Stability bounds and almost sure convergence of improved particle swarm optimization methods

Particle swarm optimization (PSO) is a member of nature-inspired metaheuristic algorithms. Its formulation is simple and does not need the computation of derivatives. It and its many variants have been applied to many different types of optimization problems across several disciplines. There have been many attempts to study the convergence properties of PSO, but a rigorous and complete proof of its almost sure convergence to the global optimum is still lacking. We propose two modified versions of PSO and prove their convergence to the global optimum. We conduct simulation studies to gain further insights into their properties and evaluate their performance relative to PSO.