In this paper we investigate three important properties (stability, local convergence, and transformation invariance) of a variant of particle swarm optimization called standard particle swarm optimization 2011. Through some experiments, we identify boundaries of coefficients for this algorithm that ensure particles converge to their equilibrium. Our experiments show that these convergence boundaries for this algorithm are: 1) dependent on the number of dimensions of the problem, 2) different from that of some other PSO variants, and 3) not affected by the stagnation assumption. We also determine boundaries for coefficients associated with different behaviors, e.g., non-oscillatory and zigzagging, of particles before convergence through analysis of particle positions in the frequency domain. In addition, we investigate the local convergence property of this algorithm and we prove that it is not locally convergent. We provide a sufficient condition and related proofs for local convergence for a formulation that represents updating rules of a large class of particle swarm optimization variants. We modify the standard particle swarm optimization 2011 in such a way that it satisfies that sufficient condition, hence, the modified algorithm is locally convergent. Also, we prove that the original standard particle swarm optimization algorithm is not sensitive to rotation, scaling, and translation of the search space.
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