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
have been formally investigated.
while it is well-studied how to let the swarm converge
to a single point in the search space,
no general 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.
To do so, we investigate 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.