Theoretical Analysis of Fitness-proportional Selection: Landscapes and Efficiency

We investigate theoretically how the fitness landscape influences the optimization process of population-based evolutionary algorithms using fitness-proportional selection. Considering the function OneMax, we show that it cannot be optimized in polynomial time with high probability regardless of the population size. This is proved by a generalization of drift analysis. For populations of at most logarithmic size, the negative result transfers to any function with unique optimum. Based on these insights, we investigate the effect of scaling the objective function in combination with a population that is not too small and show that then such algorithms compute optimal solutions for a wide range of problems in expected polynomial time. Finally, relationships with (1+λ)-EAs and (1,λ)-EAs are described.