Evolution and Optimum Seeking

A Multimembered Evolution Strategy 135
2b
N 2
N 3
s 2 < 0
0
x 2
E
s 1 > 0
N 1
Line of equal probability density
Figure 5.11: Corridor model function
= 1
"
erf
2
!
b ; xEi
p + erf
2
!#
b + xEi
p
2
Contours F(x) = const.
Downwards
x 1
Allowed region
Forbidden region
That is, the probability depends on the current position xEi of the starting point E. We
can only construct an average value for all possible situations if we know the probability
pa of certain situations occurring. It could well be that, during the minimum search,
positions near the border are occupied less often than others. The same problem of
nding the occupation probability pa has arisen already in the theoretical treatment of
the (1+1) strategy. Rechenberg (1973) discovered that
pa = 1
2 b (with respect to one of the variables xi i= 2(1)n)
which is a constant independent of the current values of the variables. We will assume
that this also holds here. Thus the average probability that one of the n ; 1 constrained
variables will remain within the corridor can be given as:
= 1
4 b
~p(jx`ij b) =
Zb
xEi=;b
"
erf
Zb
xEi=;b
!
b ; xEi
p + erf
2
pa p(jx`ij b) dxEi
!#
b + xEi
p dxEi
2