Multi-dimensional particle swarm optimization in dynamic environments

Particle swarm optimization (PSO) was proposed as an optimization technique for static environments; however, many real problems are dynamic, meaning that the environment and the characteristics of the global optimum can change in time. In this paper, we adapt recent techniques, which successfully address several major problems of PSO and exhibit a significant performance over multi-modal and non-stationary environments. In order to address the pre-mature convergence problem and improve the rate of PSO's convergence to the global optimum, Fractional Global Best Formation (FGBF) technique is used. FGBF basically collects all the best dimensional components and fractionally creates an artificial Global Best particle (aGB) that has the potential to be a better guide than the PSO's native gbest particle. To establish follow-up of 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. Finally for the multidimensional dynamic environments where the optimum dimension also changes in time, we utilize a recent PSO technique, the multi-dimensional (MD) PSO, which re-forms the native structure of the 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 pushes the frontier of the optimization problems in dynamic environments towards a global search in a multi-dimensional space, where there exists a multi-modal problem possibly in each dimension. We investigated both standalone and mutual applications of the proposed methods over the moving peaks benchmark (MPB), which originally simulates a dynamic environment in a unique (fixed) dimension. MPB is appropriately extended to accomplish the simulation of a multi-dimensional dynamic system, which contains dynamic environments active in several dimensions. An extensive set of experiments show that in traditional MPB application domain, FGBF technique applied with multi-swarms exhibits an impressive speed gain and tracks the global peak with the minimum error so far achieved with respect to the other competitive PSO-based methods, When applied over the extended MPB, MD PSO with FGBF can find optimum dimension and provide the (near-) optimal solution in this dimension. (C) 2010 Elsevier Ltd. All rights reserved.