All of the state space characteristics described in 4.3.2 for the two-operator algo-
rithm follow. In particular, the Markov chain of the three-operator algorithm is irreduc-
ible. Thus, a unique stationary distribution exists for the time-homogeneous
three-operator simple genetic algorithm, and as in the two-operator case it is completely
determined by the objective function and the algorithm parameter values.
4.3.4 Summary
The asymptotic behavior of the one-operator simple genetic algorithm is dominated
by the states which correspond to uniform populations, the one-operator absorbing states.
The algorithm necessarily arrives at some member of the absorbing state set within a
finite number of algorithm iterations (Eq. 4.8). The asymptotic probability distribution
depends upon the algorithm initial population, mo. This observation is equivalent to the
fact, established in Section 4.3.1, that the stationary distribution of the one-operator algo-
rithm is not unique.
A unique stationary distribution exists for the time-homogeneous two and
three-operator algorithm variants (with a > 0), or equivalently, their asymptotic probabil-
ity distributions are independent ofmo. However, in the a -* 0+ limit, both the two and
three-operator algorithms degenerate into the absorbing state behavior which typifies the
one-operator case (Eq. 4.17 and Eq. 4.23, 4.27). A very important question is whether the
unique stationary distributions of the two and three-operator algorithms approach limits
as a O'. Section 7 answers that question affirmatively, and in Section 8, the lower
bounds reflected in Eq. 4.21 and Eq. 4.31 are employed to arrive at a monotone decreas-
ing sequence bound on pm(k) sufficient to guarantee that the limiting distribution is
achieved asymptoticallyy) by the inhomogeneous two and three-operator Markov chains.
The analogous conditional probability arrays [Pi(i I n)] and [P2'(i I n)], whose ele-
ments are defined by Eq. 4.2 and Eq. 4.22 respectively, play a very essential role in the
following sections, especially in Section 9. Most of the results developed hereafter apply
equally to the two and three-operator algorithm variants by substituting from these