeros(NP,D); % intermediate population of perturbed vectors
mui = zeros(NP,D); % mask for intermediate population
mpo = zeros(NP,D); % mask for old population
rot = (0:1:NP-1); % rotating index array (size NP)
rotd= (0:1:D-1); % rotating index array (size D)
rt = zeros(NP); % another rotating index array
rtd = zeros(D); % rotating index array for exponential crossover
a1 = zeros(NP); % index array
a2 = zeros(NP); % index array
a3 = zeros(NP); % index array
a4 = zeros(NP); % index array
a5 = zeros(NP); % index array
ind = zeros(4);
iter = 1;
while ((iter < itermax) & (bestval > VTR))
popold = pop; % save the old population
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
pm4 = popold(a4,:); % shuffled population 4
pm5 = popold(a5,:); % shuffled population 5
for i=1:NP % population filled with the best member
bm(i,:) = bestmemit; % of the last iteration
end
mui = rand(NP,D) < CR; % all random numbers < CR are 1, 0 otherwise
if (strategy > 5)
st = strategy-5; % 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
end
end
mui = mui'; % transpose back
end
mpo = mui < 0.5; % inverse mask to mui
if (st == 1) % DE/best/1
ui = bm + F*(pm1 - pm2); % differential variation
ui = popold.*mpo + ui.*mui; % crossover
elseif (st == 2) % DE/rand/1
ui = pm3 + F*(pm1 - pm2); % differential variation
ui = popold.*mpo + ui.*mui; % crossover
elseif (st == 3) % DE/rand-to-best/1
ui = popold + F*(bm-popold) + F*(pm1 - pm2);
ui = popold.*mpo + ui.*mui; % crossover
elseif (st == 4) % DE/best/2
ui = bm + F*(pm1 - pm2 + pm3 - pm4); % differential variation
ui = popold.*mpo + ui.*mui; % crossover
elseif (st == 5) % DE/rand/2
ui = pm5 + F*(pm1 - pm2 + pm3 - pm4); % differential variation
ui = popold.*mpo + ui.*mui; % crossover
end
%-----Select which vectors are allowed to enter the new population------------
for i=1:NP
tempval = feval(fname,ui(i,:),y); % check cost of competitor
nfeval = nfeval + 1;
if (tempval <= val(i)) % if competitor is better than value in "cost array"
pop(i,:) = ui(i,:); % replace old vector with new one (for new iteration)
val(i) = tempval; % save value in "cost array"
%----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
end
end
end %---end for imember=1:NP
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)
fprintf(1,'Iteration: %d, Best: %f, F: %f, CR: %f, NP: %d\n',iter,bestval,F,CR,NP);
for n=1:D
fprintf(1,'best(%d) = %f\n',n,bestmem(n));
end
end
end
iter = iter + 1;
end %---end while ((iter < itermax) ...
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