Evolutionary algorithms (EAs) are population-based randomized search heuristics that often solve problems successfully. Here the focus is on the possible effects of changing the parent population size in a simple, but still realistic, mutation-based EA. It preserves diversity by avoiding duplicates in its population. On the one hand its behavior on well-known pseudo-Boolean example functions is investigated by means of a rigorous runtime analysis. A comparison with the expected runtime of the algorithm's variant that does not avoid duplicates demonstrates the strengths and weaknesses of maintaining diversity. On the other hand, newly developed functions are presented for which the optimizer considered that even a decrease of the population size by a single increment leads from efficient optimization to enormous runtime and overwhelming probability. This is proven for all feasible population sizes and thereby this result forms a hierarchy theorem. In order to obtain all these results new methods for the analysis of the EA are developed.
Some of the results of this article have been presented as a best paper awarded contribution to the Genetic and Evolutionary Computation Conference – GECCO 2004 (see Storch, 2004). Major parts of this article were prepared while the author worked at the Department of Computer Science 2, University of Dortmund, Germany, and was supported by the German-Israeli Foundation and the German Research Foundation.