A Size-Imbalanced Dual Population with Complementary Search for Sparse Large-Scale Multi-Modal Multi-Objective Optimization

Multi-modal multi-objective optimization problems (MMOPs), which contain multiple distinct Pareto sets, are found in many real-world applications. Their multi-modality characteristics require algorithms to be specifically designed to enhance the diversity in the decision space. However, existing methods often struggle to achieve either desired diversity or satisfactory convergence speed, particularly in large-scale scenarios. To address this issue, we propose an evolutionary framework that employs a size-imbalanced dual population to balance the convergence and diversity in the decision space. Specifically, a large main population, guided by non-zero mask vectors, searches promising regions of the decision space for fast convergence. On the other hand, a small complementary population is guided by a mask vector that is bitwise complementary to those of the main population, enabling it to focus on the unexplored areas and ensure comprehensive coverage of the search space. Periodically, the two populations and their corresponding mask vectors are updated by inter-and intra-population information exchanges, allowing them to collaboratively adjust their search directions and effectively address multi-modality challenges. Additionally, to further enhance the search efficiency, a clustering-enhanced genetic operator is designed to update different populations by adaptively applying either exploration or exploitation search phases. Last, we introduce a new evaluation metric, termed the multi-objective multi-modal index, to evaluate the multi-modality performance of algorithms in real-world MMOPs without reliance on groundtruth data. Experimental results show that the proposed method significantly outperforms five state-of-the-art algorithms on (non-) multi-modal benchmarks and real-world applications with substantially reduced computational time. (The source code is available at https://github.com/XWang1216/MMEA-IDC.)