Evolution and Optimum Seeking

130 Evolution Strategies for Numerical Optimization
This nally gives the probability function for the useful distance s0 covered in one generation,
expressed in units of u:
w(s 0 )= a
rE
e ; a au2 ; 2 u e
0
2 I ;1(au) @1 ; ae ; a 2
Zu
v=0
av2 ;
v e 2 I ;1(av) dv
Since the expectation value of this distribution is not readily obtainable, we shall determine
its maximum to give an approximation ~'. From the necessary condition
with the more concise notation
we obtain the relation
=1+ @D(u)
@u u=1; ~'=rE
@w(s 0 )
@s 0
s 0 =~'
!
=0
D(y) =ae ; a ay2
; 2 y e 2 I ;1(ay)
0
B
[D(1 ; ~'=rE)] ;2
@1 ;
Z
1; ~'=rE
v=0
1
C
1
A
;1
D(v) dvA
(5.18)
Except for the upper limit of integration, this is the same integral that made it so di cult
to obtain the exact solution for the rate of progress in the (1+1) evolution strategy (see
Rechenberg, 1973). Under the condition 1and =a 1, which means for many
variables and at a large enough distance from the optimum, Rechenberg obtained an
estimate by expanding Debye's asymptotic series representation of the Bessel function
(e.g., Jahnke-Emde-Losch, 1966) in powers of =a. Without giving here the individual
steps in the derivation, the result is
Z1
D(v) dv ' 1
"
1 ; erf
2
!#
p +
2 a 2
p
a
p 2
v=0
"
exp
; ( ; 1)2
!
; exp
2 a
2
;
2 a
!#
(5.19)
It is clear from Equation (5.4) that the rate of progress of the (1+1) strategy for the
two membered evolution varies inversely as the number of variables. Even if a higher
convergence rate is expected from the multimembered scheme, with many descendants
per parent, there will be no change in the relation to n, thenumber of parameters. In
addition to the assumptions already made regarding and =a, without further risk to
the validity of the approximate theory we can assume that 1 ; ~'=rE ' 1. Equation (5.19)
can now also be applied here.
For the partial di erential
@D(u)
@u u=1; ~'=rE
we obtain with the use of the Debye series again:
@D(u)
@u u=1; ~'=rE
= D(1 ; ~'=rE)
"
a exp
a (1 ; ~'=rE)
!
1
+
1 ; ~'=rE
#
; a (1 ; ~'=rE)