Hierarchical Heterogeneous Particle Swarm Optimization

Particle swarm optimization (PSO) has recently been modified to several versions. Heterogeneous PSO is a recent extension which includes behavioral heterogeneity of particles. Here we propose a further developed version that has hierarchical interaction patterns among heterogeneous particles, which we call hierarchical heterogeneous PSO (HHPSO). Two algorithm designs that have been developed and tested are multi-layer HHPSO (ml-HHPSO) and multi-group HHPSO (mg-HHPSO). The performances of these algorithms were measured on a set of benchmark functions and compared with standard PSO and heterogeneous PSO. The results showed that the performances of both HHPSO algorithms were significantly improved from standard PSO and heterogeneous PSO, with higher quality of optimal solutions and faster convergence speed. Particle swarm optimization (PSO), a population-based computational meta-heuristic, has been widely applied to many areas of science and engineering [6]. Many variants of PSO have demonstrated that the performance of PSO is largely influenced by particle behavior and interactions [1,5,7]. Among the recent extensions of PSO algorithms, introducing hierarchical social interaction [3] and particle heterogeneity [2] is our particular interest, because many biological collective systems seem to possess these properties to facilitate information exchange and balance different modes of collective behaviors (such as exploration and exploitation). To the extent of our knowledge, however, none of the earlier works combined hierarchical structures and heterogeneous behaviors in PSO. Here we propose a hierarchical heterogeneous PSO (HHPSO) algorithm that integrates both heterogeneous particle behaviors and hierarchical interaction patterns into PSO. The basic idea of HHPSO is to introduce a two-phase procedure that first arranges particles into hierarchical structures based on their current fitness values and then assigns different behaviors to the particles based on their ranks in the hierarchy. To investigate the effects of varying hierarchical social structures and different levels of communication on PSO algorithms, we have designed two versions of HHPSO: multi-layer HHPSO (ml-HHPSO) and multi-group HHPSO (mg-HHPSO). Multi-layer HHPSO is a basic version of HHPSO that focuses on communication between layers of particles (Fig. 1). In each iteration, particles are arranged into equally sized layers based on their current fitness values. Particles are allowed to interact with their immediate superior or inferior particles. Compared to standard PSO where particles are attracted to global and personal best solutions, particles in ml-HHPSO are also attracted to immediately superior particles (except for particles in the top layer that are attracted to the ones with higher fitness within the same layer). Figure 1: Schematic diagram of multi-layer HHPSO. Particles are structured into multiple layers based on their current fitness values; particles in each layer are attracted to those in the layer above (left). An illustrative example of particles in a search space is shown on the right, with attractive forces shown by arrows of different colors. Figure 2: Schematic diagram of multi-group HHPSO. Particles are randomly split into multiple groups, inside each of which they are further structured into three layers based on their fitness values (red: head, blue: tail, yellow: body). Groups are spatially clustered in this figure just for visual clarity, but they can overlap in real cases. Body particles are attracted to others with higher fitness within its group (black arrows). Heads are attracted to other heads with higher fitness (red arrows). Tails are attracted to all heads (blue arrows). ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems Figure 3: Average best fitness values of standard PSO, heterogeneous PSO, ml-HHPSO and mg-HHPSO for six benchmark functions [1]: (a) Ackley; (b) Quadric; (c) Rastrigin; (d) Rosenbrock; (e) Griewank; (f) Salomon. Error bars indicate the standard errors. Lower fitness means better results. Each swarm is made of 50 particles. Results are averaged over 50 independent simulation runs. (c) and (e) are plotted in a “symlog” scale of matplotlib [4] to show convergence to zero. The second version of our algorithm, multi-group HHPSO, is a more elaborated one that aims to implement both vertical (i.e., between-layer) and horizontal (i.e., between-subpopulation) interaction of particles. In mg-HHPSO, a swarm is randomly separated into equally sized groups. Within each group, the particles are further separated into a three-layer hierarchical structure based on their current fitness values. Specifically, the particle with the highest fitness is designated as the head, the particle with the lowest as the tail, and the remaining particles as the body particles, each of which follows a different behavioral rule. Heads are responsible for leading other group members to where better solutions might exist. They are also attracted to other heads with higher fitness values. Body particles are attracted to their head and other body particles with higher fitness values, which contributes to an improvement in collective fitness. Tails are attracted to all heads regardless of which groups they belong to. This behavior of the tails helps increase the basic fitness level of the whole population from the bottom up. All particles are also attracted to global and personal bests, like in standard PSO. Finally, in both mland mg-HHPSO, we implemented a mechanism for stagnancy detection, similar to the one used in [3]. Once stagnancy is detected, the stagnant particle randomly chooses a new behavioral rule rather than abides by the one determined by their ranks in the hierarchy. The performance of these two HHPSO algorithms were evaluated through comparisons with standard PSO and heterogeneous PSO on six benchmark functions in a 50dimensional search space [1] (Fig. 3). Five layers and ten groups were used in mland mg-HHPSO, respectively. Each algorithm was run 50 times for each benchmark function. Figure 3 shows that both of our HHPSO algorithms outperformed standard PSO and heterogeneous PSO. The quality of best solutions and convergence speed improved significantly for most of the benchmark functions. In conclusion, we have proposed two versions of HHPSO algorithms which demonstrate significant improvement of search processes in both solution accuracy and convergence speed. We believe that this was because the combination of hierarchical structures and heterogeneous behaviors helped the swarm maintains diverse behaviors and effective communications among particles to balance its explorative and exploitative abilities simultaneously. Future research will concentrate on validating our algorithms using additional metrics and exploring other forms of implementing hierarchy and heterogeneity in more depth. This material is based upon work supported by the US National Science Foundation under Grant No. 1319152.