A Novel Grid-based Crowding Distance for Multimodal Multi-objective Optimization

Preserving diversity in decision space plays an important role in Multimodal Multi-objective Optimization problems (MMOPs). Due to the lack of mechanisms to keep different solutions with the same fitness value, most of the available Multi-objective Evolutionary Algorithms (MOEAs) perform poorly when applied to MMOPs. To deal with these problems, this paper proposes a novel method for diversity preserving in the decision space. To this end, the concept of grid-based crowding distance for decision space is introduced. Furthermore, to keep a good diversity of solutions in both decision and objective spaces, we propose different frameworks by combining this method with crowding distance in decision space, crowding distance in objective space, and the weighted sum of both crowding distances. In order to evaluate the performance of these frameworks, we integrate them into the diversity preserving part of the NSGA-II algorithm, and compare them with the NSGA-II (as the baseline algorithm) and the state-of-the-art multimodal multi-objective optimization algorithms on ten different MMOPs with different levels of complexity.