The bound is asserted in Proposition 8.1, and its form (i.e. k-r) is asymptotically superior
to the K/log(k) bound associated with the simulated annealing algorithm. It is very note-
worthy that the same bound applies both to the two and three-operator algorithm variants.
At least in terms of the Section 8 bound, the crossover operator does not expedite
convergence.
All of the results developed in Sections 7 and 8 are obtained without explicitly solv-
ing the stationary distribution system. Section 9 attacks the problem of explicit solution.
It is a very extensive and somewhat tedious development. The product of that work is an
expression for the general term in a multivariate Taylor's series expansion of the determi-
nant form required for explicit solution of the stationary distribution equations. The
results are expressed in Eq. 9.26 and 9.27 for the general nonzero mutation probability
case, augmented by Eq. 9.28 for the zero mutation probability limit. These results stop
short of a useable answer but they do provide some insight into the nature of the solution.
Further, Section 9.5 provides some intriguing ideas for extending the work started in Sec-
tion 9.
The attempt to extrapolate the simulated annealing convergence theory onto the
genetic algorithm fails in the sense that the zero mutation probability stationary distribu-
tion limits of the two and three-operator simple genetic algorithm variants do not satisfy
the required form for extrapolation of the simulated annealing global optimality result.
However, evidence is provided which suggests that for the two-operator algorithm vari-
ant, the required behavior can be approached by increasing the population size parameter
(Eq. 9.36). The question is more complicated for the three-operator case, and as pointed
out in Section 9.4, implementation of crossover with nonzero p, may indeed preclude
convergence to global optimality even in the infinite population size limiting sense of Eq.
9.36.
The latter observation concerning crossover, along with the equivalence of the
mutation probability sequence bounds for the two and three-operator cases noted