Multimodal multiobjective problems (MMOPs) arise frequently in the real world, in which multiple Pareto-optimal solution (PS) sets correspond to the same point on the Pareto front. Traditional multiobjective evolutionary algorithms (MOEAs) show poor performance in solving MMOPs due to a lack of diversity maintenance in the decision space. Thus, recently, many multimodal MOEAs (MMEAs) have been proposed. However, for most existing MMEAs, the convergence performance in the objective space does not meet expectations. In addition, many of them cannot always obtain all equivalent Pareto solution sets. To address these issues, this study proposes an MMEA based on a weighted indicator, termed MMEA-WI. The algorithm integrates the diversity information of solutions in the decision space into an objective space performance indicator to maintain the diversity in the decision space and introduces a convergence archive to ensure a more effective approximation of the Pareto-optimal front (PF). These strategies can readily be applied to other indicator-based MOEAs. The experimental results show that MMEA-WI outperforms some state-of-the-art MMEAs on the chosen benchmark problems in terms of the inverted generational distance (IGD) and IGD in the decision space (IGDX) metrics.

中文翻译：

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基于加权指标的多模态多目标优化进化算法

多模态多目标问题 (MMOP) 在现实世界中经常出现，其中多个帕累托最优解 (PS) 集对应于帕累托前沿上的同一点。由于决策空间缺乏多样性维护，传统的多目标进化算法 (MOEA) 在解决 MMOP 方面表现不佳。因此，最近提出了许多多模态 MOEA（MMEA）。然而，对于大多数现有的 MMEA，在目标空间的收敛性能并不符合预期。此外，他们中的许多人不能总是获得所有等价的帕累托解集。为了解决这些问题，本研究提出了基于加权指标的 MMEA，称为 MMEA-WI。该算法将决策空间中解的多样性信息整合为客观空间性能指标，以保持决策空间的多样性，并引入收敛档案，以确保更有效地逼近帕累托最优前沿（PF）。这些策略可以很容易地应用于其他基于指标的 MOEA。实验结果表明，在决策空间 (IGDX) 指标中的反向代际距离 (IGD) 和 IGD 方面，MMEA-WI 在所选基准问题上优于一些最先进的 MMEA。