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基于在线感知Pareto前沿划分目标空间的多目标进化优化

封文清 巩敦卫

封文清, 巩敦卫. 基于在线感知 Pareto 前沿划分目标空间的多目标进化优化. 自动化学报, 2020, 46(8): 1628−1643 doi: 10.16383/j.aas.2018.c180323
引用本文: 封文清, 巩敦卫. 基于在线感知 Pareto 前沿划分目标空间的多目标进化优化. 自动化学报, 2020, 46(8): 1628−1643 doi: 10.16383/j.aas.2018.c180323
Feng Wen-Qing, Gong Dun-Wei. Multi-objective evolutionary optimization with objective space partition based on online perception of Pareto front. Acta Automatica Sinica, 2020, 46(8): 1628−1643 doi: 10.16383/j.aas.2018.c180323
Citation: Feng Wen-Qing, Gong Dun-Wei. Multi-objective evolutionary optimization with objective space partition based on online perception of Pareto front. Acta Automatica Sinica, 2020, 46(8): 1628−1643 doi: 10.16383/j.aas.2018.c180323

基于在线感知Pareto前沿划分目标空间的多目标进化优化

doi: 10.16383/j.aas.2018.c180323
基金项目: 

国家重点研发计划项目 2018YFB1003802-01

国家自然科学基金 61773384

国家自然科学基金 61763026

国家自然科学基金 61673404

国家自然科学基金 61573361

国家自然科学基金 61503220

国家973科技计划项目 2014CB046306-2

详细信息
    作者简介:

    封文清  中国矿业大学信息与控制工程学院博士研究生. 2013年获得黑龙江大学硕士学位.主要研究方向为多目标进化优化与应用. E-mail: fwq_cumt@163.com

    通讯作者:

    巩敦卫  中国矿业大学信息与控制工程学院教授. 1999年获得中国矿业大学博士学位.主要研究方向为进化计算与应用.本文通信作者. E-mail: dwgong@vip.163.com

Multi-objective Evolutionary Optimization With Objective Space Partition Based on Online Perception of Pareto Front

Funds: 

The National Key Research and Development Program of China 2018YFB1003802-01

National Natural Science Foundation of China 61773384

National Natural Science Foundation of China 61763026

National Natural Science Foundation of China 61673404

National Natural Science Foundation of China 61573361

National Natural Science Foundation of China 61503220

National Basic Research Program of China (973 Program) 2014CB046306-2

More Information
    Author Bio:

    FENG Wen-Qing  Ph. D. candidate at the School of Information and Control Engineering, China University of Mining and Technology. She received her master degree from Heilongjiang University in 2013. Her research interest covers multi-objective evolutionary optimization and its applications

    Corresponding author: GONG Dun-Wei  Professor at the School of Information and Control Engineering, China University of Mining and Technology. He received his Ph. D. degree from China University of Mining and Technology in 1999. His research interest covers evolutionary computation and its applications. Corresponding author of this paper
  • 摘要: 多目标进化优化是求解多目标优化问题的可行方法.但是, 由于没有准确感知并充分利用问题的Pareto前沿, 已有方法难以高效求解复杂的多目标优化问题.本文提出一种基于在线感知Pareto前沿划分目标空间的多目标进化优化方法, 以利用感知的结果, 采用有针对性的进化优化方法求解多目标优化问题.首先, 根据个体之间的拥挤距离与给定阈值的关系感知优化问题的Pareto前沿上的间断点, 并基于此将目标空间划分为若干子空间; 然后, 在每一子空间中采用MOEA/D (Multi-objective evolutionary algorithm based on decomposition)得到一个外部保存集; 最后, 基于所有外部保存集生成问题的Pareto解集.将提出的方法应用于15个基准数值函数优化问题, 并与NSGA-Ⅱ、RPEA、MOEA/D、MOEA/DPBI、MOEA/D-STM和MOEA/D-ACD等比较.结果表明, 提出的方法能够产生收敛和分布性更优的Pareto解集, 是一种非常有竞争力的方法.
    Recommended by Associate Editor WEI Qing-Lai
    1)  本文责任编委  魏庆来
  • 图  1  总体框架

    Fig.  1  General framework

    图  2  搜索间断点

    Fig.  2  Searching for discontinuous points

    图  3  网格的形成

    Fig.  3  Forming grids

    图  4  目标空间划分

    Fig.  4  Dividing the objective space

    图  5  子空间信息交互

    Fig.  5  Information interaction among subspaces

    图  6  个体分布的调整

    Fig.  6  Adjusting the distribution of individuals

    图  7  个体数量调整

    Fig.  7  Adjusting the number of individuals

    图  8  不同时$\alpha$的IGD、GD和运算时间

    Fig.  8  Curves of IGD, GD and time for different values of $\alpha$

    图  9  IGD指标随$\alpha$的变化曲线

    Fig.  9  Curves of IGD with respect to the number of generations for different values of $\alpha$

    图  10  $\alpha$取不同值时, ZDT3的Pareto前沿

    Fig.  10  Pareto front of different values of $\alpha$ for ZDT3

    表  1  不同方法求解测试函数的IGD值

    Table  1  The values of IGD obtained by different algorithms

    优化问题 NSGA-Ⅱ RPEA MOEA/D MOEA/D-PBI MOEA/D-STM MOEA/D-ACD MOEA-PPF
    ZDT1 $6.546 \times 10^{-3}\dagger$ $2.260\times 10^{-3}\dagger$ $8.708 \times 10^{-4}$ $1.608\times 10^{-3}\dagger$ $7.637 \times10^{-4}\dagger$ ${\bf 7.629 \times 10^{-4}}\dagger$ $8.668 \times 10^{-4}$
    ZDT2 $1.685\times 10^{-2}\dagger$ $4.700\times 10^{-3}\dagger$ $8.392 \times 10^{-4}$ $1.110\times10^{-3}\dagger$ $7.507 \times 10^{-4}\dagger$ ${\bf 7.502 \times 10^{-4}}\dagger$ $8.368 \times 10^{-4}$
    ZDT3 $2.647\times 10^{-3}\dagger$ $3.534\times 10^{-3}\dagger$ $2.038\times 10^{-3}\dagger$ $3.978\times 10^{-3}\dagger$ $2.060\times 10^{-3}\dagger$ $2.050\times10^{-3}\dagger$ ${\bf 1.620 \times10^{-3}}$
    ZDT4 $2.463\times 10^{-2}\dagger$ $2.198\times 10^{-3}\dagger$ $7.791 \times 10^{-4}$ $1.277\times 10^{-3}\dagger$ ${\bf 7.637 \times 10^{-4}}\dagger$ $7.819 \times10^{-4}$ $7.719 \times10^{-4}$
    ZDT6 $8.319 \times10^{-4}\dagger$ $1.177 \times10^{-1}\dagger$ $3.994 \times 10^{-4}$ $4.006 \times 10^{-4}$ ${\bf 3.811 \times10^{-4}}\dagger$ $5.225 \times 10^{-4}\dagger$ 3.988 $\times 10^{-4}$
    KUR $3.962 \times10^{-1}$ $3.284\times 10^{-2}\dagger$ 1.012 $\times 10^{-2}$ $1.029 \times10^{-2}$ $9.779\times 10^{-3}\dagger$ $9.853 \times 10^{-3}$ ${\bf 9.609 \times 10^{-3}}$
    WFG1 $6.949 \times10^{-1}\dagger$ $4.685 \times10^{-1}\dagger$ $4.553 \times10^{-3}$ $3.248 \times10^{-2}\dagger$ $8.218 \times10^{-3}$ $4.680 \times10^{-1}\dagger$ ${\bf 3.940 \times 10^{-3}}$
    WFG2 $3.968 \times 10^{-3}$ $1.049\times 10^{-3}\dagger$ $5.063 \times 10^{-3}$ $4.850 \times10^{-2}\dagger$ $4.805 \times 10^{-3}$ $5.241 \times 10^{-3}$ ${\bf 3.748 \times10^{-3}}$
    WFG3 $3.198 \times10^{-3}$ $5.316 \times10^{-3}\dagger$ $2.953 \times10^{-3}$ $4.498 \times10^{-3}\dagger$ ${\bf 2.597 \times 10^{-3}}\dagger$ $2.747\times10^{-3}\dagger$ $2.951 \times10^{-3}$
    WFG4 $2.665 \times10^{-2}\dagger$ $1.816 \times10^{-2}\dagger$ $5.724 \times10^{-3}$ $1.886 \times10^{-2}\dagger$ $8.427 \times10^{-3}\dagger$ $1.215 \times10^{-2}\dagger$ ${\bf 5.122 \times10^{-3}}$
    WFG5 $6.206 \times10^{-3}\dagger$ $6.777 \times10^{-2}\dagger$ $6.326 \times10^{-2}$ $6.852 \times10^{-2}$ ${\bf 6.179\times10^{-2}}\dagger$ $6.287 \times10^{-2}\dagger$ $6.510 \times10^{-2}$
    WFG6 $5.278 \times10^{-2}\dagger$ $7.205 \times10^{-2}\dagger$ $5.560 \times10^{-2}$ $5.597 \times10^{-2}$ $2.862 \times10^{-3}\dagger$ ${\bf 2.672\times10^{-3}}$ $5.287 \times10^{-2}$
    WFG7 $3.186 \times10^{-3}\dagger$ $3.014 \times10^{-2}\dagger$ $2.663 \times10^{-3}$ $3.850 \times10^{-3}\dagger $ ${\bf 2.585\times10^{-3}}\dagger$ $2.666 \times10^{-3}$ $2.689 \times10^{-3}$
    WFG8 $1.129 \times10^{-1}$ $1.017\times10^{-1}\dagger$ $1.033 \times10^{-1}\dagger$ $1.074 \times10^{-1}$ $9.793 \times10^{-2}\dagger$ ${\bf 8.302\times10^{-2}}\dagger$ $1.029 \times10^{-1}$
    WFG9 $1.915 \times10^{-2}$ $1.272 \times10^{-2}\dagger$ $1.533 \times10^{-2}$ $2.155 \times10^{-2}$ $1.229 \times10^{-2}$ ${\bf 8.062\times10^{-3}}$ $1.522 \times10^{-2}$
    †表示对比方法与本文方法的IGD指标具有显著差异(Mann-Whitney U分布检验, 置信水平为0.05)
    下载: 导出CSV

    表  2  不同方法获得的CR值

    Table  2  Metric CR obtained by different algorithms

    优化问题 NSGA-Ⅱ RPEA MOEA/D MOEA/D-PBI MOEA/D-STM MOEA/D-ACD MOEA-PPF
    ZDT1 0.9400 0.6860 0.9440 0.9360 0.9440 0.9440 0.9440
    ZDT2 0.9260 0.6380 0.9960 0.9940 0.9960 0.9960 0.9960
    ZDT3 0.6440 0.4340 0.5760 0.6460 0.5760 0.6720 0.7480
    ZDT4 0.9340 0.6320 0.9440 0.9340 0.9440 0.9420 0.9440
    ZDT6 0.9460 0.6640 0.9960 0.9960 0.9960 0.9960 0.9960
    KUR 0.8020 0.6060 0.7800 0.9160 0.7780 0.8940 0.9140
    WFG1 0.7320 0.3620 0.8700 0.8240 0.8480 0.4520 0.8720
    WFG2 0.6100 0.3680 0.4880 0.6060 0.4860 0.6220 0.7360
    WFG3 0.9460 0.8260 0.9920 0.9960 0.9960 0.9960 0.9960
    WFG4 0.9280 0.5820 0.9500 0.9520 0.8440 0.8340 0.9560
    WFG5 0.9020 0.5760 0.9180 0.9180 0.9540 0.9140 0.9200
    WFG6 0.9000 0.5500 0.9460 0.9580 0.9620 0.9620 0.9580
    WFG7 0.9240 0.5780 0.9660 0.9700 0.9680 0.9620 0.9680
    WFG8 0.5400 0.3500 0.9060 0.9220 0.9140 0.9180 0.9000
    WFG9 0.9220 0.6340 0.9260 0.9280 0.8920 0.9220 0.8980
    下载: 导出CSV
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  • 收稿日期:  2018-05-18
  • 录用日期:  2018-08-28
  • 刊出日期:  2020-08-26

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