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An evolutionary many-objective algorithm based on decomposition and hierarchical clustering selection

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Abstract

In recent years, many multi-objective evolutionary algorithms have been proposed to solve many-objective optimization problems with regular Pareto front. These algorithms have shown good performance in balancing convergence and diversity. However, in the high-dimensional objective space, the non-dominated solutions increases exponentially as the number of objectives increases. The metrics to evaluate algorithm performance are also computationally intensive. In particular, solving the many-objective optimization problem of the irregular Pareto front faces great challenges. Moreover, many-objective evolutionary algorithms, do not easily show their convergence and diversity through visualization, as multi-objective evolutionary algorithms do. To address these problems, a many-objective optimization algorithm based on decomposition and hierarchical clustering selection is proposed in this paper. First, a set of uniformly distributed reference vectors divides non-dominanted individuals into different sub-populations, and then candidate solutions are selected based on the aggregation function values in the sub-populations. Second, a set of adaptive reference vectors is used to rank the dominant individuals in the population and retain promising candidate solutions. Third, a hierarchical clustering selection strategy is used to enable solutions with good convergence to be selected. Finally, a diversity maintenance strategy is used to remove solutions with poor diversity. The experimental results show that the proposed algorithm EA-DAH has advantages over other comparative algorithms in many-objective optimization problems with irregular Pareto fronts.

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Notes

  1. The codes of the IGD and CPF are implemented in [35], available at https://github.com/BIMK/PlatEMO.

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Acknowledgements

This research is partly supported by the Natural Science Foundation of China (Grant No. 11871279 and 61971234), Humanity and Social Science Youth foundation of Ministry of Education of China (Grant No. 12YJCZH179), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 16KJA110001). Thanks all authors for providing the source codes of the comparison algorithms.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Yuehong Sun, Kelian Xiao, Siqiong Wang & Qiuyue Lv

  2. Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing, 210023, People’s Republic of China

    Yuehong Sun

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  1. Yuehong Sun
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Appendix

Appendix

Fig. 12
figure 12

Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ1 with 3-objective

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Fig. 13
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ1 with 4-objective

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Fig. 14
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ1 with 5-objective

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Fig. 15
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ1 with 6-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ1 with 8-objective

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Fig. 17
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ1 with 10-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ2 with 3-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ2 with 4-objective

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Fig. 20
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ2 with 5-objective

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Fig. 21
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ2 with 6-objective

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Fig. 22
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ2 with 8-objective

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Fig. 23
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ2 with 10-objective

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Fig. 24
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ3 with 3-objective

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Fig. 25
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ3 with 4-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ3 with 5-objective

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Fig. 27
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ3 with 6-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ3 with 8-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ3 with 10-objective

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Fig. 30
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ4 with 3-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ4 with 4-objective

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Fig. 32
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ4 with 5-objective

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Fig. 33
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ4 with 6-objective

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Fig. 34
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ4 with 8-objective

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Fig. 35
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ4 with 10-objective

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Fig. 36
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ5 with 3-objective

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Fig. 37
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ5 with 4-objective

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Fig. 38
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ5 with 5-objective

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Fig. 39
figure 39

Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ5 with 6-objective

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Fig. 40
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ5 with 8-objective

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Fig. 41
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ5 with 10-objective

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Fig. 42
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ6 with 3-objective

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Fig. 43
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ6 with 4-objective

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Fig. 44
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ6 with 5-objective

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Fig. 45
figure 45

Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ6 with 6-objective

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Fig. 46
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ6 with 8-objective

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Fig. 47
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ6 with 10-objective

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Fig. 48
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ7 with 3-objective

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Fig. 49
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ7 with 4-objective

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Fig. 50
figure 50

Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ7 with 5-objective

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Fig. 51
figure 51

Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ7 with 6-objectiv

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Fig. 52
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ7 with 8-objective

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Fig. 53
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on DTLZ7 with 10-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C1DTLZ1 with 3-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C1DTLZ1 with 4-objective

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Fig. 56
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C1DTLZ1 with 5-objective

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Fig. 57
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C1DTLZ1 with 6-objective

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Fig. 58
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C1DTLZ1 with 8-objective

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Fig. 59
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C1DTLZ1 with 10-objective

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Fig. 60
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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C2DTLZ2 with 3-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C2DTLZ2 with 4-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C2DTLZ2 with 5-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C2DTLZ2 with 6-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C2DTLZ2 with 8-objective

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Parallel Coordinate Plots for True PFs and PF Approximations Obtained by EA-DAH, DDEANS, RVEA, NSGA-III, VaEA, EFRRR, and MOEA/DD on C2DTLZ2 with 10-objective

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Sun, Y., Xiao, K., Wang, S. et al. An evolutionary many-objective algorithm based on decomposition and hierarchical clustering selection. Appl Intell 52, 8464–8509 (2022). https://doi.org/10.1007/s10489-021-02669-9

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