Evolution and Optimum Seeking

112 Evolution Strategies for Numerical Optimization
or after n generations
(g+1)
opt
(g)
opt
(g+n)
opt
(g)
opt
= r(g+1) k1
=1;
r (g) n
=
1 ; k1
! n
n
If n is large compared to one, and the formulae derived by Rechenberg are only valid
under this assumption, the step length factor tends to a constant:
lim
n!1
1 ; k1
n
! n
= e ;k1 ' 0:817 ' 1
1:224
The same result is obtained by considering the rate of progress as a di erential quotient
' = dr=dg, in which g represents the iteration number.
This matches the limiting case of very manyvariables because, according to Equation
(5.4) the rate of progress is inversely proportional to the number of variables. The fact
that the rate of progress ' near its maximum is quite insensitive to small changes in the
variances, together with the fact that the probability of success can only be determined
from an average over several mutations, leads to the following more precise formulation
of the 1=5 success rule for numerical optimization:
After every n mutations, check howmany successes have occurred over the
preceding 10 n mutations. If this number is less than 2 n, multiply the step
lengths by thefactor0:85 divide them by 0:85 if more than 2 n successes
occurred.
The 1=5 success rule enables the step lengths or variances of the random variations
to be controlled. One might doeven better by looking for a control mechanism with
additional di erential and integral coe cients to avoid the oscillatory behavior of a mere
proportional feedback. However, the probability of success unfortunately gives no indi-
2
cation of how appropriate are the ratios of the variances i to each other. The step
lengths can only be all reduced together, or all increased. One would sometimes rather
like to build in a scaling of the variables, i.e., to determine ratios of the step lengths to
each other. This can be achieved by a suitable formulation of the objective function, in
which new parameters are introduced in place of the original variables. The functional
dependence can be freely chosen and in the simplest case it is given by multiplicative
factors. In the formulation of the numerical procedure for the two membered evolution
strategy (Appendix B, Sect. B.1) the possibility is therefore included of specifying an
initial step length for each individual variable. The ratios of the variances to each other
remain constant during the optimum search, unless speci ed lower bounds to the step
lengths are not operating at the same time.
All digital computers handle data only in the form of a nite number of units of
information (bits). The number of signi cant gures and the range of numbers is thereby
limited. If a quantity is repeatedly divided by a factor greater than one, the stored value of