Stochastic Models of Jaya and Semi-Steady-State Jaya Algorithms

The Jaya algorithm and its variants have enjoyed great success in diverse application areas, but no theoretical analysis of the algorithm, to our knowledge, is available in the literature. In this paper we build stochastic models for analyzing Jaya and semi-steady-state Jaya algorithms. For these algorithms, the computational cost depends on how, at each iteration, the new individual fares against the existing individual. Costs must be incurred for any replacement of individuals and the subsequent update of the population-worst individual’s (and/or the population-best individual’s) index. We use the following two quantities as the main metrics for analysis: the expected number of updates in a generation of the worst individual’s index, and the corresponding expectation for updating the best individual’s index. Clearly, the higher these expectations, the costlier the algorithm. The analysis shows that for semi-steady-state Jaya (a) the maximum expected number of worst-index updates per generation is 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) 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.