Fractional Particle Swarm Optimization in Multidimensional Search Space

In this paper, we propose two novel techniques, which successfully address several major problems in the field of particle swarm optimization (PSO) and promise a significant breakthrough over complex multimodal optimization problems at high dimensions. The first one, which is the so-called multidimensional (MD) PSO, re-forms the native structure of swarm particles in such a way that they can make interdimensional passes with a dedicated dimensional PSO process. Therefore, in an MD search space, where the optimum dimension is unknown, swarm particles can seek 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. Nevertheless, MD PSO is still susceptible to premature convergences due to lack of divergence. Among many PSO variants in the literature, none yields a robust solution, particularly over multimodal complex problems at high dimensions. To address this problem, we propose the fractional global best formation (FGBF) technique, which basically collects all the best dimensional components and fractionally creates an artificial global best (aGB) particle that has the potential to be a better "guide" than the PSO's native gbest particle. This way, the potential diversity that is present among the dimensions of swarm particles can be efficiently used within the aGB particle. We investigated both individual and mutual applications of the proposed techniques over the following two well-known domains: 1) nonlinear function minimization and 2) data clustering. An extensive set of experiments shows that in both application domains, MD PSO with FGBF exhibits an impressive speed gain and converges to the global optima at the true dimension regardless of the search space dimension, swarm size, and the complexity of the problem.