Normalization of objectives plays a crucial role in evolutionary multi-objective optimization (EMO) to handle objective functions with different scales, which can be found in real-world problems. Although the effect of normalization methods on the performance of EMO algorithms has been investigated in the literature, that of preference-based EMO (PBEMO) algorithms is poorly understood. Since PBEMO aims to approximate a region of interest, its population generally does not cover the Pareto front in the objective space. This property may make normalization of objectives in PBEMO difficult. This paper investigates the effectiveness of three normalization methods in three representative PBEMO algorithms. We present a bounded archive-based method for approximating the nadir point. First, we demonstrate that the normalization methods in PBEMO perform significantly worse than that in conventional EMO in terms of approximating the ideal point, nadir point, and range of the PF. Then, we show that PBEMO requires normalization of objectives on problems with differently scaled objectives. Our results show that there is no clear "best normalization method" in PBEMO, but an external archive-based method performs relatively well.

翻译：目标函数归一化在进化多目标优化中具有关键作用，用于处理现实问题中存在的不同量级目标函数。尽管文献已研究了归一化方法对EMO算法性能的影响，但偏好型进化多目标优化算法的相关影响仍鲜有探讨。由于PBEMO旨在逼近感兴趣区域，其种群通常无法覆盖目标空间中的帕累托前沿，这一特性可能导致PBEMO中的目标归一化存在困难。本文研究了三种归一化方法在三个代表性PBEMO算法中的有效性，并提出一种基于有界存档的纳迪尔点逼近方法。首先证实，在逼近理想点、纳迪尔点及帕累托前沿范围方面，PBEMO中归一化方法的表现显著弱于传统EMO。进而证明，当处理不同量级目标函数的问题时，PBEMO需要实施目标归一化。结果表明PBEMO中不存在明确的“最优归一化方法”，但基于外部存档的方法表现相对较优。