Evolution and Optimum Seeking

A Multimembered Evolution Strategy 121
x 2
Circles : lines of
constant probability
density
(g)
E
1
(g) (g+1)
N = E
2 2
(g)
N
1
(g)
N
4
Opt.
(g) (g+1)
N = E
3 1
(g)
E
2
x
1
E : Parents
k
N : Offspring
Figure 5.4: Multimembered (2 , 4) evolution strategy
point N with vector xN =(xN1:::xNn) T is
w(E ! N) =
1
p2
! n
exp
The distance kxE ; xNk between xE and xN is
kxE ; xNk =
vu
u
t nX
i=1
; 1
2 2
nX
i=1
(xEi ; xNi) 2
(g) : Generation index
(xEi ; xNi) 2
!
(5.9)
But of this, only a part, s = f(xExN), is useful in the sense of approaching the objective.
To discover the total probability density forcovering a useful distance s, anintegration
must be performed over the locus of points for which the useful distance is s, measured
from the starting point xE. This locus is the surface of a nite region in n-dimensional
space:
Z Z
p(s) =
w(E ! N) dxN1 dxN2 ::: dxNn (5.10)
f(xExN) =s
The result of the integration depends on the weighting function f(xExN) andthus on
the topology of the objective function F (x).
So far only one random change has been considered. In the multimembered evolution
strategy, however, the average over the best of the o spring must be taken, in which
each of the o spring is to be associated with its own distance s`. We rst have to nd the
probability density w (s 0 ) for the th best descendant of a generation to cover the useful
distance s 0 .Itisacombinatorial product of