The essence of the simulated annealing theory is demonstration of (1) existence of a
unique asymptotic probability distribution (stationary distribution) for the stationary Mar-
kov chain corresponding to every strictly positive constant value of an algorithm control
parameter (absolute temperature), (2) existence of a stationary distribution limit as the
control parameter approaches zero, (3) the desired behavior of the stationary distribution
limit (i.e. optimal solution with probability one) and (4) sufficient conditions on the algo-
rithm control parameter to ensure that the nonstationary algorithm achieves (asymptoti-
cally) the limiting distribution. With the exception of (3), this work adapts that
methodology to the genetic algorithm Markov chain model employing a genetic operator
parameter (mutation probability) as the algorithm control parameter. The results include a
mutation probability control parameter bound analogous to (and asymptotically superior
to) the conventional simulated annealing parameter bounds, and a framework for repre-
senting the genetic algorithm stationary distribution components at all consistent fixed
control parameter values, including zero.
The genetic algorithm stationary distribution limit has nonzero components corre-
sponding to all solutions. Thus, the simulated annealing global optimality convergence
result does not extrapolate. However, both empirical and theoretical evidence is provided
which suggests that the desired limiting behavior can be approached by suitably adjusting
the algorithm parameters.