Basins of Attraction and Metaoptimization for Particle Swarm Optimization Methods

Particle swarm optimization (PSO) is a metaheuristic optimization method that finds near- optima by spawning particles which explore within a given search space while exploiting the best candidate solutions of the swarm. PSO algorithms emulate the behavior of, say, a flock of birds or a school of fish, and encapsulate the randomness that is present in natural processes. In this paper, we discuss different initialization schemes and meta-optimizations for PSO, its performances on various multi-minima functions, and the unique intricacies and obstacles that the method faces when attempting to produce images for basins of attraction, which are the sets of initial points that are mapped to the same minima by the method. This project compares the relative strengths and weaknesses of the Particle Swarm with other optimization methods, namely gradient-descent, in the context of basin mapping and other metrics. It was found that with proper parameterization, PSO can amply explore the search space regardless of initialization. For all functions, the swarm was capable of finding, within some tolerance, the global minimum or minima in fewer than 60 iterations by having sufficiently well chosen parameters and parameterization schemes. The shortcomings of the Particle Swarm method, however, are that its parameters often require fine-tuning for different search spaces to most efficiently optimize and that the swarm cannot produce the analytical minimum. Overall, the PSO is a highly adaptive and computationally efficient method with few initial restraints that can be readily used as the first step of any optimization task.