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Recently a continuous description of the particle swarm optimization (PSO)based on a system of stochastic differential equations was proposed by Grassiand Pareschi in arXiv:2012.05613 where the authors formally showed the linkbetween PSO and the consensus based optimization (CBO) through zero-inertialimit. This paper is devoted to solving this theoretical open problem proposedin arXiv:2012.05613 by providing a rigorous derivation of CBO from PSO throughthe limit of zero inertia, and a quantified convergence rate is obtained aswell. The proofs are based on a probabilistic approach by investigating theweak convergence of the corresponding stochastic differential equations (SDEs)of Mckean type in the continuous path space and the results are illustratedwith some numerical examples.

Zero-Inertia Limit: from Particle Swarm Optimization to Consensus Based Optimization