(6) A methodology for representing the two and three-operator stationary distri-
bution components at all consistent values of mutation probability (including
the zero mutation probability limit), and a proposed approach for extending
that methodology to produce an explicit result.
10.3 Future Direction
In order to achieve the stated goal of this work, a complete analytical framework for
the simple genetic algorithm, additional progress must be made on the stationary distribu-
tion solution effort begun in Section 9. The coefficient relationships noted in Section 9.5,
especially the coefficient identities which attend transpositions of rows and columns in
the differentiation order array and their connection with the quasi-symmetric and quasi-
alternating polynomial notions presented in Appendix C, provide a foundation for pro-
ceeding with this effort. An explicit representation of the functional form of the stationary
distribution, reduced to a rational function expression in the algorithm parameters and
objective function, would provide a very valuable theoretical tool for use in the analysis
of genetic algorithm performance, and is the ultimate goal. However, even if explicit
solution is not attainable, it may prove possible to deduce very useful bounds on the sta-
tionary distribution components from continuation of the Section 9 development.
A second promising area for continuation of this work concerns the mutation
probability parameter sequence bound provided in Section 8. It is based on very simple
lower bounds (Eq. 4.21 and 4.31) which exist for the conditional probabilities which
compose the state transition matrix, and it only employs the one-step transition matrix in
the (1 t,(P)) sequence employed to establish weak ergodicity (Section 8.2). Some pre-
liminary work not reported in Section 8 suggests that employing two-step transition
matrices in summing the (1 Ti(P)) sequence may allow a refinement of the bound to
something of the form k~'. It also appears from that preliminary work that the same bound
applies for both the two and three-operator algorithm variants.