  A particle filter is introduced to numerically approximate a solution of the
global optimization problem. The theoretical significance of this work comes
from its variational aspects: (i) the proposed particle filter is a controlled
interacting particle system where the control input represents the solution of
a mean-field type optimal control problem; and (ii) the associated density
transport is shown to be a gradient flow (steepest descent) for the optimal
value function, with respect to the Kullback--Leibler divergence. The optimal
control construction of the particle filter is a significant departure from the
classical importance sampling-resampling based approaches. There are several
practical advantages: (i) resampling, reproduction, death or birth of particles
is avoided; (ii) simulation variance can potentially be reduced by applying
feedback control principles; and (iii) the parametric approximation naturally
arises as a special case. The latter also suggests systematic approaches for
numerical approximation of the optimal control law. The theoretical results are
illustrated with numerical examples.
