Evolution and Optimum Seeking

120 Evolution Strategies for Numerical Optimization
Step 0: (Initialization)
De ne x (0)
k
x (0)
k
= x (0)
Ek =(x(0)
k1:::x (0)
kn) T for all k = 1(1) :
= x(0) Ek is the vector of the kth parent Ek, suchthat Gj(x (0)
k ) 0 for all k =1(1) and all j = 1(1)m:
Set the generation counter g =0:
Step 1: (Mutation)
Generate x (g+1)
`
= x (g+1)
k + z (g + `)
such thatGj(x (g+1)
` ) 0 j =1(1)m ` =1(1)
where k 2 [1 ]
e.g., k =
( if ` = p pinteger
`(mod ) otherwise.
x (g+1)
` = x (g+1)
N`
=(x(g+1) `1 :::x (g+1)
`n ) T is the vector of the `th o spring N`
and z (g +`) is a normally distributed random vector with n components:
Step 2: (Selection)
Sort the x (g+1)
` for all ` = 1(1) so that
F (x (g+1)
`1 ) F (x (g+1)
`2 ) for all `1 = 1(1) `2 = + 1(1)
Assign x (g+2)
k
= x (g+1)
`1 for all k`1 = 1(1) :
Increase the generation counter g g +1:
Go to step 1, unless some termination criterion is ful lled.
What happens in one generation for a (2 , 4) evolution strategy is shown schematically on
the two dimensional contour diagram of a non-linear optimization problem in Figure 5.4.
5.2.2 The Rate of Progress of the (1 , )Evolution Strategy
In this section we attempt to obtain approximately the rate of progress of the multimembered,
or ( , ) strategy{at least for = 1. For this purpose the n-dimensional
sphere and corridor models, as used by Rechenberg (1973), are employed for calculating
the progress for the (1+1) strategy.
In the two membered evolution strategy ' was the expectation value of the useful
distance covered in each mutation. It is convenient here to de ne the rate in terms of the
number of generations.
' = expectation value k^x ; x (g) k;k^x ; x (g;1) k
where ^x is the vector of the optimum and x (g) is the average vector of the parents of
generation g.
From the chosen n-dimensional normal distribution of the random vector, which has
expectation value zero and variance 2 for all independent vector components, the probability
density for going from a point E with vector xE = (xE1:::xEn) T to another