Evolution and Optimum Seeking

144 Evolution Strategies for Numerical Optimization
Since the Y (g) are all (0 2 ) normally distributed, it follows from the addition theorem
of the normal distribution (Heinhold and Gaede, 1972) that
1
n
nX
g=1
isa(0 2 =n) normally distributed random quantity. Accordingly, the two quantities
exp( = p n)arecharacteristic of the average changes (minus sign for reduction) in the
step lengths per generation. The median of w(z) isofcoursejuste 0 =1.Together with
Y (g)
Equation (5.34), our observation leads us to the requirement
or
exp ' max
n
' exp
' 'max p
n
p n
!
(5.35)
The variance 2 of the normally distributed random numbers Y , from which the lognormally
distributed random multipliers for the standard deviations (\step sizes") of the
changes in the object variables are produced, thus must vary inversely as the number of
variables. Its actual value should depend on the expected rate of convergence ' and
hence on the choice of the number of descendants .
Instead of only one common strategy parameter ,each individual can now have a
complete set of n di erent i i = 1(1)n, for every alteration in the corresponding n
object variables xi i=1(1)n. The two following schemes can be envisioned:
or
(g) (g) (g)
Ni = Ei Z i
(g)
Ni = (g)
Ei Z (g)
i
Z (g)
0
(5.36)
(5.37)
But only the second one should be taken into further consideration, because otherwise in
the case of n 1theaverage overall step size of the o spring
sN =
vu
u
t nX
i=1
2
Ni
could not be substantially di erent from that of its parent
sE =
vu
ut nX
due to the levelling e ect of the many randommultiplication events (law of large number
of events). In order to split the mutation e ects to the overall step size and the individual
step sizes one could choose
0 '
'
i=1
2
Ei
p
'
for Z0 (5.38)
2 n
' p p for all Zi i= 1(1)n (5.39)
2 n