  In this paper we consider a continuous description based on stochastic
differential equations of the popular particle swarm optimization (PSO) process
for solving global optimization problems and derive in the large particle limit
the corresponding mean-field approximation based on Vlasov-Fokker-Planck-type
equations. The disadvantage of memory effects induced by the need to store the
local best position is overcome by the introduction of an additional
differential equation describing the evolution of the local best. A
regularization process for the global best permits to formally derive the
respective mean-field description. Subsequently, in the small inertia limit, we
compute the related macroscopic hydrodynamic equations that clarify the link
with the recently introduced consensus based optimization (CBO) methods.
Several numerical examples illustrate the mean field process, the small inertia
limit and the potential of this general class of global optimization methods.
