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Reporting the empirical results of swarm and evolutionary computation algorithms is a challenging task with many possible difficulties. These difficulties stem from the stochastic nature of such algorithms, as well as their inability to guarantee an optimal solution in polynomial time. This research deals with measuring the performance of stochastic optimization algorithms, as well as the confidence intervals of the empirically obtained statistics. Traditionally, the arithmetic mean is used for measuring average performance, but we propose quantiles for measuring average, peak and bad-case performance, and give their interpretations in a relevant context for measuring the performance of the metaheuristics. In order to investigate the differences between arithmetic mean and quantiles, and to confirm possible benefits, we conducted experiments with 7 stochastic algorithms and 20 unconstrained continuous variable optimization problems. The experiments showed that median was a better measure of average performance than arithmetic mean, based on the observed solution quality. Out of 20 problem instances, a discrepancy between the arithmetic mean and median happened in 6 instances, out of which 5 were resolved in favor of median and 1 instance remained unresolved as a near tie. The arithmetic mean was completely inadequate for measuring average performance based on the observed number of function evaluations, while the 0.5 quantile (median) was suitable for that task. The quantiles also showed to be adequate for assessing peak performance and bad-case performance. In this paper, we also proposed a bootstrap method to calculate the confidence intervals of the probability of the empirically obtained quantiles. Considering the many advantages of using quantiles, including the ability to calculate probabilities of success in the case of multiple executions of the algorithm and the practically useful method of calculating confidence intervals, we recommend quantiles as the standard measure of peak, average and bad-case performance of stochastic optimization algorithms. [ABSTRACT FROM AUTHOR]

Formation control is one of the most concerning topics within the realm of swarm intelligence. This paper presents a metaheuristic approach that leverages a natural co-evolutionary strategy to solve the formation control problem for a swarm of missiles. The missile swarm is modeled by a second-order system with a heterogeneous reference target, and the exponential of the resultant error is accumulated to be the objective function such that the swarm converges to optimal equilibrium states satisfying specific formation requirements. Focusing on the issue of the local optimum and unstable evolution, we incorporate a novel model-based policy constraint and a population adaptation strategy that significantly alleviates the performance degradation of the existing natural co-evolutionary strategy in terms of slow training and instability of convergence. With application of the Molloy–Reed criterion in the field of network communication, we developed an adaptive topology method that assures connectivity under node failure, and its effectiveness is validated theoretically and experimentally. The experimental results demonstrate that the accuracy of formation flight achieved by this method is competitive with that of conventional control methods and is much more adaptable. More significantly, we show that it is feasible to treat the generic formation control problem as an optimal control problem for finding a Nash equilibrium strategy and solving it through iterative learning. [ABSTRACT FROM AUTHOR]