Date: 21 May 1999

Analysis on the Island Model Parallel Genetic Algorithms for the Genetic Drifts

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Abstract

In our former paper, we have investigated the relation among the mean convergence time, the population size, and the chromosome length of genetic algorithms (GAs). Our analyses of GAs make use of the Markov chain formalism based on the Wright-Fisher model, which is a typical and well-known model in population genetics. The Wright-Fisher model is characterized by 1-locus, 2-alleles, fixed population size, and discrete generation. For these simple characters, it is easy to evaluate the behavior of genetic process. We have also given the mean convergence time under genetic drift. Genetic drift can be well described in the Wright-Fisher model, and we have determined the stationary states of the corresponding Markov chain model and the mean convergence time to reach one of these stationary states. The island model is also well-known model in population genetics, and it is similar to one of the most typical model of parallel GAs, which require parallel computer for high performance computing. We have also derived the most effective migration rate for the island model parallel GAs with some restrictions. The obtained most effective migration rate is rather small value, i.e. one immigrant per generation, however the behaviors of the island model parallel GAs at that migration rate are not revealed yet clearly. In this paper, we discuss the mean convergence time for the island model parallel GAs from both of exact solution and numerical simulation. As expected from the Wright-Fisher model’s analysis, the mean convergence time of the island model parallel GAs is proportional to population size, and the coefficient is larger with smaller migration rate. Since to keep the diversity in population is important for effective performance of GAs, the convergence in population gives a bad influence for GAs. On the other hand, mutation and crossover operation prevent converging in GAs population. Because of the small migration rate makes converging force weak, it must be effective for GAs. This means that the island model parallel GAs is more efficient not only to use large population size with parallel computers, but also to keep the diversity in population, than usual GAs.