Evolution and Optimum Seeking

A Multimembered Evolution Strategy 125
s 0 =su
x 2
z
N
E x
E : Parent
1
N : th offspring
s = z ,1
Contours
F (x) = const.
Figure 5.5: The inclined plane model function
To the minimum
sublinearly however, probably proportional to the logarithm of .Tocompare the above
approximation ~' with the exact value ' the following integral must be evaluated:
' =
Z1
s0 p
2
exp ; s02
2 2
! "
0
1
s
1 + erf p
2
2
!#! ;1
ds 0
For small values of the integration can be performed by elementary methods, but
not for general values of . The value of ' was therefore obtained by simulation on the
computer rst for the case in which the parent survives if the best of the descendants is
worse than the parent ('sur with su = 0) and secondly for the case in which the parent
is no longer considered in the selection ('ext with su = ;1). The two results are shown
in Figure 5.6 for comparison with the approximate solution ~'. Itisimmediately striking
that for only ve o spring the extinction of the parent has hardly any e ect on the rate
of progress, i.e., for 5 it is as good as certain that at least one of the descendants
will be better than the parent. The greatest di erences between 'sur and 'ext naturally
appear when = 1. Whereas 'ext goes to zero, 'sur keeps a nite value. This can be
determined exactly. Omitting here the details of the derivation, which is straightforward,
the result is simply
'sur( =1)=p 2
The relationship to the (1+1) evolution scheme is thereby established. The di erences
between the approximate theory ( ~') and the simulation ('ext) indicate that the assumption
of the symmetry of w(s 0 ) is not correct. The discrepancy with regard to '= seems
to tend to a constant value as increases. While the approximate theory is shown by this