Particle Swarm Optimization for 1D Problems
Particle swarm optimization (PSO) comes from the pioneering work of Kennedy and Eberhart [1, 2]. PSO algorithms mimic the social behavior patterns of organisms that live and interact within large groups, such as swarms of bees. This optimization technique is used to find the minimum of the following 1D test function: , with ; you can vary the parameters and . For the global minimum of , perfect agreement is found using either the Mathematica built-in command NMinimize (the green square) or PSO (the red dots) with the appropriate number of iterations. You can vary the number of iterations as well as the swarm size. However, notice the complementary effect of these parameters. Here, the problem is one-dimensional, but extension to multidimensional problems using the program is straightforward.
 J. Kennedy and R. Eberhart, "Particle Swarm Optimization," in Proceedings of the 1995 International Conference on Neural Networks, Vol. 4, New York: IEEE Press, 1995 pp. 1942–1948. doi:10.1109/ICNN.1995.488968.
 J. Kennedy and R. Eberhart, Swarm Intelligence, San Francisco: Morgan Kaufmann, 2001.