Effect of Objective Normalization and Penalty Parameter on Penalty Boundary Intersection Decomposition-Based Evolutionary Many-objective Optimization Algorithms

An objective normalization strategy is essential in any evolutionary multi- or manyobjective 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 paper, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBIbased 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 paper makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study.