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基于平均距离聚类的NSGA-Ⅱ

崔志华 张茂清 常宇 张江江 王晖 张文生

崔志华, 张茂清, 常宇, 张江江, 王晖, 张文生. 基于平均距离聚类的NSGA-Ⅱ.自动化学报, 2021, 47(5): 1171-1182 doi: 10.16383/j.aas.c180540
引用本文: 崔志华, 张茂清, 常宇, 张江江, 王晖, 张文生. 基于平均距离聚类的NSGA-Ⅱ.自动化学报, 2021, 47(5): 1171-1182 doi: 10.16383/j.aas.c180540
Cui Zhi-Hua, Zhang Mao-Qing, Chang Yu, Zhang Jiang-Jiang, Wang Hui, Zhang Wen-Sheng. NSGA-Ⅱ with average distance clustering. Acta Automatica Sinica, 2021, 47(5): 1171-1182 doi: 10.16383/j.aas.c180540
Citation: Cui Zhi-Hua, Zhang Mao-Qing, Chang Yu, Zhang Jiang-Jiang, Wang Hui, Zhang Wen-Sheng. NSGA-Ⅱ with average distance clustering. Acta Automatica Sinica, 2021, 47(5): 1171-1182 doi: 10.16383/j.aas.c180540

基于平均距离聚类的NSGA-Ⅱ

doi: 10.16383/j.aas.c180540
基金项目: 

国家自然科学基金 61806138

国家自然科学基金 U1636220

国家自然科学基金 61663028

山西省自然科学基金 201801D121127

江西省杰出青年人才计划 20171BCB23075

江西省自然科学基金 20171BAB202035

详细信息
    作者简介:

    张茂清  同济大学博士研究生. 主要研究方向为多目标优化, 高维多目标优化, 深度学习以及应用. E-mail: maoqing_zhang@163.com

    常宇  太原科技大学硕士研究生. 主要研究方向为动态高维多目标优化. E-mail: yuchang78dd@163.com

    张江江  太原科技大学硕士研究生. 主要研究方向为高维多目标优化. E-mail: jiangofyouth@163.com

    王晖  南昌工程学院副教授. 主要研究方向为进化计算, 群智能优化和大规模优化. E-mail: huiwang@nit.edu.cn

    张文生  中国科学院自动化研究所研究员. 主要研究方向为大数据知识挖掘, 人工智能, 机器学习, 嵌入式视频图像处理. E-mail: wensheng.zhang@ia.ac.cn

    通讯作者:

    崔志华  太原科技大学教授. 主要研究方向为智能计算, 随机算法和组合优化. 本文通信作者. E-mail: zhihua.cui@hotmail.com

NSGA-Ⅱ With Average Distance Clustering

Funds: 

National Natural Science Foundation of China 61806138

National Natural Science Foundation of China U1636220

National Natural Science Foundation of China 61663028

Natural Science Foundation of Shanxi Province 201801D121127

Distinguished Young Talents Plan of Jiangxi Province 20171BCB23075

Natural Science Foundation of Jiangxi Province 20171BAB202035

More Information
    Author Bio:

    ZHANG Mao-Qing   Ph.D. candidate at Tongji University. His research interest covers multi-objective optimization, many-objective optimization, and deep learning and their applications

    CHANG Yu   Master student at Taiyuan University of Science and Technology. Her main research interest is dynamic many-objective optimization

    ZHANG Jiang-Jiang   Master student at Taiyuan University of Science and Technology. His main research interest is many-objective optimization

    WANG Hui   Associate professor at Nanchang Institute of Technology. His research interest covers evolutionary computing, swarm intelligence optimization, and large-scale optimization

    ZHANG Wen-Sheng   Professor at the Institute of Automation, Chinese Academy of Sciences. His research interest covers large data knowledge mining, artificial intelligence, machine learning, and embedded video image processing

    Corresponding author: CUI Zhi-Hua   Professor at Taiyuan University of Science and Technology. His research interest covers intelligent computing, stochastic algorithms, and combinatorial optimization. Corresponding author of this paper
  • 摘要: 拥挤度距离是一种用于度量解集多样性的指标. 然而, 在许多情况下, 该指标无法有效区分多样性较优个体. 其原因为拥挤度距离主要利用每个位置的局部信息. 为解决该问题, 基于整个种群全局位置信息, 本文设计了基于平均距离聚类的多样性度量指标, 并进一步提出了基于平均距离聚类的NSGA-Ⅱ. 该算法利用平均距离将种群划分为若干个大致均匀分布的小种群, 然后分别在各小种群内执行选择、交叉和变异等操作. 实验结果表明, 本文所提算法可以有效地保持种群多样性.
    Recommended by Associate Editor ZHANG Min-Ling
    1)  本文责任编委 张敏灵
  • 图  1  熔化率与比例、积分系数之间的关系

    Fig.  1  Comparing melting rate with proportion and integral parameters of the consarc controller

    图  2  式(5)中参数说明

    Fig.  2  Illustration of parameter in (5)

    图  3  小种群划分示意图

    Fig.  3  Illustration of small populations

    图  4  交叉算子示意图

    Fig.  4  Illustration of crossover operator

    图  5  ADCNSGA-Ⅱ算法流程图

    Fig.  5  The flowchart of ADCNSGA-Ⅱ

    图  6  测试结果对比

    Fig.  6  Comparisons of obtained solutions

    表  1  参数$S$对算法ADCNSGA-Ⅱ性能影响

    Table  1  Influence of parameter $S$ on ADCNSGA-Ⅱ

    测试函数指标$S=1$$S=2$$S=3$$S=4$$S=5$$S=6$
    SCHmean$(GD)$$ \bf{5.87\times 10^{-2}} $$6.47\times 10^{-2}$$6.68\times 10^{-2}$$6.77\times 10^{-2}$$6.67\times 10^{-2}$$6.69\times 10^{-2}$
    std$(GD)$$8.14\times 10^{-3}$$ \bf{3.84\times 10^{-3}}$$5.03\times 10^{-3}$$5.11\times 10^{-3}$$5.28\times 10^{-3}$$4.15\times 10^{-3}$
    mean$(SP)$$ \bf{9.87\times 10^{-3}}$$1.18\times 10^{-2}$$1.77\times 10^{-2}$$1.89\times 10^{-2}$$2.28\times 10^{-2}$$1.77\times 10^{-2}$
    std$(SP) $$6.23\times 10^{-3}$$ \bf{2.62\times 10^{-3}}$$1.12\times 10^{-2}$$1.07\times 10^{-2}$$2.16\times 10^{-2}$$1.01\times 10^{-2}$
    ZDT1mean$(GD) $$ \bf{2.05\times 10^{-6}}$$4.07\times 10^{-6}$$4.71\times 10^{-5}$$3.28\times 10^{-5}$$7.25\times 10^{-5}$$7.71\times 10^{-5}$
    std$(GD)$$ \bf{1.11\times 10^{-5}}$$2.10\times 10^{-5}$$8.90\times 10^{-5}$$6.66\times 10^{-5}$$1.32\times 10^{-4}$$1.28\times 10^{-4}$
    mean$(SP)$$ \bf{4.48\times 10^{-3}}$$5.34\times 10^{-3}$$5.94\times 10^{-3}$$6.48\times 10^{-3}$$5.87\times 10^{-3}$$5.42\times 10^{-3}$
    std$(SP)$$1.59\times 10^{-3}$$1.31\times 10^{-3}$$ \bf{1.20\times 10^{-3}}$$1.92\times 10^{-3}$$2.57\times 10^{-3}$$1.78\times 10^{-3}$
    ZDT2mean$(GD)$$ \bf{1.32\times 10^{-5}}$$3.80\times 10^{-5}$$5.17\times 10^{-5}$$5.51\times 10^{-5}$$8.31\times 10^{-5}$$8.55\times 10^{-5}$
    std$(GD)$$ \bf{6.55\times 10^{-5}}$$9.60\times 10^{-5}$$9.26\times 10^{-5}$$7.18\times 10^{-5}$$9.89\times 10^{-5}$$1.21\times 10^{-4}$
    mean$(SP)$$ \bf{4.47\times 10^{-3}}$$5.40\times 10^{-3}$$6.32\times 10^{-3}$$6.91\times 10^{-3}$$6.78\times 10^{-3}$$6.79\times 10^{-3}$
    std$(SP)$$1.79\times 10^{-3}$$ \bf{1.02\times 10^{-3}}$$1.44\times 10^{-3}$$2.60\times 10^{-3}$$3.32\times 10^{-3}$$2.46\times 10^{-3}$
    ZDT3mean$(GD)$$ \bf{1.09\times 10^{-6}}$$6.07\times 10^{-6}$$2.48\times 10^{-5}$$3.57\times 10^{-5}$$1.38\times 10^{-5}$$5.13\times 10^{-5}$
    std$(GD)$$ \bf{5.91\times 10^{-6}}$$2.61\times 10^{-5}$$5.56\times 10^{-5}$$5.54\times 10^{-5}$$2.20\times 10^{-5}$$5.81\times 10^{-5}$
    mean$(SP)$$ \bf{4.93\times 10^{-3}}$$6.20\times 10^{-3}$$6.01\times 10^{-3}$$7.84\times 10^{-3}$$6.85\times 10^{-3}$$8.04\times 10^{-3}$
    std$(SP)$$1.89\times 10^{-3}$$ \bf{1.21\times 10^{-3}}$$2.12\times 10^{-3}$$3.61\times 10^{-3}$$2.22\times 10^{-3}$$2.91\times 10^{-3}$
    ZDT4mean$(GD)$$1.51\times 10^{-5}$$8.46\times 10^{-6}$$2.03\times 10^{-5}$$ \bf{4.54\times 10^{-6}}$$2.67\times 10^{-5}$$7.20\times 10^{-6}$
    std$(GD)$$4.49\times 10^{-5}$$2.13\times 10^{-5}$$4.46\times 10^{-5}$$ \bf{1.25\times 10^{-5}}$$6.31\times 10^{-5}$$1.76\times 10^{-5}$
    mean$(SP)$$ \bf{3.29\times 10^{-3}}$$5.21\times 10^{-3}$$6.01\times 10^{-3}$$5.98\times 10^{-3}$$6.43\times 10^{-3}$$6.43\times 10^{-3}$
    std$(SP)$$1.64\times 10^{-3}$$ \bf{1.20\times 10^{-3}}$$1.75\times 10^{-3}$$1.65\times 10^{-3}$$2.01\times 10^{-3}$$3.32\times 10^{-3}$
    ZDT6mean$(GD)$$ \bf{1.11\times 10^{-2}}$$2.07\times 10^{-2}$$2.07\times 10^{-2}$$3.27\times 10^{-2}$$9.92\times 10^{-3}$$2.29\times 10^{-2}$
    std$(GD)$$ \bf{2.34\times 10^{-2}}$$3.86\times 10^{-2}$$4.77\times 10^{-2}$$5.42\times 10^{-2}$$3.61\times 10^{-2}$$4.08\times 10^{-2}$
    mean$(SP)$$\bf{1.77\times 10^{-2}}$$6.14\times 10^{-2}$$5.52\times 10^{-2}$$7.81\times 10^{-2}$$ {1.09\times 10^{-2}}$$4.55\times 10^{-2}$
    std$(SP)$$5.80\times 10^{-2}$$1.64\times 10^{-1}$$1.26\times 10^{-1}$$1.18\times 10^{-1}$$ \bf{3.05\times 10^{-2}}$$1.14\times 10^{-1}$
    下载: 导出CSV

    表  2  Friedman测试结果

    Table  2  Comparison results of Friedman test

    参数秩均值
    $S=1$1.92
    $S=2$2.40
    $S=3$3.67
    $S=4$4.38
    $S=5$4.23
    $S=6$4.42
    下载: 导出CSV

    表  3  Wilcoxon检测测试结果

    Table  3  Comparison results of Wilcoxon test

    $S=1$对比p值
    $S=2$0.0765
    $S=3$0.0018
    $S=4$0.0003
    $S=5$0.0258
    $S=6$0.0003
    下载: 导出CSV

    表  4  实验性能均值和方差对比

    Table  4  Means and variances of the performance metrics

    测试函数指标PNIASPEA2NSGA-Ⅱg-NSGA-ⅡADCNSGA-Ⅱ
    SCHmean$(GD)$$ \bf{3.81\times 10^{-2}} $$4.41\times 10^{-2}$$6.17\times 10^{-2}$$4.43\times 10^{-2}$$5.87\times 10^{-2}$
    std$(GD)$$ \bf{1.93\times 10^{-3}}$$1.14\times 10^{-3}$$2.96\times 10^{-3}$$1.61\times 10^{-2}$$8.14\times 10^{-3}$
    mean$(SP)$$ \bf{4.96\times 10^{-3}}$$7.01\times 10^{-3}$$1.75\times 10^{-2}$$3.25\times 10^{-2}$$9.87\times 10^{-3}$
    std$(SP)$$ \bf{3.66\times 10^{-3}}$$1.61\times 10^{-3}$$1.02\times 10^{-2}$$1.45\times 10^{-1}$$6.32\times 10^{-3}$
    ZDT1mean$(GD)$$6.56\times 10^{-4}$$2.80\times 10^{-5}$$2.58\times 10^{-4}$$1.73\times 10^{-3}$$ \bf{2.05\times 10^{-6}}$
    std$(GD)$$1.64\times 10^{-4}$$1.34\times 10^{-5}$$1.17\times 10^{-4}$$5.72\times 10^{-3}$$ \bf{1.11\times 10^{-5}}$
    mean$(SP)$$1.04\times 10^{-3}$$1.94\times 10^{-3}$$5.09\times 10^{-3}$$1.01\times 10^{-3}$$ \bf{4.48\times 10^{-3}}$
    std$(SP)$$1.06\times 10^{-3}$$3.38\times 10^{-4}$$3.92\times 10^{-3}$$7.45\times 10^{-3}$$ \bf{1.59\times10^{-3}}$
    ZDT2mean$(GD)$$8.77\times 10^{-4}$$ \bf{1.17\times 10^{-5}}$$8.42\times 10^{-5}$$5.98\times 10^{-4}$$1.32\times 10^{-5}$
    std$(GD)$$1.18\times 10^{-3}$$ \bf{1.17\times 10^{-5}}$$1.32\times 10^{-4}$$6.03\times 10^{-4}$$6.55\times 10^{-5}$
    mean$(SP)$$2.37\times 10^{-3}$$ \bf{2.26\times 10^{-3}}$$2.81\times 10^{-3}$$2.40\times 10^{-3}$$4.47\times 10^{-3}$
    std$(SP)$$3.17\times 10^{-3}$$ \bf{3.59\times 10^{-3}}$$4, 36\times 10^{-3}$$2.70\times 10^{-3}$$1.79\times 10^{-3}$
    ZDT3mean$(GD)$$1.89\times 10^{-4}$$4.43\times 10^{-6}$$1.40\times 10^{-4}$$6.22\times 10^{-5}$$ \bf{1.09\times 10^{-6}}$
    std$(GD)$$1.01\times 10^{-4}$$1.71\times 10^{-6}$$9.98\times 10^{-5}$$4.98\times 10^{-5}$$ \bf{5.91\times 10^{-6}}$
    mean$(SP)$$4.09\times 10^{-4}$$1.89\times 10^{-3}$$5.56\times 10^{-3}$$1.10\times 10^{-3}$$ \bf{4.93\times 10^{-3}}$
    std$(SP)$$5.47\times 10^{-4}$$5.99\times 10^{-4}$$3.81\times 10^{-3}$$8.23\times 10^{-4}$$ \bf{1.89\times10^{-3}}$
    ZDT4mean$(GD)$$1.25\times 10^{-2}$$6.95\times 10^{-2}$$2.68\times 10^{-1}$$1.62\times 10^{-2}$$\bf{1.51\times 10^{-5}}$
    std$(GD)$$1.03\times 10^{-2}$$4.19\times 10^{-2}$$1.30\times 10^{-1}$$6.48\times 10^{-1}$$\bf{4.49\times 10^{-5}}$
    mean$(SP)$$\bf{1.43\times 10^{-3}}$$4.37\times 10^{-3}$$5.09\times 10^{-3}$$1.70\times 10^{-2}$$3.29\times 10^{-3}$
    std$(SP)$$3.21\times 10^{-3}$$4.64\times 10^{-3}$$1.11\times 10^{-2}$$2.24\times 10^{-2}$$\bf{1.64\times10^{-3}}$
    ZDT6mean$(GD)$$1.78\times 10^{-3}$$\bf{1.33\times 10^{-3}}$$5.49\times 10^{-3}$$1.47\times 10^{-2}$$1.11\times 10^{-2}$
    std$(GD)$$2.31\times 10^{-4}$$\bf{1.84\times 10^{-4}}$$1.27\times 10^{-3}$$4.00\times 10^{-3}$$2.34\times 10^{-2}$
    mean$(SP)$$1.35\times 10^{-3}$$1.45\times 10^{-3}$$4.43\times 10^{-3}$$\bf{8.74\times 10^{-4}}$$1.77\times 10^{-2}$
    std$(SP)$$9.87\times 10^{-4}$$\bf{5.74\times 10^{-4}}$$2.05\times 10^{-3}$$7.14\times 10^{-4}$$5.80\times 10^{-2}$
    下载: 导出CSV
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出版历程
  • 收稿日期:  2018-08-09
  • 录用日期:  2019-01-22
  • 刊出日期:  2021-05-21

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