Page 353 - Applied Numerical Methods Using MATLAB

342 OPTIMIZATION

function [xo,fo] = genetic(f,x0,l,u,Np,Nb,Pc,Pm,eta,kmax)
% Genetic Algorithm to minimize f(x) s.t. l <= x <= u
N = length(x0);
if nargin < 10, kmax = 100; end %# of iterations(generations)
if nargin < 9|eta > 1|eta <= 0, eta = 1; end %learning rate(0 < eta < 1)
if nargin < 8, Pm = 0.01; end %probability of mutation
if nargin < 7, Pc = 0.5; end %probability of crossover
if nargin < 6, Nb = 8*ones(1,N); end %# of genes(bits) for each variable
if nargin < 5, Np = 10; end %population size(number of chromosomes)
%Initialize the population pool
NNb = sum(Nb);
xo = x0(:)’; l = l(:)’; u = u(:)’;
fo = feval(f,xo);
X(1,:) = xo;
for n = 2:Np, X(n,:) = l + rand(size(x0)).*(u - l); end %Eq.(7.1.26)
P = gen_encode(X,Nb,l,u); %Eq.(7.1.27)
for k = 1:kmax
X = gen_decode(P,Nb,l,u); %Eq.(7.1.28)
for n = 1:Np, fX(n) = feval(f,X(n,:)); end
[fxb,nb] = min(fX); %Selection of the fittest
if fxb < fo, fo = fxb; xo = X(nb,:); end
fX1 = max(fxs) - fX; %make the nonnegative fitness vector by Eq.(7.1.29)
fXm = fX1(nb);
if fXm < eps, return; end %terminate if all the chromosomes are equal
%Reproduction of next generation
for n = 1:Np
X(n,:) = X(n,:) + eta*(fXm - fX1(n))/fXm*(X(nb,:) - X(n,:)); %Eq.(7.1.30)
end
P = gen_encode(X,Nb,l,u);
%Mating/Crossover
is = shuffle([1:Np]);
for n = 1:2:Np - 1
if rand < Pc, P(is(n:n + 1),:) = crossover(P(is(n:n + 1),:),Nb); end
end
%Mutation
P = mutation(P,Nb,Pm);
end
function P = gen_encode(X,Nb,l,u)
% encode a population(X) of state into an array(P) of binary strings
Np=size(X,1); %population size
N = length(Nb); %dimension of the variable(state)
for n = 1:Np
b2=0;
for m = 1:N
b1 = b2+1; b2 = b2 + Nb(m);
Xnm =(2^Nb(m)- 1)*(X(n,m) - l(m))/(u(m) - l(m)); %Eq.(7.1.27)
P(n,b1:b2) = dec2bin(Xnm,Nb(m)); %encoding to binary strings
end
end
function X = gen_decode(P,Nb,l,u)
% decode an array of binary strings(P) into a population(X) of state
Np = size(P,1); %population size
N = length(Nb); %dimension of the variable(state)
for n = 1:Np
b2=0;
for m = 1:N
b1 = b2 + 1; b2 = b1 + Nb(m) - 1; %Eq.(7.1.28)
X(n,m) = bin2dec(P(n,b1:b2))*(u(m) - l(m))/(2^Nb(m) - 1) + l(m);
end
end