A second conjecture concerns the limiting distribution behavior as a function of the
parameter M. The computed limiting distribution entropy results displayed in Section 5.5
suggest that the limiting distribution is dominated by optimal solutions for M sufficiently
large. That supposition is supported by the results in Eq. 9.33 and 9.35. In the M=2 case,
it follows immediately from Eq. 9.33 that qo(02)/qo(20) = [R(1)/R(0)]2. For M=3 and R(1)
< R(0) it is straightforward to show that a corresponding bounding relationship exists, i.e.
qo(03)/qo(30) < [R(I )/R(0)]3. This suggests that the ratio of the probabilities of the uni-
form population states corresponding to i and j with R(i) < R(j) behaves at or better than
[R(i)/R(j)]M -- 0 Eq. 9.36
for M sufficiently large. If this supposition is indeed correct, then the desired limiting dis-
tribution behavior for the two-operator simple genetic algorithm (i.e. probability zero for
sub-optimal solutions) can be approached as closely as required by selecting M
sufficiently large.
The corresponding general case (i.e. L>1) three-operator counterparts of Eq. 9.32
and 9.34 are expressed in terms of the P2(i I n)' array (Eq. 4.22). Thus, the numerator
polynomial counterparts of Eq. 9.33 and 9.35 are expressed in terms of complex polyno-
mial functions of the reward function values, and consequently it may be that no general
case three-operator counterpart of Eq. 9.36 exists. (It is noted that the design of the
reward functions employed in Section 5, in which only length 0-2 schema dependence is
incorporated, tends to minimize crossover disruption, which may account for the progres-
sion toward optimality indicated by the three-operator results recorded in Figures 5-7
through 5-18). The simulated annealing global optimality may thus extrapolate onto the
simple genetic algorithm only in the p, -> 0 and M -- oo limiting sense.
9.5 Extending the Stationary Distribution Representation
Eq. 9.26 and 9.27 represent an exact expression of the value of the determinant
-I P- II = | P-II -I Pi;-i, and with Propositions 6.7 and 6.8 constitute an exact repre-
sentation of the components of the stationary distribution of the two and three-operator