Constraint-handling techniques for particle swarm optimization algorithms

Population-based methods such as Evolutionary Algorithms (EAs) and Particle Swarm Optimization (PSO) have proven their ability to cope with a variety of remarkably different problems, regardless of whether they are or are not linear, convex, differentiable or smooth.
In addition, they are able to handle problems of notably higher complexity than traditional methods. The main procedure consists of successively updating a population of candidate solutions, performing a parallel exploration instead of traditional sequential exploration (usually unable to overcome local pathologies). While the origins of the PSO method are linked to bird flock simulations, it is a stochastic optimization method in the sense that it relies on random coefficients to introduce creativity, and a bottom-up artificial intelligence-based approach in the sense that its intelligent behaviour emerges at a level higher than the individuals’ rather than being deterministically programmed. As opposed to EAs, the PSO involves no operator design and few coefficients to be tuned. Since this paper does not intend to study such tuning, general-purpose settings are taken from previous studies. The plain PSO algorithm is only able to deal with unconstrained problems, so that some technique needs to be incorporated to handle constraints. A popular one is the penalization method, which turns the original constrained problem into an unconstrained one by penalizing the function value associated with infeasible solutions. Other techniques can be specifically designed for PSO, such as the preserving feasibility and the bisection methods. Given that these strategies present advantages and disadvantages when compared to one another, there is no obvious best constraint-handling technique (CHT) for all problems, and modifications and new techniques are constantly proposed in the literature. The aim here is to develop and compare different CHTs suitable for PSOs, which are incorporated into an algorithm with general-purpose settings. The comparisons between the different CHTs are performed keeping the remaining features of the algorithm the same, while comparisons to other authors’ results are offered as a frame of reference for the optimizer as a whole. Three basic techniques: the penalization, preserving feasibility and bisection methods –as well as modifications that aim to overcome their respective weaknesses– are discussed and tested on two suites of benchmark problems. Three neighbourhood sizes are also considered in the experiments.