Evolution and Optimum Seeking

146 Evolution Strategies for Numerical Optimization
or
"d
(Fw ; Fb)
where "c and "d are to be de ned such that
"c > 0
1+"d > 1
)
X
k=1
F (x (g)
k )
according to the computational accuracy
Either absolutely or relatively, the objective function values of the parents in a generation
must fall closely together before convergence is accepted. The reason for basing the
criterion on function values, rather than variable values or step lengths, has already been
discussed in connection with the (1+1) strategy (see Sect. 5.1.3).
5.2.5 Scaling of the Variables by Recombination
The ( , ) method opens up the possibility of imitating a further principle of organic
evolution, which is of particular interest from the point ofviewofnumerical optimization
problems, namely sexual propagation. By combining the genes of twoparents a new source
of variation is added to point mutation. The fact that only a few primitive organisms do
without this mechanism of recombination leads us to expect that it is very favorable for
evolution. Instead of one vector x (g)
E now there are distinct vectors x (g)
k for k = 1(1)
in a population. In biology, the totality of all genes in a generation is known as a gene
pool. Among the concerns of population genetics (e.g., Wilson and Bossert, 1973) is the
frequency distribution of certain alleles in a population, the so-called gene frequencies.
Until now, we did not argue on that level of detail, nor did we godown to the oor of only
four nucleic acids in order to model, for example, the mutation process within evolution
strategies. This might beworthwhile for quaternary optimization, but not in our case of
continuous parameters. It would be a tedious task to model all the intermediate processes
from nucleic acids to proteins, cell, organs, etc., taking into account the genetic code and
the whole epigenetic apparatus. We shall now apply the principle of recombination to
numerical optimization with continuous parameters, once again in a simpli ed fashion.
In our population of parents we have stored di erent values of each component
xi i = 1(1)n. From this gene pool we now draw one of the values of xi for each
i = 1(1)n. The draw should be random so that the probability that an xi comes from any
particular parent(k) of the is just 1= for all k = 1(1) . The variable vector constructed
in this way forms the starting point for the subsequent variation of the components. The
Figure 5.15 should help to clarify that kind of global recombination.
By imitating recombination in this way wehave, so as to speak, replaced bisexuality
by multisexuality. This was less for reasons of principle than as a result of practical
considerations of programming. A crude test yielded only a slight further increase in
the rate of progress in changing from the bisexual to the multisexual scheme, whereas
appreciable acceleration was achieved by introducing the bisexual in place of the asexual
scheme, which allowed no recombination. A more detailed and exact comparison has yet
to be carried out. Without some guidance from theory it is hard to choose the correct
initial step lengths and rates of change of step lengths for each of the di erent algorithms.