An objective normalization strategy is essential in any evolutionary multiobjective or many-objective optimization (EMO or EMaO) algorithm, due to the distance calculations between objective vectors required to compute diversity and convergence of population members. For the decomposition-based EMO/EMaO algorithms involving the Penalty Boundary Intersection (PBI) metric, normalization is an important matter due to the computation of two distance metrics. In this article, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBI-based MOEA/D and a proposed PBI-based NSGA-III procedure. Although the effect is well recognized in the literature, few theoretical studies have been done so far to understand its true nature and the choice of a suitable penalty parameter value for an arbitrary problem. The developed theoretical results have been corroborated with extensive experimental results on three to 15-objective convex and non-convex instances of DTLZ and WFG problems. The article, makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study.

中文翻译：

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目标归一化和惩罚参数对基于惩罚边界交点分解的进化多目标优化算法的影响

由于计算种群成员的多样性和收敛性所需的目标向量之间的距离计算，客观归一化策略在任何进化多目标或多目标优化（EMO 或 EMaO）算法中都是必不可少的。对于涉及惩罚边界交叉 (PBI) 度量的基于分解的 EMO/EMaO 算法，由于计算两个距离度量，归一化是一个重要问题。在本文中，我们对归一化过程中的不稳定性对基于 PBI 的 MOEA/D 性能的影响进行了理论分析，并提出了基于 PBI 的 NSGA-III 程序。虽然效果在文献中得到了很好的认可，到目前为止，很少有理论研究来了解其真实性质以及为任意问题选择合适的惩罚参数值。已开发的理论结果已被 DTLZ 和 WFG 问题的 3 到 15 个目标凸和非凸实例的广泛实验结果所证实。这篇文章对从研究中得出的基于 PBI 的分解算法做出了重要的理论结论。