Investigating Normalization in Preference-based Evolutionary Multi-objective Optimization Using a Reference Point

Abstract: 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.