Theory of the Simple Genetic Algorithm with α-Selection

Genetic algorithms are random heuristic search (RHS) algorithms with a wide range of applications in adaptation and optimisation problems. The most advanced approach for a general theory of genetic algorithms is offered by the dynamical system model which describes the stochastic trajectory of a population under the dynamics of a genetic algorithm with the help of an underlying deterministic heuristic function and its fixed points. However, even for the simple genetic algorithm (SGA) with fitness-proportional selection, crossover and mutation the determination of the population trajectory and the fixed points of the heuristic function is unfeasible for practical problem sizes. In order to simplify the mathematical analysis α-selection is introduced in this paper. Based on this selection scheme it is possible to derive the dynamical system model and the fixed points in closed form. Although the heuristic function is not compatible with the equivalence relation imposed by schemata in the strict sense a simple coarse-grained system model with a single exogenous parameter is derivable for a given schemata family. In addition to the theoretical analysis experimental results are presented which confirm the theoretical predictions.