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