Convergence of simple genetic algorithms for the two-bit problem

Genetic algorithms (GAs) are stochastic evolutionary algorithms with applications to a variety of optimization problems. The most characteristic novel feature of the GA is that it is based on analogies with the principles of natural biological systems such as natural selection, recombination and mutation. Solving the optimization problem with the GA includes an open fundamental theoretical issue called GA-convergence: (a) what parameters and initial points make the GA converge on a single point? (b) is a GA convergence point an optimal point to the optimization problem? This paper solves these GA-convergence issues mathematically rigorously for the simplest case where the GA consists of two operators analogical with natural selection and recombination, and the optimization problem is the so-called two-bit problem (TBP), the variable consisting of two bits only. Specific results on issue (a) coincide with previous experimental results in GA, while those on issue (b) are significantly new not only in GA but also in population genetics. Moreover, the paper provides a perspective on the generalization of the results for solving the n-bit problem with ordinary simple GAs consisting of the simplest GA plus mutation.