Stability and l2 gain analysis for the particle swarm optimization algorithm

Stability of the particle swarm optimization algorithm is analyzed without any simplifying assumptions made in the previous works. To evaluate the convergence speed of the algorithm, the decay rate is introduced, and a method for finding the largest lower bound of the decay rate is presented. Moreover, it is pointed out that the l² gain of the algorithm can be used to measure exploration ability of the algorithm, and a method for finding of the smallest upper bound of the l² gain is provided. The above methods are based on linear matrix inequality techniques and therefore are carried out efficiently by using convex optimization tools. Numerical examples are given to show that the analysis methods are reasonable and effective to select the parameters in the algorithm.