Evolution and Optimum Seeking

A Multimembered Evolution Strategy 129
For the distance covered towards the objective, s`, the portion is now calculated that
contributes to an improvement of the objective function, i.e., in this case the radial difference
s` = rE ; r` (see Fig. 5.8). The locus of all points N` for which s` is the same is
the surface of the n-dimensional hypersphere about the origin with radius r` = rE ; s`.
Accordingly the total probability density that a mutation (index `) starting from point
E will cover the distance s` is the n-fold line integral:
w(s`) =
Z Z
rE ; r` = s`
1
p2
! n
exp ; 1
r2
2 2 ` + r 2
E ; 2 rE x`1 dx`1 :::dx`n
By transforming to spherical coordinates one obtains a simple integral
w(s`) =
1
p2
! n
n;1
2
; n;1
2
exp
; r2 E + r2 `
2 2
!
r n;1
`
2Z
=0
exp rE r` cos
2
The remaining integral can be expressed as a modi ed Bessel function:
w(s`) =
r n
2
`
r1; n
2
E
2
exp
; r2
E
+ r2
`
2 2
!
I n 2 ;1
rE r`
To simplify the notation we nowintroduce the following de nitions:
We thereby obtain
w(s`) = a
rE
= n
2 a= r2 E
2
v= r`
a
av2
; ;
e 2 v e 2 I ;1(av) with s` = rE (1 ; v)
rE
2
sin n;2
In order to use Equation (5.15) to calculate the total probability that the best of
descendants will cover the distance
the following quantities are still required:
with
and
s 0 =max
` fs` j ` = 1(1) g = rE ; r 0
w(s` = s 0 )= a
r 0
rE
rE
e ; a au2 ; 2 u e 2 I ;1(au)
= u and s 0 = rE (1 ; u)
p(s` s 0 )
=1; s0
=1; uR
R
s`=rE
v=0
w(s`) ds`
a av2
; ; ae 2 v e 2 I ;1(av) dv
d