Multi-dimensional particle swarm optimization in dynamic environments

Particle swarm optimization (PSO) has been initially 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. Thanks to its stochastic and population based nature, PSO may avoid becoming 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 or worse it may even not converge to any local optimum at all. In dynamic environments the optimization task becomes even more difficult, since after an environment change the earlier global optimum might become just a local optimum, and if the swarm is converged to that optimum, it is likely that new real optimum will not be found. For the same reason, local optima cannot be just discarded, because they can be later transformed into global optima. 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 premature convergence problem and improve the rate of PSO’s convergence to 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. In this way the potential diversity that is present among the dimensions of swarm particles can be efficiently used within the aGB 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. Finally for the multi-dimensional dynamic environments where the optimum dimension too changes in time, we utilize a recent PSO technique, the Multi-Dimensional (MD) PSO, which 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 pushes the frontier of the optimization problems in dynamic environments significantly, towards a global search in a multidimensional space, where there exists a multi-modal problem possibly in each dimension. Henceforth MD PSO removes the requirement of working only in a fixed dimension, which is a common drawback for the family of swarm optimizers. 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 (uni-dimensional) 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 ... Multi-dimensional Search via Fractional Multi-swarms in Dynamic Environments Serkan Kiranyaz, Jenni Pulkkinen and Moncef Gabbouj Multi-dimensional search via Fractional Particle Swarm Optimization in Dynamic Environments 2 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 (highest peak) in this dimension.