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                    模型輔助的計算費時進化高維多目標優化

                    孫超利 李貞 金耀初

                    孫超利, 李貞, 金耀初. 模型輔助的計算費時進化高維多目標優化. 自動化學報, 2021, x(x): 1?10 doi: 10.16383/j.aas.c200969
                    引用本文: 孫超利, 李貞, 金耀初. 模型輔助的計算費時進化高維多目標優化. 自動化學報, 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

                    模型輔助的計算費時進化高維多目標優化

                    doi: 10.16383/j.aas.c200969
                    基金項目: 國家自然基金(61876123), 山西省自然科學基金(201901D111262, 201901D111264)
                    詳細信息
                      作者簡介:

                      孫超利:教授, 太原科技大學博士生導師. 主要研究方向為計算智能, 機器學習. E-mail: chaoli.sun@tyust.edu.cn

                      李貞:太原科技大學碩士研究生. 主要研究為方向計算智能, 機器學習. E-mail: s20180522@stu.tyust.edu.cn

                      金耀初:教授, 英國薩里大學博士生導師. 主要研究方向為計算智能、機器學習、計算生物學和計算神經科學等交叉學科的理論研究和工程應用. 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: 代理模型能夠輔助進化算法在計算資源有限的情況下加快找到問題的最優解集, 因此建立高效的代理模型輔助多目標進化搜索逐漸受到了人們的重視. 然而, 隨著目標數量的增加, 對每個目標分別建立高斯過程模型時個體整體估值的不確定度會隨之增加. 因此, 本文通過對模型最優解集的搜索探索原問題潛在的非支配解集, 并基于個體的收斂性, 種群的多樣性和估值的不確定度, 提出了一種新的期望提高計算方法, 用于輔助從潛在的非支配解集中選擇使用真實目標函數計算的個體, 從而更新代理模型, 使其能夠在有限的計算資源下更有效地輔助優化算法找到好的非支配解集. 在7個DTLZ 基準測試問題上的實驗對比結果表明, 本文算法在求解計算費時高維多目標優化問題上是有效的, 且具有較強的競爭力.
                    • 圖  1  不同模型評價次數下算法的性能結果對比圖

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

                      圖  2  不同算法在DTLZ1上的性能結果對比

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

                      表  1  SAExp-EMO和ParEGO在3個和4個目標函數的DTLZ測試問題上獲得的平均IGD統計結果, 其中最好的結果以粗體表示.

                      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
                      下載: 導出CSV

                      表  2  SAExp-EMO、RVEA、K-RVEA和CSEA得到的平均IGD值, 其中最好的結果以粗體表示.

                      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
                      下載: 導出CSV
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                    出版歷程
                    • 收稿日期:  2020-11-22
                    • 錄用日期:  2021-03-19
                    • 網絡出版日期:  2021-07-28

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