Local Regularity Model for Multimodal Multiobjective Optimization

Multimodal multiobjective optimization aims to provide diversified acceptable decisions (ADs), including global optimal solution with consistent objective evaluations and local optimal solutions (LOSs) with acceptable objective evaluations. However, the discrimination of LOSs highly depends on the distribution of candidate solutions, which may result in the catastrophic elimination of LOSs to damage the diversity in the decision space. To address this problem, a local regularity model (LRM) method is proposed to improve the distribution of candidate solutions. There are three novelties of LRM. First, a hierarchical principal component analysis is developed to extract principal components for different nondominated sets. Then, the distribution features of different nondominated sets are described in segments by a small number of candidate solutions to construct LRM. Second, a self-organization strategy, based on the feature correlation and neighborhood violation analysis, is proposed to improve local fitting ability. Then, LRM are efficiently constructed to estimate the manifold of ADs. Third, a probability reproduction strategy is developed to reconstruct the population by LRM. Then, the population is reconstructed to enhance the distribution of candidate solutions in the decision space. Finally, the proposed optimization method is integrated into the popular multimodal multiobjective optimization algorithm to demonstrate its effectiveness in terms of the benchmark multimodal multiobjective optimization problem test suite.