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模型辅助的计算费时进化高维多目标优化

孙超利 李贞 金耀初

孙超利, 李贞, 金耀初. 模型辅助的计算费时进化高维多目标优化. 自动化学报, 2021, x(x): 1−10 doi: 10.16383/j.aas.c200969
引用本文: 孙超利, 李贞, 金耀初. 模型辅助的计算费时进化高维多目标优化. 自动化学报, 2021, x(x): 1−10 doi: 10.16383/j.aas.c200969
Sun Chao-Li, LI Zhen, JIN Yao-Chu. Surrogate-assisted expensive evolutionary many-objective optimization. Acta Automatica Sinica, 2021, x(x): 1−10 doi: 10.16383/j.aas.c200969
Citation: Sun Chao-Li, LI Zhen, JIN Yao-Chu. Surrogate-assisted expensive evolutionary many-objective optimization. Acta Automatica Sinica, 2021, x(x): 1−10 doi: 10.16383/j.aas.c200969

模型辅助的计算费时进化高维多目标优化

doi: 10.16383/j.aas.c200969
基金项目: 国家自然基金(61876123), 山西省自然科学基金(201901D111262, 201901D111264)
详细信息
    作者简介:

    孙超利:教授, 太原科技大学博士生导师. 主要研究方向为计算智能, 机器学习. E-mail: chaoli.sun@tyust.edu.cn

    李贞:太原科技大学硕士研究生. 主要研究为方向计算智能, 机器学习. E-mail: s20180522@stu.tyust.edu.cn

    金耀初:教授, 英国萨里大学博士生导师. 主要研究方向为计算智能、机器学习、计算生物学和计算神经科学等交叉学科的理论研究和工程应用. 本文通信作者. E-mail: yaochu.jin@surrey.ac.uk

Surrogate-assisted Expensive Evolutionary Many-objective Optimization

Funds: This work was supported in part by National Natural Science Foundation of China (61876123), Natural Science Foundation of Shanxi Province (201901D111262, 201901D111264)
More Information
    Author Bio:

    SUN Chao-Li Professor, Doctoral tutor of Taiyuan University of Science and Technology. Her main research interests are computational intelligence and machine learning

    LI Zhen Postgraduate student at Taiyuan University of Science and Technology. Her research interests include computational intelligence and machine learning

    JIN Yao-Chu Professor, Doctoral tutor of University of Surrey, UK. His research interests lie in interdisciplinary areas that bridge the gap between computational intelligence and machine learning, computational neuroscience, and computational biology. Corresponding author of this paper

  • 摘要: 代理模型能够辅助进化算法在计算资源有限的情况下加快找到问题的最优解集, 因此建立高效的代理模型辅助多目标进化搜索逐渐受到了人们的重视. 然而, 随着目标数量的增加, 对每个目标分别建立高斯过程模型时个体整体估值的不确定度会随之增加. 因此, 本文通过对模型最优解集的搜索探索原问题潜在的非支配解集, 并基于个体的收敛性, 种群的多样性和估值的不确定度, 提出了一种新的期望提高计算方法, 用于辅助从潜在的非支配解集中选择使用真实目标函数计算的个体, 从而更新代理模型, 使其能够在有限的计算资源下更有效地辅助优化算法找到好的非支配解集. 在7个DTLZ 基准测试问题上的实验对比结果表明, 本文算法在求解计算费时高维多目标优化问题上是有效的, 且具有较强的竞争力.
  • 图  1  不同模型评价次数下算法的性能结果对比图

    Fig.  1  Performance comparison of the proposed method with different number of evaluations on surrogate model.

    图  2  不同算法在DTLZ1上的性能结果对比

    Fig.  2  Performance comparison of different methods on three-objective DTLZ1 problem.

    表  1  SAExp-EMO和ParEGO在3个和4个目标函数的DTLZ测试问题上获得的平均IGD统计结果, 其中最好的结果以粗体表示.

    Table  1  Average IGD statistical results of SAExp-EMO and ParEGO on DTLZ test problems of 3 and 4 objective functions, with the best results shown in bold.

    Problem objs ParEGO SAExp-EMO
    DTLZ1 3 $4.84\times 10^{1}(8.51\times 10^{0})$− $\bf{1.09\times10^{1}(4.16\times 10^{0})}$
    4 $5.49\times10^{1}(1.09\times10^{1})$− $\bf{1.25\times10^{1}(5.06\times10^{1})}$
    DTLZ2 3 $4.76\times10^{-1}(3.63\times10^{-2})$− $\bf{1.72\times10^{-1}(4.60\times10^{-2})} $
    4 $5.77\times10^{-1}(2.98\times10^{-2})$− $\bf{3.65\times10^{-1}(4.40\times10^{-1})}$
    DTLZ3 3 $4.61\times10^{0}(5.38\times10^{1})$− $\bf{1.65\times10^{2}(6.07\times10^{1})} $
    4 $4.45\times10^{2}(7.57\times10^{1})$− $\bf{2.40\times10^{2}(1.19\times10^{2})}$
    DTLZ4 3 $7.80\times10^{-1}(7.38\times10^{-2})$$\approx$ $\bf{5.59\times10^{-1}(4.80\times10^{-2})}$
    4 $8.92\times10^{-1}(9.00\times10^{-1})$− $\bf{7.12\times10^{-1}(1.48\times10^{\rm{-}1})} $
    DTLZ5 3 $3.74\times10^{-1}(7.78\times10^{-2})$− $\bf{4.01\times10^{-2}(7.00\times10^{-2})}$
    4 $4.09\times10^{-1}(4.52\times10^{-2})$− $\bf{7.80\times10^{-2}(0.00\times10^{0})}$
    DTLZ6 3 $8.04\times10^{0}(2.44\times10^{-1})$− $\bf{3.63\times10^{0}(2.61\times10^{0})}$
    4 $8.16\times10^{0}(2.52\times10^{-1})$− $\bf{3.84\times10^{0}(4.61\times10^{-1})}$
    DTLZ7 3 $7.28\times10^{0}(2.16\times10^{0})$− $\bf{7.70\times10^{-1}(1.35\times10^{-1})}$
    4 $ 1.11\times10^{1}(7.97\times10^{-1})$− $\bf{1.09\times10^{0}(3.18\times10^{-1})}$
    +/−/≈ 0/13/1
    下载: 导出CSV

    表  2  SAExp-EMO、RVEA、K-RVEA和CSEA得到的平均IGD值, 其中最好的结果以粗体表示.

    Table  2  Average IGD values obtained by SAExp-EMO, RVEA, K-RVEA and CSEA, with the best results shown in bold

    problem objective RVEA K-RVEA CSEA SAExp-EMO
    DTLZ1 3 $3.65\times10^{1}(1.10\times10^{1})$− $2.48\times10^{1}(8.56\times10^{0})$− $1.97\times10^{1}(5.82\times10^{0})$− $\bf{1.33\times10^{1}(4.53\times10^{0})} $
    4 $3.18\times10^{1}(1.03\times10^{1})$− $3.01\times10^{1}(1.18\times10^{1})$− $1.71\times10^{1}(5.31\times10^{0})$− $\bf{1.35\times10^{1}(5.03\times10^{0})} $
    6 $2.96\times10^{1}(8.16\times10^{0})$− $3.18\times10^{1}(6.94\times10^{0})$− $1.43\times10^{1}(6.68\times10^{0})$− $\bf{1.15\times10^{1}(6.29\times10^{0})}$
    8 $2.00\times10^{1}(9.31\times10^{0})$− $3.22\times10^{1}(1.12\times10^{1})$$\approx$ $1.44\times10^{1}(6.01\times10^{0})$− $\bf{1.17\times10^{1}(4.46\times10^{0})} $
    10 $2.15\times10^{1}(8.45\times10^{0})$− $2.48\times10^{1}(9.28\times10^{0})$− $1.45\times10^{1}(5.70\times10^{0})$− $\bf{1.28\times10^{1}(5.45\times10^{0})}$
    DTLZ2 3 $4.09\times10^{-1}(3.22\times10^{-2})$− $2.66\times10^{-1}(4.88\times10^{-2})$− $2.69\times10^{-1}(1.13\times10^{-1})$− $\bf{1.38\times10^{-1}(6.13\times10^{-2} )}$
    4 $5.16\times10^{-1}(3.61\times10^{-2})$− $3.95\times10^{-1}(4.94\times10^{-2})$− $4.76\times10^{-1}(1.04\times10^{-1})$− $\bf{3.27\times10^{-1}(5.53\times10^{-2})}$
    6 $6.97\times10^{-1}(6.73\times10^{-2})$− $5.93\times10^{-1}(4.96\times10^{-2})$− $\bf{5.76\times10^{-1}(4.01\times10^{-2})}$$\approx$ ${6.15\times10^{-1}(4.93\times10^{-2})}$
    8 $7.93\times10^{-1}(3.69\times10^{-2})$− $6.54\times10^{-1}(4.95\times10^{-2})$− $7.57\times10^{-1}(3.52\times10^{-2})$− $\bf{5.45\times10^{-1}(2.42\times10^{-1})}$
    10 $9.54\times10^{-1}(5.16\times10^{-2})$− $7.36\times10^{-1}(4.59\times10^{-2})$− $8.44\times10^{-1}(5.65\times10^{-2})$− $\bf{6.08\times10^{-1}(3.09\times10^{-1})}$
    DTLZ3 3 $4.18\times10^{2}(6.66\times10^{1})$− $3.38\times10^{2}(7.51\times10^{1})$− $2.12\times10^{2}(4.37\times10^{1})$− $\bf{1.13\times10^{2}(2.96\times10^{1})}$
    4 $4.17\times10^{2}(7.54\times10^{1})$− $3.56\times10^{2}(7.56\times10^{1})$− $2.17\times10^{2}(4.94\times10^{1})$− $\bf{1.26\times10^{2}(6.16\times10^{1})}$
    6 $3.85\times10^{2}(7.05\times10^{1})$− $3.45\times10^{2}(7.90\times10^{1})$− $2.09\times10^{2}(5.44\times10^{1})$− $\bf{1.46\times10^{2}(7.89\times10^{1})}$
    8 $3.57\times10^{2}(7.05\times10^{1})$− $3.38\times10^{2}(5.74\times10^{1})$− $2.08\times10^{2}(5.09\times10^{1})$− $\bf{1.49\times10^{2}(7.88\times10^{1} )}$
    10 $3.77\times10^{2}(1.02\times10^{2})$− $3.24\times10^{2}(7.92\times10^{1})$− $2.18\times10^{2}(5.85\times10^{1})$ $\approx$ $\bf{1.10\times10^{2}(2.87\times10^{1})}$
    DTLZ4 3 $5.58\times10^{-1}(6.90\times10^{-2})$− $\bf{4.17\times10^{-1}(1.12\times10^{-1})}$ $\approx$ $7.22\times10^{-1}(1.53\times10^{-1})$− $4.81\times10^{-1}(1.40\times10^{-1})$
    4 $6.96\times10^{-1}(8.80\times10^{-2})$ $\approx$ $5.46\times10^{-1}(1.13\times10^{-1})$+ $\bf{5.43\times10^{-1}(1.02\times10^{-1})}$+ $6.64\times10^{-1}(1.45\times10^{-1})$
    6 $8.53\times10^{-1}(8.13\times10^{-2})$− $6.84\times10^{-1}(8.64\times10^{-2})$+ $\bf{5.74\times10^{-1}(1.01\times10^{-1})}$ $\approx$ $8.52\times10^{-1}(8.99\times10^{-2})$
    8 $9.32\times10^{-1}(7.75\times10^{-2})$− $8.34\times10^{-1}(9.14\times10^{-2})$+ $\bf{7.39\times10^{-1}(3.42\times10^{-2})}$+ $8.36\times10^{-1}(1.76\times10^{-1})$
    10 $1.03\times10^{0}(7.03\times10^{-2})$− $8.89\times10^{-1}(6.96\times10^{-2})$ $\approx$ $\bf{8.12\times10^{-1}(4.54\times10^{-2})}$+ $8.17\times10^{-1}(2.43\times10^{-1})$
    DTLZ5 3 $3.45\times10^{-1}(4.41\times10^{-2})$− $1.81\times10^{-1}(4.44\times10^{-2})$− $1.46\times10^{-1}(4.29\times10^{-2})$− $\bf{3.84\times10^{-2}(8.64\times10^{-3})}$
    4 $3.79\times10^{-1}(7.42\times10^{-2})$− $1.90\times10^{-1}(3.12\times10^{-2})$− $2.00\times10^{-1}(4.29\times10^{-2})$− $\bf{6.98\times10^{-2}(1.43\times10^{-2})}$
    6 $4.28\times10^{-1}(6.52\times10^{-2})$− $2.29\times10^{-1}(3.40\times10^{-1})$ $\approx$ $2.17\times10^{-1}(7.87\times10^{-1})$− $\bf{1.30\times10^{-1}(4.04\times10^{-2})}$
    8 $4.26\times10^{-1}(5.83\times10^{-2})$− $2.19\times10^{-1}(4.87\times10^{-2})$− $2.43\times10^{-1}(6.08\times10^{-2})$− $\bf{8.31\times10^{-2}(2.32\times10^{-2})}$
    10 $4.06\times10^{-1}(1.02\times10^{-1})$− $2.23\times10^{-1}(5.87\times10^{-2})$ $\approx$ $2.54\times10^{-1}(5.35\times10^{-2})$− $\bf{9.97\times10^{-2}(4.07\times10^{-2})} $
    DTLZ6 3 $7.94\times10^{0}(2.75\times10^{-1})$− $4.42\times10^{0}(5.40\times10^{-1})$− $4.54\times10^{0}(5.84\times10^{-1})$− $\bf{3.07\times10^{0}(7.11\times10^{-1})}$
    4 $8.02\times10^{0}(2.61\times10^{-1})$− $4.35\times10^{0}(4.66\times10^{-1})$− $6.99\times10^{0}(7.87\times10^{-1})$− $\bf{3.46\times10^{0}(4.82\times10^{-1})}$
    6 $8.19\times10^{0}(3.42\times10^{-1})$− $4.58\times10^{0}(7.79\times10^{-1})$− $7.11\times10^{0}(1.76\times10^{-1})$− $\bf{4.19\times10^{0}(6.93\times10^{-1})} $
    8 $8.18\times10^{0}(2.75\times10^{-1})$− $5.78\times10^{0}(4.49\times10^{-1})$− ${7.21\times10^{0}(5.28\times10^{-1})}$$\approx$ $\bf{3.60\times10^{0}(5.11\times10^{-1})}$
    10 $8.22\times10^{0}(4.07\times10^{-1})$− $6.32e\times10^{0}(6.35\times10^{-1})$ $\approx$ $7.44\times10^{0}(4.18\times10^{-1})$ $\approx$ $\bf{3.95\times10^{0}(1.16\times10^{0})}$
    DTLZ7 3 $6.85\times10^{0}(7.29\times10^{-1})$− $1.15\times10^{0}(1.69\times10^{0})$− $4.02\times10^{0}(4.82\times10^{0})$− $\bf{5.36\times10^{-1}(2.26\times10^{-1})}$
    4 $8.81\times10^{0}(1.31\times10^{0})$− $2.14\times10^{0}(3.24\times10^{0})$ $\approx$ $7.54\times10^{0}(9.33\times10^{-1})$ $\approx$ $\bf{6.92\times10^{-1}(1.37\times10^{-1})}$
    6 $1.29\times10^{1}(1.44\times10^{0})$− $3.49\times10^{0}(2.76\times10^{0})$− $1.36\times10^{1}(1.65\times10^{0})$− $ \bf{1.13\times10^{0}(2.42\times10^{-1})}$
    8 $1.72\times10^{1}(2.18\times10^{0})$− $4.18\times10^{0}(2.18\times10^{0})$− $2.26\times10^{1}(2.27\times10^{0})$− $\bf{6.82\times10^{-1}(1.38\times10^{-1})}$
    10 $2.18\times10^{1}(3.56\times10^{0})$− $7.83\times10^{0}(3.32\times10^{0})$− $2.86\times10^{1}(2.30\times10^{0})$− $\bf{1.19\times10^{0}(5.68\times10^{-1})} $
    +/−/≈ 0/34/1 3/25/7 3/26/6
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-11-22
  • 录用日期:  2021-03-19
  • 网络出版日期:  2021-07-28

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