Several stability analyses and stable regions of particle swarm optimization (PSO) have been proposed before. The assumption of stagnation and different definitions of stability are adopted in these analyses. In this paper, the order-2 stability of PSO is analyzed based on a weak stagnation assumption. A new definition of stability is proposed and an order-2 stable region is obtained. Several existing stable analyses for canonical PSO are compared, especially their definitions of stability and the corresponding stable regions. It is shown that the classical stagnation assumption is too strict and not necessary. Moreover, among all these definitions of stability, it is shown that our definition requires the weakest conditions, and additional conditions bring no benefit. Finally, numerical experiments are reported to show that the obtained stable region is meaningful. A new parameter combination of PSO is also shown to be good, even better than some known best parameter combinations.

The popular performance profiles and data profiles for benchmarking deterministic optimization algorithms are extended to benchmark stochastic algorithms for global optimization problems. A general confidence interval is employed to replace the significance test, which is popular in traditional benchmarking methods but suffering more and more criticisms. Through computing confidence bounds of the general confidence interval and visualizing them with performance profiles and (or) data profiles, our benchmarking method can be used to compare stochastic optimization algorithms by graphs. Compared with traditional benchmarking methods, our method is synthetic statistically and therefore is suitable for large sets of benchmark problems. Compared with some sample-mean-based benchmarking methods, e.g., the method adopted in black-box-optimization-benchmarking workshop/competition, our method considers not only sample means but also sample variances. The most important property of our method is that it is a distribution-free method, i.e., it does not depend on any distribution assumption of the population. This makes it a promising benchmarking method for stochastic optimization algorithms. Some examples are provided to illustrate how to use our method to compare stochastic optimization algorithms.