Evolution and Optimum Seeking

126 Evolution Strategies for Numerical Optimization
2
1
0
ϕ
1
Rate of progress for σ = 1
ϕ
ext
(λ)
ϕ
sur
(λ)
ϕ ( λ) λ
(1+1)
5 10 15 20 25
Simulation with
“extinction”
Simulation with
“survival”
(1, ) approximate theory
Theory
Number of offspring
Figure 5.6: Rate of progress for the inclined plane model
comparison to be poor for making exact quantitative predictions, it nevertheless correctly
reproduces the qualitative relation between the rate of progress and the number of descendants
in a generation. The probability distributions w(s 0 ) are illustrated in Figure 5.7
for ve di erent values of 2f1 3 10 30 100g, according to Equation (5.16).
For the inclined plane model the question of an optimal step length does not arise. The
rate of progress increases linearly with the step length. Another question that does arise,
however, is how tochoose the optimal number of o spring per parent in a generation.
The immediate answer is: the bigger is, the faster the evolution advances. But in
nature, since resources are limited (territory, food, etc.) it is not possible to increase
the number of descendants arbitrarily. Likewise in applications of the strategy to solving
problems on the digital computer, the requirements for computation time impose limits.
The computers in common use today can only work in a serial rather than parallel way.
Thus all the mutations must be produced one after the other, and the more descendants
the longer the computation time. We should therefore turn our attention instead to nding
the optimum value of '= . In the case where the parent survives if it is not bettered by
any descendant, we have the trivial solution
opt =1
The corresponding value for the (1 , ) strategy is, however, larger. With Equation (5.17)
λ