Evolution and Optimum Seeking

150 Evolution Strategies for Numerical Optimization
In the choice of (number of parents) and (number of descendants) there is no
need to ensure that is exactly divisible by . The association of descendants to parents
is made by a random selection of uniformly distributed random integers from the range
[1 ]. It is only necessary that exceeds by a su cient margin that on average at least
one descendant can be better than its parent. From the results of Section 5.2.3 a suitable
choice would be for example 6 .
The transformation from [0 1]evenly distributed random numbers to (0 2 ) normally
distributed pseudorandom numbers is carried out in the same way as in subroutine EVOL
of the (1+1) strategy (see Sect. 5.1.5). The log-normally distributed variance multipliers
are produced by the exponential function. The step lengths (standard deviations of the
individual random components) can initially be speci ed individually. During the subsequent
process of generation they satisfy the constraints
(g)
i
(g)
i
"a
)
and "b jx (g)
where
and
for all i = 1(1)n
i j
)
"a > 0
according to the computational accuracy
1+"b > 1
can be speci ed in advance.
The parameter which in uences the average rate of change of the step lengths
should be given a value roughly proportional to 1= p nincaseoftwofactors (the case
to be preferred), a global and an individual one, the values given in Section 5.2.3 are
recommended. The constant of proportionality depends mainly on another adjustable
feature, = , whichmaybe called the selection pressure. For a (10 , 100) strategy it should
be set at about unity to allow the fastest convergence of simple optimization problems like
the hypersphere. With increasing this value ' can be changed sublinearly according
to
p '
' e
(compare Equation (5.22)).
(0)
If the initial step lengths i are chosen to be too large, what may havebeen an
especially well situated starting point x (0) can be thrown away. Nevertheless, this step
backwards in the rst generation works in favor of reaching a global minimum among
several local minima. In principle, for > 1eachofthedi erent starting vectors
x (0)
k 2 IRn and (0)
k 2 IRn k = 1(1) can be speci ed. In the present program this
di erentiation of the parent generation is carried out automatically the x (0)
k are produced
from x (0) by addition of (0 ( (0) ) 2 ) normally distributed random vectors. The (0) (0)
k =
are initially equal for all parents.
The convergence criterion is described in Section 5.2.4. It is based on the di erence
in objective function values between the current best and worst parents of a generation.
As accuracy parameters, an absolute and a relativequantity ("c and "d) must be speci ed
(compare Sect. 5.1.3). Furthermore, an upper bound on the computation time for the
search can be given so that whatever the outcome results can be output from the main
program (see also Sect. 5.1.5).
Inequality constraints are treated as described for subroutine EVOL (Sect. 5.1.4) so