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.

For a many-objective optimization problem with redundant objectives, we propose two novel objective reduction algorithms for linearly and, nonlinearly degenerate Pareto fronts. They are called LHA and NLHA respectively. The main idea of the proposed algorithms is to use a hyperplane with non-negative sparse coefficients to roughly approximate the structure of the PF. This approach is quite different from the previous objective reduction algorithms that are based on correlation or dominance structure. Especially in NLHA, in order to reduce the approximation error, we transform a nonlinearly degenerate Pareto front into a nearly linearly degenerate Pareto front via a power transformation. In addition, an objective reduction framework integrating a magnitude adjustment mechanism and a performance metric σ * are also proposed here. Finally, to demonstrate the performance of the proposed algorithms, comparative experiments are done with two correlation-based algorithms, LPCA and NLMVUPCA, and with two dominance-structure-based algorithms, PCSEA and greedy δ - MOSS, on three benchmark problems: DTLZ5(I,M), MAOP(I,M), and WFG3(I,M). Experimental results show that the proposed algorithms are more effective.