Particle swarm optimization (PSO) is a popular nature-inspired
meta-heuristic for solving continuous optimization problems.
Although this technique is widely used, up to now
only some partial aspects of the method like
trajectories, runtime aspects,
the initial behavior in a bounded search space, and
parameter selection have been formally investigated.
while it is well-studied how to let the swarm converge
to a single point in the search space,
no theoretical statements about this point or
on the best position any particle has found
have been known.
For a very general class of objective functions,
we provide for the first time results about the quality of the
We show that a slightly adapted PSO almost surely finds
a local optimum by investigating the newly defined
of the swarm.
The potential drops when
the swarm approaches the point of convergence,
but increases if the swarm remains close to a point
that is not a local optimum, meaning that the swarm
charges potential and continues its movement.