Evolution and Optimum Seeking

A Multimembered Evolution Strategy 137
outcome would be extinction of the population and the rate of progress would no longer
be de ned. The probability of extinction of the population is given by the product of the
lethal probabilities:
pstop =(1; v n;1 )
To be able to optimize further in such situations let us adopt the following procedure: If
all the mutations lead to forbidden points, the parent willsurvive and produce another
generation of descendants. Thus for this generation the rate of progress takes the value
zero. Equation (5.25) then only holds for s 0 6= 0 and we must reformulate the probability
of advancing by s 0 in one generation as follows:
where
w(s 0 )= ~w(s 0 )+ pstop
=
( 0 if s 0 6=0
1 if s 0 =0
The distribution w(s 0 ) is no longer continuous, and even if w 0 (s 0 ) is symmetricwe cannot
assume that the maximum of the distribution is a useful approximation to the average
rate of progress (Fig. 5.12). The following condition must be satis ed:
Z1
s 0 =;1
w(s 0 ) ds 0 =
Z1
s 0 =;1
~w(s 0 ) ds 0 + wstop =1 (5.26)
We can think of w(s 0 ) as a superposition of two density distributions, with conditional
mathematical expectation values
and
and with associated frequencies
and
p1 =
'1 =
Z1
s 0 =;1
Z1
s 0 =;1
'2 =0
s 0 ~w(s 0 ) ds 0
~w(s 0 ) ds 0 =1; pstop
p2 = pstop
The events belonging to the two density distributions are mutually exclusiveandby virtue
of Equation (5.26) they make up together a complete set of events. The expectation value
is then given by (e.g., Gnedenko, 1970 Sweschnikow ,1970).
' =
Z1
s 0 =;1
s 0 w(s 0 ) ds 0 = '1 p1 + '2 p2 = '1 (1 ; pstop)