Theory of the Simple Genetic Algorithm with α-Selection, Uniform Crossover and Bitwise Mutation

Genetic algorithms (GA) are instances of random heuristic search (RHS) which mimic biological evolution and molecular genetics in simplified form. These random heuristic search algorithms can be theoretically described by an infinite population model with the help of a deterministic dynamical system model by which the stochastic trajectory of a population can be characterized using a deterministic heuristic function and its fixed points. For practical problem sizes the determination of the fixed points is unfeasible even for the simple genetic algorithm (SGA) with fitness-proportional selection, crossover and bitwise mutation. The recently introduced simple genetic algorithm with α-selection allows the analytical calculation of the unique fixed point of the corresponding intrinsic system model. In this paper, an overview of the theoretical results for the simple genetic algorithm with α-selection and its intrinsic system model is given. The unique fixed point of the intrinsic system model is derived and its compatibility with the equivalence relation imposed by schemata is shown. In addition to the theoretical analysis experimental results for the simple genetic algorithm with α-selection, uniform crossover and bitwise mutation are presented showing a close agreement to the theoretical predictions. Key–Words: Simple genetic algorithm, α-selection, random heuristic search, dynamical system model, infinite population model, intrinsic system model, schemata