Running-time analysis of evolutionary programming based on Lebesgue measure of searching space

There have been many studies on the runtime analysis of evolutionary algorithms in discrete optimization, and however, relatively few homologous results have been obtained on continuous optimization, such as evolutionary programming (EP). This paper presents an analysis of the running time (as approximated by the mean first hitting time) of two EP algorithms based on Gaussian and Cauchy mutations, using an absorbing Markov process model. Given a constant variation, we analyze the running-time upper bound of special Gaussian mutation EP and Cauchy mutation EP, respectively. Our analysis shows that the upper bounds are impacted by individual number, problem dimension number, searching range, and the Lebesgue measure of the optimal neighborhood. Furthermore, we provide conditions whereby the mean running time of the considered EPs can be no more than a polynomial of n. The condition is that the Lebesgue measure of the optimal neighborhood is larger than a combinatorial computation of an exponential and the given polynomial of n. In the end, we present a case study on sphere function, and the experiment validates the theoretical result in the case study.