SECTION 9
REPRESENTATION OF THE STATIONARY DISTRIBUTION SOLUTION
9.1 Overview
Previous sections of this work establish some key results required for extrapolation
of the simulated annealing convergence theory onto the nonstationary Markov chain
model of the simple genetic algorithm. Specifically, existence of a unique stationary dis-
tribution for the time-homogeneous two and three-operator algorithms is established in
Sections 4.3.2 and 4.3.3, and in Section 6, the existence argument is formulated into a
Cramer's rule expression for the stationary distribution components. Sections 7 and 8
continue that development by establishing the existence of a limiting distribution as the
mutation probability parameter approaches zero and a mutation probability sequence
bound sufficient to achieve it. However, the empirical results in Section 5.5 suggest a
complication, confirmed in Section 7, associated with the form of the limiting distribu-
tion. The limiting distribution behavior necessary for extending the simulated annealing
global optimality result does not obtain because the limiting distribution is nonzero for all
states with uniform population (one-operator absorbing states), including those for sub-
optimal solutions. The limiting distribution entropy results reported at the conclusion of
Section 5.5 support the intuitive notion that increasing the population size parameter
should bias the limiting distribution toward the desired behavior. However, to pursue that
notion further requires closer examination of the stationary distribution equations and the
requirements for their solution. This section begins that task. It is a very extensive devel-
opment and it stops short of explicit solution. However, it provides some insight into the
nature of the solution and additionally, it defines a promising approach to continuing the
work started here.