algorithm for which the mutation probability sequence both observes this bound and con-
verges to zero achieves asymptoticallyy) the limiting probability distribution defined in
Propositions 7.5 and 7.6.
8.4 Comparison With the Simulated Annealing Parameter Bound
It is instructive to compare the mutation probability sequence bound developed here
with the annealing schedule bounds reviewed in Section 2.4.2.2, both of which are of the
form K/log(k). Let p(k) be defined as the ratio
p(k) = p(k)/T(k)
where p,(k) is selected as the bound developed herein and T(k) is selected as the bound
provided by either Eq. 2.12 or Eq. 2.13. That is
p(k)= k-"'/[K/log(k)] Eq. 8.5
2
1 1
= log(k)/k'.
Thus, decreasing values of p(k) imply that the genetic algorithm convergence rate is
superior asymptoticallyy) to that of the simulated annealing algorithm.
Now, let k = exp(x), or equivalently x = log(k). Substituting into Eq. 8.5 yields
p(k) = x exp .
Then, since for all positive constants y, the limit of x exp(-yx) as x -> o is zero, it follows
that
lim p(k)= Ig(/] = 0. Eq. 8.6
Thus, the nonstationary simple genetic algorithm provides an asymptotically superior
convergence rate.