Abstract
The frequently used basic version of MOEA/D (multi-objective evolutionary algorithm based on decomposition) has no normalization mechanism of the objective space, whereas the normalization was discussed in the original MOEA/D paper. As a result, MOEA/D shows difficulties in finding a set of uniformly distributed solutions over the entire Pareto front when each objective has a totally different range of objective values. Recent variants of MOEA/D have normalization mechanisms for handling such a scaling issue. In this paper, we examine the effect of the normalization of the objective space on the performance of MOEA/D through computational experiments. A simple normalization mechanism is used to examine the performance of MOEA/D with and without normalization. These two types of MOEA/D are also compared with recently proposed many-objective algorithms: NSGA-III, MOEA/DD, and theta -DEA. In addition to the frequently used many-objective test problems DTLZ and WFG, we use their minus versions. We also propose two variants of the DTLZ test problems for examining the effect of the normalization in MOEA/D. Test problems in one variant have objective functions with totally different ranges. The other variant has a kind of deceptive nature, where the range of each objective is the same on the Pareto front but totally different over the entire feasible region. Computational experiments on those test problems clearly show the necessity of the normalization. It is also shown that the normalization has both positive and negative effects on the performance of MOEA/D. These observations suggest that the influence of the normalization is strongly problem dependent.
Highlights
Many-objective optimization has received a lot of attention in the evolutionary multi-objective optimization (EMO) community, where optimization of four or more objectives is called many-objective optimization [14,15,23]
The normalization of the objective space is needed in WFG4-9, whereas it is not needed in DTLZ1-4
This is because MOEA/DD has no normalization mechanism, whereas the search in NSGA-III and θ -DEA is performed in the normalized objective space
Summary
Many-objective optimization has received a lot of attention in the evolutionary multi-objective optimization (EMO) community, where optimization of four or more objectives is called many-objective optimization [14,15,23]. Many-objective problems present a number of challenges [10,19] to the EMO community such as the deterioration in search ability of Pareto dominance-based algorithms [6,29] and the increase in computation time of hypervolume-based algorithms [2,3]. For many-objective problems, it has been demonstrated in the literature [10,12] that MOEA/D [27] works well in comparison with Pareto dominance-based and hypervolume-based algorithms in terms of their search ability and computation time. A number of EMO algorithms have been proposed for many-objective problems based on the same or similar framework as MOEA/D (e.g., NSGA-III [5], MOEA/DD [16], I-DBEA [1], and θ -DEA [26]).
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