Evolution and Optimum Seeking

A Multimembered Evolution Strategy 139
With the new quantities ' and , Equation (5.24) for v becomes
p !
2 n
v = erf ; p
2
"
1 ; exp
n
; 2 n2
!#
2
Since the argument of the error function increases as n, thenumber of variables, the
approximation
erf(y) ' 1 ; 1
p exp (;y
y 2 )
can be used to give
and with
nally
v =1; n p 2
lim
n!1
1+ 1
n
v 1;n =exp
The desired relation = (' )isthus
=1+
p ~'
p 2
exp
2
4
~'
p2
in which, from Equation (5.27),
~' =
! 2 3
5
"
erf
n
p 2
~'
p2
'
for n 1
= e
!
!
1 ; h 1 ; exp ;p 2
+ 2 exp
i
p 2
!
#
; 1
(5.29)
Pairs of values obtained iteratively are shown in Figure 5.13 together with simulation
results for the cases of \survival" and \extinction" of the parent (n = 100 b = 100,
average over 10 000 successful generations).
As in the case of the sphere model, the deviations can be attributed to the simplifying
assumptions made in deriving the approximate theory. For = 1' is always zero if
the parent is not included in the selection. The transition to the inclined plane model is
correctly reproduced in this respect. Negative rates of progress cannot occur.
The position of the maxima ' max = ' ( = opt) at constant are obtained in the
same way as for the sphere model. The condition to be added to Equation (5.29) is
c exp (; + )[1; exp (; + )] ;1 ; 1
h erf(' + )+2exp( + ) ; 1 ih 1+2' +2 i + 2
p ' + exp (;' +2 )
in which the following new quantities are introduced again for compactness:
+
!
!
+2exp( + ) ! =0
(5.30)