Locating Potentially Disjoint Feasible Regions of a Search Space with a Particle Swarm Optimizer

In constraint optimization problems set in continuous spaces, a feasible search space may consist of many disjoint regions and the global optimal solution might be within any of them. Thus, locating these feasible regions (as many as possible, ideally all of them) is of great importance. In this chapter, we introduce niching techniques that have been studied in connection with multimodal optimization for locating feasible regions, rather than for finding different local optima. One of the successful niching techniques was based on the particle swarm optimizer (PSO) with a specific topology, called nonoverlapping topology, where the swarm was divided into several nonoverlapping sub-swarms. Earlier studies have shown that PSO with such nonoverlapping topology, with a small number of particles in each sub-swarm, is quite effective in locating different local optima if the number of dimensions is small (up to 8). However, its performance drops rapidly when the number of dimensions grows. First, a new PSO, called mutation linear PSO, MLPSO, is proposed. This algorithm is effective in locating different local optima when the number of dimensions grows. MLPSO is applied to optimization problems with up to 50 dimensions, and its results in locating different local optima are compared with earlier algorithms. Second, we incorporate a constraint handling technique into MLPSO; this variant is called EMLPSO. We test different topologies of EMLPSO and evaluate them in terms of locating feasible regions when they are applied to constraint optimization problems with up to 30 dimensions. The results of this test show that the new method with nonoverlapping topology with small swarm size in each sub-swarm performs better in terms of locating different feasible regions in comparison to other topologies, such as the global best topology and the ring topology.