Evolution and Optimum Seeking

122 Evolution Strategies for Numerical Optimization
The probability density w(s`1 = s 0 ) that a particular descendant N`1 gets exactly
s 0 closer to the objective
The probability p(s`2 >s 0 ) that a descendant N`2 advances further than s 0
The probability p(s`3 s 0 )
Y
` +1 =1
` +162f`1`2:::` g
; X+2
` 2 =1
`26=`1
(
p(s`2 >s 0 )
X
` =` ;1 +1
` 62f`1`2:::` ;2g
p(s` +1 s 0 )
p(s`3 >s 0 )
(5.11)
w(s 0 )= 1 X
w (s 0 ) (5.12)
' =
Z1
s 0 =su
=1
s 0 w(s 0 ) ds 0
(5.13)
The meaning of su will be described later.
To evaluate ', besides and , all components of the position vectors of all parents of
the generation must be given, together with the values of for producing each descendant.
If ' is to become independent of a particular initial con guration, it is necessary to
de ne representative oraverage values of the relative positions of the parents, which are
established during the optimization as a function of the topology. Todosowould require
setting up and solving an integral equation. This has not yet been achieved.
To be able to say something nevertheless about the rate of convergence some simplifying
assumptions will be made. All parents will be represented by a single position vector
xk, and the standard deviations `i will be assumed equal for all components i = 1(1)n
and for the descendants ` = 1(1) . Equation (5.11) thereby simpli es to
Since
w (s 0 )=
; 1
; 1
!
w(s` = s 0 )[p(s` s 0 )] ;1
p(s` >s 0 )+p(s`