Evolution and Optimum Seeking

A Multimembered Evolution Strategy 143
Model function
Inclined plane
Sphere
weopt
1
2
0:27
1
weopt 2
3.7
opt
2.5
4.7
Corridor
1
2e 5.4 6.0
How should the step lengths now be altered? We shall rst consider only a single
variance 2 for changes in all the variables. In the production of the random changes,
the standard deviation is always a positive factor. It is therefore reasonable to generate
new step lengths from the old by amultiplicative rather than additive process, according
to the scheme
(g)
N = (g)
E Z (g)
(5.32)
The median of the random distribution for the quantity Z must equal one to satisfy the
condition that there is no deterministic drift without selection. Furthermore an increase
of the step length should occur with the same frequency as a decrease more precisely,the
probability of occurrence of a particular random value must be the same as that of its reciprocal.
The third requirement is that small changes should occur more often than large
ones. All three requirements are satis ed by the log-normal distribution. Random quantities
obeying this distribution are obtained from (0 2 ) normally distributed numbers Y
by the process
Z = e Y (5.33)
The probability distribution for Z is then
w(z) = 1
p 2
1
z exp
; (ln z)2
2 2
!
The next question concerns the choice of ,andwe shall answer it, in the same way as
for the (1+1) strategy, with reference to the rate of change of step lengths that maintains
the maximum rate of progress in the sphere model. Regarding ' as a di erential quotient
;dr=dg leads to the relation (see Sect. 5.1.2)
(g+1)
opt
(g)
opt
=exp ; ' max
n
(5.34)
for the optimal step lengths of two consecutive generations, where ' max now has a different,
larger value that depends on and . The actual size of the average changes in
the variances, using the proposed mutation scheme based on Equations (5.32) and (5.33),
depends on the topology of the objective function and the number of parents and descendants.
If n, thenumber of variables, is large, the optimal variance will only change
slightly from generation to generation. We will therefore assume that the selection in
any generation is more or less indi erent to reductions and increases in the step length.
We thereby obtain the multiplicative change in the random quantity X, averaged over n
generations:
X =
0
@ nY
g=1
Z (g)
1
A
1
n
= exp
0
@ 1
n
nX
g=1
Y (g)
1
A