Multi-dimensional particle swarm optimization for dynamic environments

The particle swarm optimization (PSO) was introduced as a population based stochastic search and optimization process for static environments; however, many real problems are dynamic, meaning that the environment and the characteristics of the global optimum can change over time. Thanks to its stochastic and population based nature, PSO can avoid being trapped in local optima and find the global optimum. However, this is never guaranteed and as the complexity of the problem rises, it becomes more probable that the PSO algorithm gets trapped into a local optimum due to premature convergence. In this paper, we propose novel techniques, which successfully address several major problems in the field of particle swarm optimization (PSO) and promise efficient and robust solutions for multi-dimensional and dynamic problems. The first one, so-called multi-dimensional (MD) PSO, re-forms the native structure of swarm particles in such a way that they can make inter-dimensional passes with a dedicated dimensional PSO process. Therefore, in a multi-dimensional search space where the optimum dimension is unknown, swarm particles can seek for both positional and dimensional optima. This eventually removes the necessity of setting a fixed dimension a priori, which is a common drawback for the family of swarm optimizers. To address the premature convergence problem, we then propose fractional global best formation (FGBF) technique, which basically collects all the best dimensional components and fractionally creates an artificial global-best particle (aGB) that has the potential to be a better ldquoguiderdquo than the PSOs native gbest particle. To establish follow-up of (current) local optima, we then introduce a novel multi-swarm algorithm, which enables each swarm to converge to a different optimum and use FGBF technique distinctively. We then propose a multi-dimensional extension of the moving peaks benchmark (MPB), which is a publicly available for testing optimization algorithms in a multi-modal dynamic environment. In this extended benchmark an extensive set of experiments show that MD PSO using FGBF technique with multi-swarms exhibits an impressive performance and tracks the global maximum peak with the minimum error.