Diversity and frame invariance characteristics in Particle Swarm Optimization with and without digital pheromones

Academic problems for testing optimization methods are widely criticized for not being a good representative of real world problems. Due to the unavailability of publishable proprietary information from industries they collaborate with, researchers tend to simulate complex problems by using ‘n-dimensional’ multimodal and/or multi-objective academic test problems for evaluating optimization methods developed. Most of these benchmarking test problems can be decomposed and solved as ‘n’ 1-dimensional optimization problems, rendering them as ineffective representation of real-world problems. However, studies show that coordinate rotation of test problems through an arbitrary angle makes design variables dependent on each other and cannot easily be decomposed into simpler problem chunks. Test problems formulated with coordinate rotation therefore will represent a realistic test bed for evaluating the performance of an optimization routine. However with coordinate rotation, the complexity of the problems potentially increases from O(n n ) to O(exp(n ln n)) imposing performance loss on the optimization method that solves the problem. In this paper, the authors attempted to investigate whether coordinate rotation affects the performance characteristics of the digital pheromone implementation of Particle Swarm Optimization (PSO). In particular, two characteristics - swarm diversity with different random number schemes for the velocity vector, and frame invariance with rotational problems are studied and reported. In other words, the authors intended to evaluate whether PSO with digital pheromones is truly capable of solving complex problems.