  This paper derives, under minimal modelling assumptions, a simple to use
theorem for obtaining both order-$1$ and order-$2$ stability criteria for a
common class of particle swarm optimization (PSO) variants. Specifically, PSO
variants that can be rewritten as a finite sum of stochastically weighted
difference vectors between a particle's position and swarm informers are
covered by the theorem. Additionally, the use of the derived theorem allows a
PSO practitioner to obtain stability criteria that contains no artificial
restriction on the relationship between control coefficients. Almost all
previous PSO stability results have provided stability criteria under the
restriction that the social and cognitive control coefficients are equal; such
restrictions are not present when using the derived theorem. Using the derived
theorem, as demonstration of its ease of use, stability criteria are derived
without the imposed restriction on the relation between the control
coefficients for three popular PSO variants.
