Comments on "Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. I. Basic properties of selection and mutation" [with reply]

In this paper, Gao points out two crucial errors in the proof of the theorem on the convergence of genetic algorithms (GAs) in the above paper by Qi-Palmieri (ibid., vol.5 (1994)). He presents two counter examples and comments that the errors may cause misleading on the convergence nature of GAs. He demonstrates that the sequence of mutation probabilities that increases the probability mass of the average set does not necessarily increase the mean fitness. In reply, Qi-Palmieri points out that the Theorem 3 simply states that it is possible to find a sequence of mutation densities that still guarantees convergence. The proof of Theorem 3 may be incomplete, but the result is unquestionably solid. They conclude that Gao´s counterexamples may not add much insight into the nature of the problem, and they would rather encourage constructive contributions to the difficult, still open problems of the evolutionary paradigm.