Statistical stability analysis for particle swarm optimization dynamics with random coefficients

Particle swarm optimization (PSO), a meta-heuristic global optimization method, has attracted special interest for its simple algorithm and high searching ability. The updating formula of PSO involves coefficients with random numbers as parameters to enhance diversification ability in searching for the global optimum. However, the randomness makes stability of the searching points difficult to analyze mathematically, and the users need to adjust the parameter values by trial and error. In this paper, stability of the stochastic dynamics of PSO is analyzed mathematically and an exact stability condition taking the randomness into consideration is presented with an index called the "statistical eigenvalue," which is a new concept for evaluating the degree of stability of PSO dynamics. The accuracy and effectiveness of the proposed stability discrimination using the presented index are certified in numerical simulation for simple examples.