Multiobjective evolutionary algorithms (MOEAs) with higher population diversity have been extensively presented in literature studies and shown great potential in the approximate Pareto front (PF). Especially, in the recent development of MOEAs, the reference line method is increasingly favored due to its diversity enhancement nature and auxiliary selection mechanism based on the uniformly distributed reference line. However, the existing reference line method ignores the nadir point and consequently causes the Pareto incompatibility problem, which makes the algorithm convergence worse. To address this issue, a multiobjective evolutionary algorithm based on the adaptive cross-reference line method, called MOEA-CRL, is proposed under the framework of the indicator-based MOEAs. Based on the dominant penalty distance (DPD) indicator, the cross-reference line method can not only solve the Pareto incompatibility problem but also enhance the population diversity on the convex PF and improve the performances of MOEA-CRL for irregular PF. In addition, the MOEA-CRL adjusts the distribution of the cross-reference lines directly defined by the DPD indicator according to the contributing solutions. Therefore, the adaptation of cross-reference lines will not be affected by the population size and the uniform distribution of cross-reference lines can be maintained. The MOEA-CRL is examined and compared with other MOEAs on several benchmark problems. The experimental results show that the MOEA-CRL is superior to several advanced MOEAs, especially on the convex PF. The MOEA-CRL exhibits the flexibility in population size setting and the great versatility in various multiobjective optimization problems (MOPs) and many-objective optimization problems (MaOPs).

中文翻译：

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基于交叉参考线方法的多目标进化算法可增强种群多样性。

具有较高种群多样性的多目标进化算法（MOEA）已在文献研究中广泛提出，并在近似帕累托前沿（PF）中显示出巨大潜力。特别地，在MOEA的最新发展中，参考线方法由于其多样性增强的特性和基于均匀分布的参考线的辅助选择机制而越来越受到青睐。然而，现有的参考线方法忽略了最低点，从而导致帕累托不兼容问题，这使得算法的收敛性更差。为了解决这个问题，在基于指标的MOEAs框架下，提出了一种基于自适应交叉参考线方法的多目标进化算法MOEA-CRL。根据显性惩罚距离（DPD）指标，交叉参考线法不仅可以解决帕累托不相容的问题，而且可以提高凸PF上的种群多样性，提高MOEA-CRL在不规则PF上的性能。此外，MOEA-CRL根据提供的解决方案调整由DPD指标直接定义的交叉参考线的分布。因此，交叉参考线的适应将不受总体大小的影响，并且可以保持交叉参考线的均匀分布。对MOEA-CRL进行了检查，并将其与其他MOEA进行了一些基准问题比较。实验结果表明，MOEA-CRL优于几种先进的MOEA，特别是在凸PF上。