Evolution and Optimum Seeking

A Multimembered Evolution Strategy 133
2.0
1.0
1
Maximal universal rate
of progress
ϕ ( σ )
max opt
(1+1) - theory
Numer of offspring
5 10 15 20 25 30
Figure 5.10: Maximal rate of progress for the sphere model
Points on the curve ' max = ' ( = opt) can only be obtained iteratively. To
express = (' max), the non-linear system of equations consisting of Equations (5.20)
and (5.21) must be solved. The results as obtained with the multimembered evolution
strategy are shown in Figure 5.10. A convenient formula can only be obtained by assuming
' + ' + i.e., 2 ' max ' 2
opt
This estimate is actually not far wrong, since the second term in Equation (5.21) goes to
zero. We thus nd
' 1+ q 'max exp('max ) 1 + erf 1 q
'max (5.22)
2
a relation with comparable structure to the result for the inclined plane.
Finally we ask whether ' max= has a maximum, as in the inclined plane case. If the
parent can survive the o spring, opt = 1 here too if not the condition
p 1
opt =2
2 +('+ + + ) 2 exp[(' + + + ) 2 ][1 + erf( + )] ' +
must be added to Equations (5.20) and (5.21). The solution, obtained iteratively, is:
opt ' 4:7 (as an integer: opt =5)
λ
(5.23)