Particle swarm optimization (PSO) is a population-based stochastic recursion procedure, which simulates the social behavior of a swarm of ants or a school of fish. Based upon the general representation of individual particles, this paper introduces a decreasing coefficient to the updating principle, so that PSO can be viewed as a regular stochastic approximation algorithm. To improve exploration ability, a random velocity is added to the velocity updating in order to balance exploration behavior and convergence rate with respect to different optimization problems. To emphasize the role of this additional velocity, the modified PSO paradigm is named PSO with controllable random exploration velocity (PSO-CREV). Its convergence is proved using Lyapunov theory on stochastic process. From the proof, some properties brought by the stochastic components are obtained such as "divergence before convergence" and "controllable exploration." Finally, a series of benchmarks is proposed to verify the feasibility of PSO-CREV. © 2007 IEEE.