Stochastic models of Jaya and semi-steady-state Jaya algorithms

We build stochastic models for analyzing Jaya and semi-steady-state Jaya algorithms. The analysis shows that for semi-steady-state Jaya (a) the maximum expected value of the number of worst-index updates per generation is a paltry 1.7 regardless of the population size; (b) regardless of the population size, the expectation of the number of best-index updates per generation decreases monotonically with generations; (c) exact upper bounds as well as asymptotics of the expected best-update counts can be obtained for specific distributions; the upper bound is 0.5 for normal and logistic distributions, $\ln 2$ for the uniform distribution, and $e^{-\gamma} \ln 2$ for the exponential distribution, where $\gamma$ is the Euler-Mascheroni constant; the asymptotic is $e^{-\gamma} \ln 2$ for logistic and exponential distributions and $\ln 2$ for the uniform distribution (the asymptotic cannot be obtained analytically for the normal distribution). The models lead to the derivation of computational complexities of Jaya and semi-steady-state Jaya. The theoretical analysis is supported with empirical results on a benchmark suite. The insights provided by our stochastic models should help design new, improved population-based search/optimization heuristics.