5
the empirical results presented in Section 5 and some results developed later in Section 9
suggest that the required limiting behavior can be approached as closely as desired by
adjusting the algorithm parameters appropriately.
Section 9 attacks the problem of explicitly solving the system which results from
the Cramer's Rule formulation of the stationary distribution of the time-homogeneous
two and three-operator algorithms. It is a very extensive development which yields an
expression for the coefficient of the general term in the Taylor's series expansion of the
required determinants. It is based upon the highly symmetrical nature of the state trans-
ition matrix, as alluded to earlier.
The results of Section 9 are not reduced to a directly useable explicit solution.
Nevertheless, they do provide significant insight into the functional form of the stationary
distribution components. Furthermore, Section 9.5 points out some very significant iden-
tities which exist among the coefficients of the Taylor's series and suggests a method for
continuing the Section 9 development based upon the algebra of symmetric and
alternating polynomials. Explicit solution of the stationary distribution equations is the
major incomplete task required for extrapolation of the simulated annealing convergence
theory onto the genetic algorithm.
Section 10 summarizes this work and recapitulates the significant results. It also
proposes continuation of two parts of this research: (1) pursuit of the stationary distribu-
tion solution and (2) refinement of the mutation probability control parameter bound.
An appropriate mathematical framework for examining both the simulated
annealing and genetic algorithms is the theory of Markov chains. Appendix A is included
to summarize some essential definitions and theorems. Appendix B is devoted to the
Perron-Frobenius theorem, which is fundamental to the study of nonnegative matrices in
general and Markov chains in particular. Several important Markov chain theorems are
specializations of it and the key developments in Sections 6 and 7 require its application.
All of the Appendix A and Appendix B results are provided without proof or elaboration,