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Many-objective optimization problems (MaOPs), which are problems requiring optimization of more than three conflicting objectives, have recently attracted widespread interest in the evolutionary multi-objective optimization (EMO) community. NSGA-II (Deb et al., 2002), one of the most well known EMO methods based on the principle of Pareto dominance selection, has been used to solve various multi-objective optimization problems (MOPs). It has achieved great success in solving various MOPs, including (Lu et al., 2020; S. Zhu & Xu, 2018) a wide range of application cases. The concept of Pareto dominance, an intuitive and qualitative notion of compromise, has been commonly adopted to distinguish the quality of solutions for traditional two- or three-dimensional MOPs. However, the effectiveness of Pareto-based multi-objective evolutionary algorithms in solving multi-objective evolutionary problems has dramatically decreased. The main challenge of these methods is the loss of Pareto-based selection pressure toward the true Pareto front (PF) while the number of objectives M grows (Santos & Takahashi, 2016), i.e., the solutions become incomparable due to dominance resistance, and the difficulty of balancing convergence with diversity (Deb & Jain, 2013). To address this problem, various methods to enhance Pareto dominance have been proposed to solve MaOPs, which can be broadly classified into two categories.
The first category is concerned with developing new dominance relationships, and the basic idea is to increase the probability that two candidate solutions on MaOP are comparable. In the existing literature there are many approaches to developing new dominance relations, such as expanding the dominance region (J. Liu et al., 2022; Sato et al., 2007; C. Zhu et al., 2015; S. Zhu et al., 2022), latticing the object space (Laumanns et al., 2002; Yang et al., 2013), using fuzzy logic (Wang & Jiang, 2007; He et al., 2013; Qasim et al., 2022), defining dominance relations with weight vectors (Yuan et al., 2016; Elarbi et al., 2017), etc.
The second category is characterized by a combination of Pareto dominance and additional selection criteria. This method first uses Pareto dominance to eliminate a few poorer candidate solutions, and then uses a quadratic selection criterion to distinguish the non-comparable candidates. Existing methods of this type use three main basic ideas. The first idea is to create new criteria to bias the criteria with better convergence and diversity among the non-comparable candidate solutions, such as KnEA (Zhang et al., 2014), VaEA (Xiang et al., 2017), and AGEMOEA-II (Panichella, 2022). The second idea uses performance metrics to distinguish the quality of non-comparable candidate solutions by selecting the candidate solutions with higher contribution, such as HypE (Bader & Zitzler, 2011) and ARMOEA (Tian et al., 2017a). The third idea is to combine Pareto dominance with decomposition-based algorithms such as MOEA/DD (Li et al., 2014) and FDEA (S. Liu et al., 2016).