The quasispecies regime for the simple genetic algorithm with ranking selection

Abstract

We study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by ℓ \ell the length of the chromosomes, by m m the population size, by p C p_C the crossover probability and by p M p_M the mutation probability. We introduce a parameter σ \sigma , called the strength of the ranking selection, which measures the selection intensity of the fittest chromosome. We show that the dynamics of the genetic algorithm depends in a critical way on the parameter \[ π = σ ( 1 − p C ) ( 1 − p M ) ℓ . \pi \,=\,\sigma (1-p_C)(1-p_M)^\ell \,. \] If π > 1 \pi >1 , then the genetic algorithm operates in a disordered regime: an advantageous mutant disappears with probability larger than 1 − 1 / m β 1-1/m^\beta , where β \beta is a positive exponent. If π > 1 \pi >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 ∗ p^* (which does not depend on m 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: ∙ \bullet π = σ ( 1 − p C ) ( 1 − p M ) ℓ \pi =\sigma (1-p_C)(1-p_M)^\ell should be slightly larger than 1 1 ; ∙ \bullet p M p_M should be of order 1 / ℓ 1/\ell ; ∙ \bullet m m should be larger than ℓ ln ⁡ ℓ \ell \ln \ell ; ∙ \bullet the running time should be at most of exponential order in m m . The first condition requires that ℓ p M + p C > ln ⁡ σ \ell p_M +p_C> \ln \sigma . 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.