In recent years, numerous efficient and effective multimodal multiobjective evolutionary algorithms (MMOEAs) have been developed to search for multiple equivalent sets of Pareto optimal solutions simultaneously. However, some of the MMOEAs prefer convergent individuals over diversified individuals to construct the mating pool, and the individuals with slightly better decision space distribution may be replaced by significantly better objective space distribution. Therefore, the diversity in the decision space may become deteriorated, in spite of the decision and objective diversities have been taken into account simultaneously in most MMOEAs. Because the Pareto optimal subsets may have various shapes and locations in the decision space, it is very difficult to drive the individuals converged to every Pareto subregion with a uniform density. Some of the Pareto subregions may be overly crowded, while others are rather sparsely distributed. Consequently, many existing MMOEAs obtain Pareto subregions with imbalanced density. In this article, we present a two-stage double niched evolution strategy, namely DN-MMOES, to search for the equivalent global Pareto optimal solutions which can address the above challenges effectively and efficiently. The proposed DN-MMOES solves the multimodal multiobjective optimization problem (MMOP) in two stages. The first stage adopts the niching strategy in the decision space, while the second stage adapts double niching strategy in both spaces. Moreover, an effective decision density self-adaptive strategy is designed for improving the imbalanced decision space density. The proposed algorithm is compared against eight state-of-the-art MMOEAs. The inverted generational distance union (IGDunion) performance indicator is proposed to fairly compare two competing MMOEAs as a whole. The experimental results show that DN-MMOES provides a better performance to search for the complete Pareto Subsets and Pareto Front on IDMP and CEC 2019 MMOPs test suite.

中文翻译：

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多模态多目标优化的两阶段双壁龛进化策略

近年来，已经开发了许多高效和有效的多模态多目标进化算法（MMOEA）来同时搜索多个等效的帕累托最优解集。然而，一些 MMOEA 更喜欢收敛个体而不是多样化个体来构建交配池，决策空间分布稍好的个体可能会被明显更好的目标空间分布所取代。因此，尽管在大多数 MMOEA 中同时考虑了决策和客观多样性，但决策空间的多样性可能会恶化。由于帕累托最优子集在决策空间中可能具有各种形状和位置，因此很难将个体以均匀的密度收敛到每个帕累托子区域。一些帕累托子区域可能过于拥挤，而另一些则分布相当稀疏。因此，许多现有的 MMOEA 获得密度不平衡的帕累托子区域。在本文中，我们提出了一种两阶段双生态位进化策略，即 DN-MMOES，以寻找可以有效解决上述挑战的等效全局帕累托最优解。所提出的 DN-MMOES 分两个阶段解决多模态多目标优化问题 (MMOP)。第一阶段在决策空间采用小生境策略，第二阶段在两个空间均采用双生境策略。此外，设计了一种有效的决策密度自适应策略来改善不平衡的决策空间密度。将所提出的算法与八个最先进的 MMOEA 进行比较。提出了反向代际距离联合 (IGDunion) 性能指标，以公平地将两个相互竞争的 MMOEA 作为一个整体进行比较。实验结果表明，DN-MMOES 在 IDMP 和 CEC 2019 MMOPs 测试套件上搜索完整的 Pareto Subsets 和 Pareto Front 提供了更好的性能。