高级检索

  • ISSN 1006-3080
  • CN 31-1691/TQ

基于统计信息反馈的分步多目标优化

王学武 谢祖洪 周昕 顾幸生

王学武, 谢祖洪, 周昕, 顾幸生. 基于统计信息反馈的分步多目标优化[J]. 华东理工大学学报(自然科学版). doi: 10.14135/j.cnki.1006-3080.20210427004
引用本文: 王学武, 谢祖洪, 周昕, 顾幸生. 基于统计信息反馈的分步多目标优化[J]. 华东理工大学学报(自然科学版). doi: 10.14135/j.cnki.1006-3080.20210427004
WANG Xuewu, XIE Zuhong, ZHOU Xin, GU Xingsheng. A Stepwise Multi-Objective Evolutionary Optimization Algorithm Based on Statistical Feedback Information[J]. Journal of East China University of Science and Technology. doi: 10.14135/j.cnki.1006-3080.20210427004
Citation: WANG Xuewu, XIE Zuhong, ZHOU Xin, GU Xingsheng. A Stepwise Multi-Objective Evolutionary Optimization Algorithm Based on Statistical Feedback Information[J]. Journal of East China University of Science and Technology. doi: 10.14135/j.cnki.1006-3080.20210427004

基于统计信息反馈的分步多目标优化

doi: 10.14135/j.cnki.1006-3080.20210427004
基金项目: 国家自然科学基金(62076095,61973120)
详细信息
    作者简介:

    王学武(1972−),男,陕西合阳人,博士,副教授,主要研究方向为智能优化算法、焊接机器人智能化技术。E-mail:wangxuew@ecust.edu.cn

  • 中图分类号: TP301

A Stepwise Multi-Objective Evolutionary Optimization Algorithm Based on Statistical Feedback Information

  • 摘要: 传统的多目标进化算法通常协同考虑解的分布性和收敛性,在搜索初期会生成大量的支配解,造成计算资源的浪费,甚至导致算法不收敛。本文提出了一种基于统计信息反馈的分步多目标优化算法。将算法分为单目标探索阶段、单目标到多目标的过渡阶段、群体划分局部优化阶段3个阶段,根据每个阶段的性质设计任务和策略,以增强算法的收敛性和分布性。在第二、三阶段中,根据目标函数值将解划分为不同群体,分别对不同区域解的信息进行统计分析,再根据反馈统计信息指导亲本选择过程,改善解的分布性和收敛性。在DTLZ和WFG系列问题上进行测试,并与其他多目标进化算法进行了比较,实验结果验证了本文算法在复杂、难收敛问题上的优势。

     

  • 图  1  部分三维测试案例第一阶段解集分布情况

    Figure  1.  Distribution of solution sets in the first stage of some 3-objective test cases

    图  2  三维优化问题的种群划分

    Figure  2.  Population division of for 3-objective optimization problems

    图  3  三维PF线性拟合

    Figure  3.  Linear fitting of three dimensional PF

    图  4  SFI-SMOEA下部分测试案例的解分布

    Figure  4.  Distribution of solutions of some test cases by SFI-SMOEA

    图  5  不同迭代次数下5种算法的IGD平均值

    Figure  5.  IGD mean values of five algorithms in different iterations

    表  1  DTLZ测试案例的HV平均值(标准差)

    Table  1.   HV HV mean values eval standard deviation of DTLZ test cases

    ProblemMMaxGen/NSFI-SMOEANSGA-IIIA-NSGA-IIIMOEAD-M2M$ {{I}_{\rm{SDE}}}^{+} $
    DTLZ134.0×102/
    9.10×102
    8.416×10−1
    (1.497×10−3)
    8.421×10−1
    (2.001×10−3)
    =8.419×10−1
    (2.577×10−3)
    =7.427×10−1
    (1.102×10−1)
    +8.363×10−1
    (2.589×10−3)
    =
    56.0×102/
    2.10×102
    9.745×10−1
    (4.124×10−4)
    9.733×10−1
    (2.365×10−3)
    =9.736×10−1
    (1.146×10−3)
    =5.457×10−1
    (3.027×10−1)
    +9.622×10−1
    (4.697×10−3)
    =
    87.5×102/
    1.56×102
    9.948×10−1
    (1.092×10−2)
    9.286×10−1
    (3.331×10−1)
    +9.810×10−1
    (6.743×10−2)
    =7.893×10−2
    (1.623×10−1)
    +9.911×10−1
    (1.956×10−3)
    =
    101.0×103/
    2.75×102
    9.977×10−1
    (8.365×10−3)
    9.913×10−1
    (5.393×10−2)
    =9.890×10−1
    (6.708×10−2)
    =6.896×10−4
    (1.929×10−3)
    +9.977×10−1
    (4.470×10−4)
    =
    DTLZ232.5×102/
    9.10×10
    5.615×10−1
    (3.599×10−4)
    5.598×10−1
    (9.525×10−4)
    =5.595×10−1
    (8.317×10−4)
    =4.920×10−1
    (1.467×10−2)
    +5.609×10−1
    (1.276×10−3)
    =
    53.5×102/
    2.10×102
    7.544×10−1
    (7.154×10−2)
    7.848×10−1
    (2.554×10−3)
    =7.847×10−1
    (3.607×10−3)
    =6.731×10−1
    (2.226×10−2)
    +7.937×10−1
    (2.756×10−3)
    $ - $
    85.0×102/
    1.56×102
    8.512×10−1
    (5.478×10−2)
    8.958×10−1
    (5.215×10−2)
    $ - $8.965×10−1
    (4.590×10−2)
    $ - $4.135×10−1
    (5.687×10−2)
    +9.279×10−1
    (2.179×10−3)
    $ - $
    107.5×102/
    2.75×102
    9.437×10−1
    (2.251×10−2)
    9.452×10−1
    (3.214×10−2)
    =9.515×10−1
    (2.054×10−2)
    =3.861×10−1
    (4.420×10−2)
    +9.676×10−1
    (1.006×10−3)
    =
    DTLZ331.0×103/
    9.10×10
    5.550×10−1
    (3.682×10−3)
    5.541×10−1
    (1.097×10−2)
    =5.533×10−1
    (1.734×10−2)
    =4.111×10−1
    (6.717×10−2)
    +5.591×10−1
    (2.016×10−3)
    =
    51.0×103/
    2.10×102
    7.346×10−1
    (8.182×10−2)
    7.558×10−1
    (5.893×10−2)
    =7.105×10−1
    (4.296×10−1)
    =8.671×10−3
    (2.735×10−2)
    +7.857×10−1
    (4.998×10−3)
    $ - $
    81.0×103/
    1.56×102
    7.030×10−1
    (1.827×10−1)
    5.172×10−1
    (4.469×10−1)
    +3.998×10−1
    (4.523×10−1)
    +0
    (0)
    +9.158×10−1
    (4.272×10−3)
    $ - $
    101.5×103/
    2.75×102
    9.082×10−1
    (5.438×10−2)
    6.015×10−1
    (4.884×10−1)
    +5.705×10−1
    (4.843×10−1)
    +0
    (0)
    +9.601×10−1
    (2.481×10−3)
    $ - $
    DTLZ436.0×102/
    9.10×10
    5.621×10−1
    (1.903×10)
    5.393×10−1
    (1.216×10−1)
    =5.502×10−1
    (1.241×10−1)
    =5.269×10−1
    (8.830×10−3)
    +5.606×10−1
    (1.710×10−3)
    =
    51.0×102/
    2.10×102
    6.658×10−1
    (4.748×10−2)
    7.823×10−1
    (3.793×10−3)
    $ - $7.825×10−1
    (4.318×10−3)
    $ - $7.392×10−1
    (1.630×10−2)
    $ - $7.835×10−1
    (1.907×10−2)
    $ - $
    81.2×103/
    1.56×102
    8.303×10−1
    (3.473×10−2)
    9.018×10−1
    (1.007×10−2)
    $ - $8.967×10−1
    (4.417×10−2)
    $ - $5.304×10−1
    (1.175×10−1)
    +9.173×10−1
    (2.434×10−3)
    $ - $
    102.0×103/
    2.75×102
    9.165×10−1
    (1.207×10−2)
    9.546×10−1
    (2.356×10−2)
    =9.563×10−1
    (2.867×10−3)
    =2.574×10−1
    (1.101×10−1)
    +9.599×10−1
    (1.804×10−3)
    =
    DTLZ535.0×102/
    9.10×10
    1.951×10−1
    (1.031×10−3)
    1.946×10−1
    (2.050×10−3)
    =1.963×10−1
    (6.548×10)
    =1.603×10−1
    (1.141×10−2)
    +1.990×10−1
    (4.448×10)
    =
    57.0×102/
    2.10×102
    1.106×10−1
    (6.386×10−3)
    9.426×10−2
    (4.446×10−2)
    +9.669×10−2
    (3.372×10−2)
    +4.333×10−3
    (6.450×10−3)
    +1.017×10−1
    (3.423×10−3)
    +
    81.0×103/
    1.56×102
    9.797×10−2
    (2.355×10−3)
    8.345×10−2
    (1.713×10−2)
    +7.248×10−2
    (3.558×10−2)
    +3.994E-05
    (1.019×10)
    +8.431×10−2
    (4.097×10−3)
    +
    DTLZ5101.5×102/
    2.75×102
    9.375×10−2
    (1.537×10−3)
    7.161×10−2
    (2.357×10−2)
    +6.389×10−2
    (3.165×10−2)
    +7.903E-07
    (3.527E-06)
    +7.963×10−2
    (4.839×10−3)
    +
    DTLZ635.0×102/
    9.10×10
    1.947×10−1
    (1.033×10−3)
    1.945×10−1
    (1.973×10−3)
    =1.963×10−1
    (1.783×10−3)
    =1.573×10−1
    (1.162×10−2)
    +1.994×10−1
    (2.960×10)
    =
    57.0×102/
    2.10×102
    9.856×10−2
    (2.298×10−2)
    1.073×10−2
    (4.968×10−2)
    +7.753×10−3
    (5.040×10−2)
    +3.224×10−2
    (3.858×10−2)
    +1.031×10−1
    (4.141×10−3)
    =
    81.0×103/
    1.56×102
    9.086×10−2
    (1.947×10)
    0
    (0)
    +0
    (0)
    +0
    (0)
    +9.111×10−2
    (5.464×10)
    =
    101.5×103/
    2.75×102
    9.127×10−2
    (1.458×10−3)
    0
    (0)
    +0
    (0)
    +0
    (0)
    +9.079×10−2
    (7.243×10)
    =
    DTLZ734.0×102/
    9.10×10
    2.737×10−1
    (7.333×10)
    2.736×10−1
    (1.772×10−3)
    =2.728×10−1
    (1.813×10−3)
    =7.320×10−2
    (2.996×10−2)
    +2.709×10−1
    (3.234×10−3)
    =
    56.0×102/
    2.10×102
    2.488×10−1
    (2.628×10−3)
    2.488×10−1
    (6.801×10−3)
    =2.489×10−1
    (6.955×10−3)
    =1.420×10−3
    (4.945×10−3)
    +2.685×10−1
    (3.800×10−3)
    $ - $
    87.5×102/
    1.56×102
    2.426×10−1
    (3.601×10−3)
    2.400×10−1
    (9.077×10−3)
    =2.379×10−1
    (8.254×10−3)
    =0
    (0)
    +2.514×10−1
    (1.056×10−2)
    =
    101.0×103/
    2.75×102
    2.318×10−1
    (4.078×10−3)
    2.309×10−1
    (5.554×10−3)
    =2.319×10−1
    (4.590×10−3)
    =0
    (0)
    +2.356×10−1
    (8.897×10−3)
    =
    下载: 导出CSV

    表  2  WFG测试案例的HV平均值(标准差)

    Table  2.   HV mean values eval standard deviation of WFG test cases

    ProblemMMaxGen(N)SFI-SMOEANSGA-IIIA-NSGA-IIIMOEAD-M2M$ {{I}_{SDE}}^{+} $
    WFG133.3×102
    (1.05×102)
    9.474×10−1
    (1.831×10−4)
    9.469×10−1
    (1.220×10−4)
    =9.469×10−1
    (1.187×10−4)
    =8.804×10−1
    (1.762×10−2)
    +9.388×10−1
    (4.500×10−3)
    =
    53.0×102
    (1.26×102)
    9.911×10−1
    (5.460×10−3)
    9.984×10−1
    (5.005×10−5)
    =9.984×10−1
    (5.761×10−5)
    =9.862×10−1
    (3.874×10−3)
    =9.885×10−1
    (3.578×10−3)
    =
    83.0×102
    (1.56×102)
    9.942×10−1
    (7.706×10−3)
    1.000 E00
    (1.385×10−5)
    =9.999×10−1
    (2.870×10−5)
    =8.973×10−1
    (2.897×10−2)
    +9.944×10−1
    (2.090×10−3)
    =
    103.0×102
    (2.75×102)
    9.980×10−1
    (1.238×10−3)
    1.000E00
    (3.269E-06)
    =1.000E00
    (6.450E-06)
    =9.577×10−1
    (1.416×10−2)
    =9.961×10−1
    (1.755×10−3)
    =
    WFG233.3×102
    (1.05×102)
    8.240×10−1
    (6.888×10−2)
    9.171×10−1
    (3.122×10−2)
    $ - $8.815×10−1
    (6.542×10−2)
    $ - $8.448×10−1
    (4.213×10−2)
    =8.709×10−1
    (7.307×10−2)
    $ - $
    53.0×102
    (1.26×102)
    8.879×10−1
    (8.739×10−2)
    9.312×10−1
    (7.735×10−2)
    =9.323×10−1
    (7.696×10−2)
    =8.808×10−1
    (3.002×10−2)
    =9.153×10−1
    (8.205×10−2)
    =
    83.0×102
    (1.56×102)
    9.887×10−1
    (3.961×10−3)
    9.861×10−1
    (5.917×10−3)
    =9.899×10−1
    (5.019×10−3)
    =8.362×10−1
    (3.088×10−2)
    +9.877×10−1
    (2.895×10−3)
    =
    103.0×102
    (2.75×102)
    9.950×10−1
    (2.114×10−3)
    9.915×10−1
    (4.584×10−3)
    =9.947×10−1
    (3.010×10−3)
    =8.100×10−1
    (3.991×10−2)
    +9.931×10−1
    (1.482×10−3)
    =
    WFG333.3×102
    (1.05×102)
    3.980×10−1
    (5.022×10−3)
    3.860×10−1
    (4.335×10−3)
    =3.737×10−1
    (1.174×10−2)
    +2.945×10−1
    (2.386×10−2)
    +4.020×10−1
    (2.465×10−3)
    =
    53.0×102
    (1.26×102)
    1.998×10−1
    (2.277×10−2)
    1.041×10−1
    (2.111×10−2)
    +1.114×10−1
    (2.461×10−2)
    +0
    (0)
    +1.749×10−1
    (2.193×10−2)
    +
    83.0×102
    (1.56×102)
    7.198×10−2
    (1.396×10−2)
    1.032×10−3
    (3.575×10−3)
    +1.314×10−2
    (1.378×10−2)
    +0
    (0)
    +7.594×10−3
    (1.060×10−2)
    +
    103.0×102
    (2.75×102)
    4.191×10−2
    (1.448×10−2)
    0
    (0)
    +0
    (0)
    +0
    (0)
    +0
    (0)
    +
    WFG433.3×102
    (1.05×102)
    5.494×10−1
    (2.565×10−3)
    5.295×10−1
    (3.160×10−3)
    =5.297×10−1
    (3.848×10−3)
    =4.601×10−1
    (1.115×10−2)
    +5.539×10−1
    (1.641×10−3)
    =
    53.0×102
    (1.26×102)
    6.961×10−1
    (5.400×10−2)
    6.937×10−1
    (1.848×10−2)
    =6.844×10−1
    (2.650×10−2)
    =4.974×10−1
    (2.981×10−2)
    +7.598×10−1
    (5.238×10−3)
    $ - $
    83.0×102
    (1.56×102)
    8.644×10−1
    (2.390×10−2)
    8.234×10−1
    (1.090×10−2)
    =8.105×10−1
    (1.684×10−2)
    +3.216×10−1
    (3.790×10−2)
    +8.783×10−1
    (7.568×10−3)
    =
    103.0×102
    (2.75×102)
    9.280×10−1
    (9.413×10−3)
    8.624×10−1
    (7.823×10−3)
    +8.574×10−1
    (1.386×10−2)
    +3.416×10−1
    (5.150×10−2)
    +9.046×10−1
    (4.736×10−3)
    =
    WFG533.3×102
    (1.05×102)
    5.070×10−1
    (1.833×10−3)
    5.007×10−1
    (1.362×10−3)
    =4.952×10−1
    (1.627×10−3)
    =4.480×10−1
    (1.034×10−2)
    +5.121×10−1
    (2.019×10−3)
    =
    53.0×102
    (1.26×102)
    7.201×10−1
    (5.015×10−3)
    6.926×10−1
    (4.568×10−3)
    =6.821×10−1
    (6.613×10−3)
    +4.664×10−1
    (2.064×10−2)
    +7.167×10−1
    (2.596×10−3)
    =
    83.0×102
    (1.56×102)
    8.530×10−1
    (4.077×10−3)
    8.248×10−1
    (4.949×10−3)
    =7.728×10−1
    (9.466×10−3)
    +3.024×10−1
    (5.253×10−2)
    +8.517×10−1
    (3.720×10−3)
    =
    103.0×102
    (2.75×102)
    8.934×10−1
    (4.473×10−3)
    8.650×10−1
    (3.803×10−3)
    =8.238×10−1
    (5.870×10−3)
    +2.741×10−1
    (3.943×10−2)
    +8.783×10−1
    (3.244×10−3)
    =
    WFG633.3×102
    (1.05×102)
    5.058×10−1
    (1.535×10−2)
    4.868×10−1
    (7.270×10−3)
    =4.875×10−1
    (1.009×10−2)
    =4.041×10−1
    (1.812×10−2)
    +5.068×10−1
    (1.389×10−2)
    =
    53.0×102
    (1.26×102)
    6.131×10−1
    (5.748×10−2)
    6.814×10−1
    (1.377×10−2)
    $ - $6.848×10−1
    (1.930×10−2)
    $ - $4.749×10−1
    (2.656×10−2)
    +7.222×10−1
    (1.699×10−2)
    $ - $
    83.0×102
    (1.56×102)
    7.961×10−1
    (3.371×10−2)
    7.998×10−1
    (1.708×10−2)
    =7.941×10−1
    (1.477×10−2)
    =2.855×10−1
    (6.135×10−2)
    +8.439×10−1
    (1.629×10−2)
    $ - $
    103.0×102
    (2.75×102)
    8.605×10−1
    (2.959×10−2)
    8.451×10−1
    (1.209×10−2)
    =8.308×10−1
    (1.035×10−2)
    =2.199×10−1
    (4.819×10−2)
    +8.702×10−1
    (1.486×10−2)
    =
    WFG733.3×102
    (1.05×102)
    5.500×10−1
    (1.733×10−2)
    5.495×10−1
    (2.171×10−3)
    =5.496×10−1
    (1.579×10−3)
    =4.401×10−1
    (1.694×10−2)
    +5.599×10−1
    (1.514×10−3)
    =
    53.0×102
    (1.26×102)
    6.485×10−1
    (7.465×10−2)
    7.468×10−1
    (5.783×10−3)
    $ - $7.447×10−1
    (5.560×10−3)
    $ - $4.241×10−1
    (4.287×10−2)
    +7.838×10−1
    (1.949×10−3)
    $ - $
    83.0×102
    (1.56×102)
    8.572×10−1
    (2.024×10−2)
    8.832×10−1
    (5.514×10−3)
    =8.624×10−1
    (1.150×10−2)
    =2.363×10−1
    (5.387×10−2)
    +9.160×10−1
    (3.851×10−3)
    $ - $
    103.0×102
    (2.75×102)
    9.364×10−1
    (6.372×10−3)
    9.307×10−1
    (6.327×10−3)
    =9.206×10−1
    (1.305×10−2)
    =2.337×10−1
    (4.920×10−2)
    +9.501×10−1
    (3.173×10−3)
    =
    下载: 导出CSV
  • [1] 王丽萍, 丰美玲, 邱启仓, 等. 偏好多目标进化算法研究综述[J]. 计算机学报, 2019, 42(6): 1289-1315. doi: 10.11897/SP.J.1016.2019.01289
    [2] COELLO C A C. Evolutionary multi-objective optimization: A historical view of the field[J]. IEEE Computational Intelligence Magazine, 2006, 1(1): 28-36. doi: 10.1109/MCI.2006.1597059
    [3] TRIVEDI A, SRINIVASAN D, SANYAL K, et al. A survey of multiobjective evolutionary algorithms based on dDecomposition[J]. IEEE Transactions on Evolutionary Computation, 2017, 21(3): 440-462.
    [4] ZHANG Q F, LI H. MOEA/D: A multiobjective evolutionary algorithm based on decomposition[J]. IEEE Transactions on Evolutionary Computation, 2007, 21(3): 712-731.
    [5] LI H, ZHANG Q F. Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II[J]. IEEE Transactions on Evolutionary Computation, 2009, 13(2): 284-302. doi: 10.1109/TEVC.2008.925798
    [6] WANG R, ZHANG Q F, ZHANG T. Decomposition-based algorithms using Pareto adaptive scalarizing methods[J]. IEEE Transactions on Evolutionary Computation, 2016, 20(6): 821-837. doi: 10.1109/TEVC.2016.2521175
    [7] QIAO J F, ZHOU H B, YANG C L, et al. A decomposition-based multiobjective evolutionary algorithm with angle-based adaptive penalty[J]. Applied Soft Computing Journal, 2019, 74: 190-205. doi: 10.1016/j.asoc.2018.10.028
    [8] DEB K, PRATAP A, AGARWAL S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197. doi: 10.1109/4235.996017
    [9] ZITZLER E, LAUMANNS M, THIELE L. SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization[C]//Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems. Athens. Greece: [s. n. ], 2001: 19-21.
    [10] CORNE D W, JERRAM N R, KNOWLES J D, et al. PESA-II: Region-based selection in evolutionary multiobjective optimization[C]//Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation. USA: ACM, 2001: 283-290.
    [11] ZITZLER E, KÜNZLI S. Indicator-based selection in multiobjective search[C]//International Conference on Parallel Problem Solving from Nature. Berlin, Heidelberg: Springer, 2004: 832-842.
    [12] BEUME N, NAUJOKS B, EMMERICH M. SMS-EMOA: Multiobjective selection based on dominated hypervolume[J]. European Journal of Operational Research, 2007, 181(3): 1653-1669. doi: 10.1016/j.ejor.2006.08.008
    [13] BADER J, ZITZLER E. HypE: An algorithm for fast hypervolume-based many-objective optimization[J]. Evolutionary Computation, 2011, 19(1): 45-76. doi: 10.1162/EVCO_a_00009
    [14] DEB K, JAIN H. An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, Part I: Solving problems with box constraints[J]. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 577-601. doi: 10.1109/TEVC.2013.2281535
    [15] SATO H. Analysis of inverted PBI and comparison with other scalarizing functions in decomposition based MOEAs[J]. Journal of Heurs, 2015, 21(6): 819-849. doi: 10.1007/s10732-015-9301-6
    [16] TRINADH P, RAMMOHAN M, NAGARATNAM S P. I SDE+—An indicator for multi and many-objective optimization[J]. IEEE Transactions on Evolutionary Computation, 2019, 23(2): 346-352. doi: 10.1109/TEVC.2018.2848921
    [17] SUN J, ZHANG H, ZHANG Q F, et al. Balancing exploration and exploitation in multiobjective evolutionary optimization[J]. Information Sciences, 2019, 497: 129-148. doi: 10.1016/j.ins.2019.05.046
    [18] GOLDBERG D E. Genetic Algorithm in Search Optimization and Machine Learning[M]. USA: Addison-Wesley Longman Publishing Co. Inc, 1989: 95-99.
    [19] 王学武, 夏泽龙, 顾幸生. 基于事件触发的自适应邻域多目标进化算法[J]. 华南理工大学学报(自然科学版), 2019, 47(4): 99-106.
    [20] LIN I K. A concordance correlation coefficient to evaluate reproducibility[J]. Biometrics, 1989, 45(1): 255-268. doi: 10.2307/2532051
    [21] DAS I, DENNIS J E. Normal-bounday intersection: A new method for generating Pareto optimal points in multicriteria optimization problems[J]. SIAM Journal on Optimization, 1998, 8(3): 631-657. doi: 10.1137/S1052623496307510
    [22] DEB K, THIELE L, LAUMANNS M, et al. Scalable test problems for evolutionary multiobjective optimization[M]// Evolutionary Multiobjective Optimization. London: Springer, 2005: 105-145.
    [23] JAIN H, DEB K. An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, Part II: Handling constraints and extending to an adaptive approach[J]. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 602-622. doi: 10.1109/TEVC.2013.2281534
    [24] LIU H, GU F, ZHANG Q F. Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems[J]. IEEE Transactions on Evolutionary Computation, 2014, 18(3): 450-455. doi: 10.1109/TEVC.2013.2281533
    [25] CORNE D W, KNOWLES J D, OATES M J. The Pareto envelope-based selection algorithm for multiobjective optimization[C]//International Conference on Parallel Problem Solving from Nature. Berlin, Heidelberg: Springer, 2000: 839-848.
  • 加载中
图(5) / 表(2)
计量
  • 文章访问数:  50
  • HTML全文浏览量:  33
  • PDF下载量:  2
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-04-27
  • 网络出版日期:  2021-07-27

目录

    /

    返回文章
    返回