A Cascade Clustering-Based Two-Stage Evolutionary Algorithm for Large-Scale Multimodal Multi-Objective Optimisation

Multimodal multi-objective evolutionary algorithms (MMEAs) tailored for multimodal multi-objective optimisation problems (MMOPs) have achieved considerable advancements in recent years. However, the adaptation of MMEAs to large-scale MMOPs with sparse Pareto optimal solutions and the attainment of all equivalent Pareto optimal solution sets (PSs) remain challenging mainly due to the obstacles of the dimensionality curse, the unknown multimodality and the unknown sparsity. Therefore, this study proposes a cascade clustering-based two-stage evolutionary algorithm to address these issues. In particular, this study employs a cascade clustering technique to distinguish distinct PSs accurately from current subpopulations. Simultaneously, a two-stage optimisation approach determined by cascade clustering is adopted to meticulously uphold multimodality detection and ensure accurate unimodal evolution. Moreover, a balanced evolution strategy is incorporated into the two-stage evolutionary process to promote the independent accurate evolution of each modality and achieve the evolutionary balance among different modalities. Furthermore, the scores of the decision variables updated by local guidance vectors are used to guide the generation of subpopulations for facilitating convergence in their respective search directions. Compared with six state-of-the-art multimodal algorithms, the proposed method achieves the first rank across all SMMOP test suites in terms of IGDX, while significantly outperforming competitors on the majority of SMMOP test suites in both IGD and HV indicators.