Particle Swarm Optimization - A Survey <s> Stability in a Stochastic Particle System <s> In recent years population based methods such as genetic algorithms, evolutionary programming, evolution strategies and genetic programming have been increasingly employed to solve a variety of optimisation problems. Recently, another novel population based optimisation algorithm - namely the particle swarm optimisation (PSO) algorithm, was introduced by R. Eberhart and J. Kennedy (1995). Although the PSO algorithm possesses some attractive properties, its solution quality has been somewhat inferior to other evolutionary optimisation algorithms (P. Angeline, 1998). We propose a number of techniques to improve the standard PSO algorithm. Similar techniques have been employed in the context of self organising maps and neural-gas networks (T. Kohonen, 1990; T.M. Martinez et al., 1994). <s> BIB001 </s> Particle Swarm Optimization - A Survey <s> Stability in a Stochastic Particle System <s> Previous stability analysis of the particle swarm optimizer was restricted to the assumption that all parameters are nonrandom, in effect a deterministic particle swarm optimizer. We analyze the stability of the particle dynamics without this restrictive assumption using Lyapunov stability analysis and the concept of passive systems. Sufficient conditions for stability are derived, and an illustrative example is given. Simulation results confirm the prediction from theory that stability of the particle dynamics requires increasing the maximum value of the random parameter when the inertia factor is reduced. <s> BIB002
The analysis that lead to a now-common practice of using the constriction parameter in  assumed a deterministic system with no randomness. A recent work by Kadirkamanathan et al. BIB002 analyzes the stability of PSO with stochastic particle dynamics including the scalar randomness (ρ) in Eq. (7), by way of Lyapunov stability analysis. They derived the condition of asymptotic stability as, where 0 < φ < K holds for φ in Eq. BIB001 . This result gives a guide for selecting the parameters φ and w for stability and convergence.