The quasispecies regime for the simple genetic algorithm with ranking selection
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We study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by ℓ the length of the chromosomes, by m the population size, by p_C the crossover probability and by p_M the mutation probability. We introduce a parameter σ, called the selection drift, which measures the selection intensity of the fittest chromosome. We show that the dynamics of the genetic algorithm depend in a critical way on the parameter π = σ(1-p_C)(1-p_M)^ℓ . If π<1, then the genetic algorithm operates in a disordered regime: an advantageous mutant disappears with probability larger than 1-1/m^β, where β is a positive exponent. If π>1, then the genetic algorithm operates in a quasispecies regime: an advantageous mutant invades a positive fraction of the population with probability larger than a constant p^* (which does not depend on m). We estimate next the probability of the occurrence of a catastrophe (the whole population falls below a fitness level which was previously reached by a positive fraction of the population). The asymptotic results suggest the following rules: π=σ(1-p_C)(1-p_M)^ℓ should be slightly larger than 1; p_M should be of order 1/ℓ; m should be larger than ℓℓ; the running time should be of exponential order in m. The first condition requires that ℓ p_M +p_C< σ. These conclusions must be taken with great care: they come from an asymptotic regime, and it is a formidable task to understand the relevance of this regime for a real-world problem. At least, we hope that these conclusions provide interesting guidelines for the practical implementation of the simple genetic algorithm.
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