The Dynamics of a Genetic Algorithm on a Model Hard Optimization Problem

A model of a hard optimization problem suggested in the literature is considered. The dynamics of a genetic algorithm (GA) using ranking selection, mutation, and uniform crossover are completely modeled on this problem and generalized to any symmetrical concave function of unitation. Full finite population effects are taken into account allowing a novel analytical comparison of roulette wheel and stochastic universal sampling. Closed form expressions are derived for the equilibrium population distribution of this model. The first passage time to move from a local to a global minimum in a two-potential well landscape is calculated. A comparison is made with a stochastic hill climber and a GA without crossover. The GA with crossover is shown to perform orders of magnitude faster giving some insights into the nature of GA search and the crossover operator.