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灰狼与郊狼混合优化算法及其聚类优化

张新明 姜云 刘尚旺 刘国奇 窦智 刘艳

张新明, 姜云, 刘尚旺, 刘国奇, 窦智, 刘艳. 灰狼与郊狼混合优化算法及其聚类优化. 自动化学报, 2020, 46(x): 1−20 doi: 10.16383/j.aas.c190617
引用本文: 张新明, 姜云, 刘尚旺, 刘国奇, 窦智, 刘艳. 灰狼与郊狼混合优化算法及其聚类优化. 自动化学报, 2020, 46(x): 1−20 doi: 10.16383/j.aas.c190617
Zhang Xin-Ming, Jiang Yun, Liu Shang-Wang, Liu Guo-Qi, Dou Zhi, Liu Yan. Hybrid coyote optimization algorithm with grey wolf optimizer and its application to clustering optimization. Acta Automatica Sinica, 2020, 46(x): 1−20 doi: 10.16383/j.aas.c190617
Citation: Zhang Xin-Ming, Jiang Yun, Liu Shang-Wang, Liu Guo-Qi, Dou Zhi, Liu Yan. Hybrid coyote optimization algorithm with grey wolf optimizer and its application to clustering optimization. Acta Automatica Sinica, 2020, 46(x): 1−20 doi: 10.16383/j.aas.c190617

灰狼与郊狼混合优化算法及其聚类优化

doi: 10.16383/j.aas.c190617
基金项目: 国家自然科学基金(61901160, U1904123), 河南省高等学校重点科研项目(19A520026)资助
详细信息
    作者简介:

    张新明:河南师范大学教授. 主要研究领域为智能优化算法、图像去噪、图像增强和图像分割. 本文通信作者.Email: xinmingzhang@126.com

    姜云:河南师范大学研究生. 主要研究领域为智能优化算法和图像分割.Email: JiangYun951120@163.com

    刘尚旺:河南师范大学副教授. 主要研究领域为图像处理和计算机视觉.Email: shwl08@126.com

    刘国奇:河南师范大学副教授. 主要研究领域为图像分割和偏微分方程.Email: liuguoqi080408@163.com

    窦智:河南师范大学讲师. 主要研究领域为算法及数字图像处理.Email: 619534345@163.com

    刘艳:河南师范大学实验师. 主要研究领域为优化算法和图像分割. Email: Liu_yan122@sina.com

Hybrid coyote optimization algorithm with grey wolf optimizer and its application to clustering optimization

Funds: Supported by National Natural Science Foundation of China (61901160, U1904123), Key Research Project of Higher Education Institutions of Henan Province, China (19A520026)
  • 摘要: 郊狼优化算法(Coyote Optimization Algorithm, COA)是最近提出的一种新颖且具有较大应用潜力的群智能优化算法, 具有独特的搜索机制和能较好解决全局优化问题等优势, 但在处理复杂优化问题时存在搜索效率低、可操作性差和收敛速度慢等不足. 为弥补其不足, 并借鉴灰狼优化算法(Grey Wolf Optimizer, GWO)的优势, 提出了一种COA与GWO的混合算法(Hybrid COA with GWO, HCOAG). 首先提出了一种改进的COA (Improved COA, ICOA), 即将一种高斯全局趋优成长算子替换原算法的成长算子以提高搜索效率和收敛速度, 并提出一种动态调整组内郊狼数方案, 使得算法的搜索能力和可操作性都得到增强; 然后提出了一种简化操作的GWO (Simplified GWO, SGWO), 以提高算法的可操作性和降低其计算复杂度; 最后采用正弦交叉策略将ICOA与SGWO二者融合进一步提高算法的优化性能. 大量的经典函数和CEC2017复杂函数优化以及K-Means聚类优化的实验结果表明, 与COA相比, HCOAG具有更高的搜索效率、更强的可操作性和更快的收敛速度, 与其他先进的对比算法相比, HCOAG具有更好的优化性能, 能更好地解决聚类优化问题.
  • 图  1  GWO的流程图

    Fig.  1  Flow chart of GWO

    图  2  组数Np与组内郊狼数Nc参数的分配图

    Fig.  2  Disposition graph of two parameters Nc and Np

    图  3  GWO与SGWO的等级情况对比

    Fig.  3  Comparison of hierarchies of GWO and SGWO

    图  4  HCOAG流程图

    Fig.  4  Flow chart of HCOAG

    图  5  HCOAG与对比算法在6个经典函数上的收敛图

    Fig.  5  Convergence curves of HCOAG and the comparison algorithms on the 6 classical benchmark functions

    图  6  HCOAG、COA、MEGWO、DEBBO、TLBO和HFPSO的收敛图

    Fig.  6  Convergence curves of HCOAG, COA, MEGWO, DEBBO, TLBO and HFPSO

    图  7  HCOAG与COA、GWO在不同类别函数上的平均时间对比图

    Fig.  7  Comparison bars of average time of HCOAG, COA and GWO on different kinds of functions

    表  1  HCOAG与其不完全算法的结果对比

    Table  1  Comparison results of HCOAG and its incomplete algorithms

    函数 HCOAG COA GWO HCOAG5 HCOAG10 ICOA SGWO
    F1 Mean 7.4494×10−4 1.2099×103 1.2813×109 4.1072×10−4 1.9800×10−3 1.0737×102 3.3279×103
    Std 1.4801×10−3 1.2998×103 9.6388×108 8.5916×10−4 2.9438×10−3 1.0569×102 4.3271×103
    Rank 2 5 7 1 3 4 6
    F2 Mean 1.1941×101 2.9013×1021 3.1831×1032 4.8580×103 1.8078×101 8.6764×1015 3.3582×1014
    Std 2.4077×101 1.1462×1022 1.5894×1033 3.3985×104 5.0208×101 3.5675×1016 1.3200×1015
    Rank 1 6 7 3 2 5 4
    F3 Mean 9.5410×10−1 6.0573×104 2.8342×104 7.6972×10−1 1.0995×100 3.3032×104 8.7276×102
    Std 1.9288×100 1.0177×104 9.2323×103 9.8032×10−1 1.6794×100 6.8409×103 7.2376×102
    Rank 2 7 5 1 3 6 4
    F4 Mean 1.8113×101 8.4041×101 2.0825×102 2.0841×101 2.8446×101 4.8248×101 1.0495×102
    Std 2.7696×101 8.5306×100 8.4445×101 3.0540×101 3.1826×101 3.3517×101 2.4806×101
    Rank 1 5 7 2 3 4 6
    F5 Mean 2.8433×101 5.2890×101 9.6116×101 3.5884×101 3.0204×101 3.4844×101 3.1488×101
    Std 6.8886×100 1.5025×101 3.2690×101 1.0115×101 8.8983×100 1.0983×101 9.2242×100
    Rank 1 6 7 5 2 4 3
    F6 Mean 1.7483×10−7 1.6399×10−5 6.3664×100 9.4452×10−7 1.5005×10−6 2.8782×10−4 2.0381×10−2
    Std 4.7524×10−7 9.6428×10−6 3.1596×100 2.4080×10−6 5.9643×10−6 1.6254×10−4 2.7102×10−2
    Rank 1 4 7 2 3 5 6
    F7 Mean 6.1055×101 7.5148×101 1.4460×102 6.7082×101 5.9300×101 6.7675×101 6.4025×101
    Std 1.0851×101 1.3762×101 4.6314×101 1.1241×101 9.4998×100 1.1520×101 1.1856×101
    Rank 2 6 7 4 1 5 3
    F8 Mean 3.2489×101 5.6110×101 8.4662×101 3.6085×101 2.9446×101 3.6138×101 3.1775×101
    Std 1.2272×101 1.8774×101 2.5270×101 8.8063×100 7.9048×100 1.0081×101 7.8884×100
    Rank 3 6 7 4 1 5 2
    F9 Mean 2.7362×10−1 5.6225×10−1 5.5392×102 5.2270×10−1 2.5931×10−1 8.8559×10−2 7.4159×100
    Std 4.8298×10−1 1.0209×100 3.2695×102 8.7374×10−1 4.6957×10−1 1.5656×10−1 6.7112×100
    Rank 3 5 7 4 2 1 6
    F10 Mean 2.2671×103 2.7575×103 3.1862×103 2.5574×103 2.3435×103 2.1380×103 2.1424×103
    Std 6.1427×102 4.6685×102 9.7886×102 5.3524×102 6.0670×102 5.6098×102 4.0691×102
    Rank 3 6 7 5 4 1 2
    F11 Mean 2.1678×101 4.1143×101 4.9771×102 2.9822×101 2.6698×101 2.2685×101 1.0908×102
    Std 2.0907×101 2.7367×101 6.4235×102 2.6128×101 2.4059×101 2.0893×101 3.8218×101
    Rank 1 5 7 4 3 2 6
    F12 Mean 9.8943×103 1.2532×105 4.0285×107 1.2577×104 1.0657×104 1.3660×105 2.0716×105
    Std 6.0932×103 1.2555×105 7.3849×107 6.4424×103 6.4606×103 9.3092×104 1.9967×105
    F12 Rank 1 4 7 3 2 5 6
    F13 Mean 1.9749×103 2.0357×104 2.8073×106 3.0265×103 3.1829×103 3.6293×102 1.3271×104
    Std 3.8565×103 2.6333×104 1.6225×107 6.2901×103 8.0719×103 8.9975×101 1.5283×104
    Rank 2 6 7 3 4 1 5
    F14 Mean 8.6436×101 8.0070×101 1.3112×105 7.7150×101 1.0134×102 5.6726×101 1.4132×104
    Std 4.3766×101 1.9915×101 2.3335×105 6.0071×101 9.2585×101 1.4850×101 1.7944×104
    Rank 4 3 7 2 5 1 6
    F15 Mean 1.8396×103 2.0792×103 3.3658×105 7.1579×102 1.7386×103 6.9111×101 6.6116×103
    Std 2.9044×103 7.9984×103 7.9125×105 1.2272×103 2.9477×103 1.9083×101 8.3961×103
    Rank 4 5 7 2 3 1 6
    F16 Mean 3.0243×102 7.9869×102 8.1416×102 3.3715×102 3.0883×102 4.6416×102 5.0252×102
    Std 2.0550×102 2.8651×102 2.6440×102 2.1415×102 1.8878×102 2.6962×102 2.4545×102
    Rank 1 6 7 3 2 4 5
    F17 Mean 4.7111×101 2.2439×102 2.7004×102 6.4809×101 5.3120×101 3.7365×101 1.3984×102
    Std 4.0925×101 1.3518×102 1.3820×102 5.3991×101 4.7801×101 4.0654×101 8.0664×101
    Rank 2 6 7 4 3 1 5
    F18 Mean 6.1013×104 6.9910×104 7.1643×105 5.1875×104 5.0376×104 3.9930×104 1.8454×105
    Std 5.7031×104 1.0210×105 8.2799×105 3.6270×104 3.7592×104 2.0034×104 1.7045×105
    Rank 4 5 7 3 2 1 6
    F19 Mean 3.4042×101 4.4886×103 4.6400×105 3.4163×101 3.0105×102 2.4678×101 5.4815×103
    Std 2.0528×101 1.3325×104 5.4998×105 3.9687×101 1.1322×103 7.1259×100 4.9859×103
    Rank 2 5 7 3 4 1 6
    F20 Mean 9.6665×101 2.4290×102 3.6059×102 1.0084×102 1.1637×102 1.0389×102 2.0165×102
    Std 7.7834×101 1.4995×102 1.0264×102 6.6726×101 7.5890×101 8.7694×101 9.6673×101
    Rank 1 6 7 2 4 3 5
    F21 Mean 2.3023×102 2.5626×102 2.8298×102 2.3713×102 2.3135×102 2.3724×102 2.3289×102
    Std 8.5095×100 1.6800×101 2.5684×101 1.0815×101 9.8453×100 1.1261×101 9.9428×100
    Rank 1 6 7 4 2 5 3
    F22 Mean 1.0010×102 1.9999×103 8.0434×102 1.0005×102 1.0005×102 1.0005×102 1.2974×102
    Std 4.8096×10−1 1.5970×103 1.1113×103 3.4354×10−1 3.4444×10−1 3.4443×10−1 2.0703×102
    Rank 4 7 6 1 3 2 5
    F23 Mean 3.7755×102 4.1635×102 4.7029×102 3.8706×102 3.7831×102 3.8441×102 3.8854×102
    Std 1.0911×101 1.6898×101 2.9324×101 1.3271×101 8.1817×100 1.0543×101 1.3692×101
    Rank 1 6 7 4 2 3 5
    F24 Mean 4.4827×102 5.4044×102 5.2489×102 4.5484×102 4.4530×102 4.5862×102 4.5671×102
    Std 1.0757×101 4.5778×101 3.3902×101 1.1783×101 1.1461×101 1.2203×101 1.1956×101
    Rank 2 7 6 3 1 5 4
    F25 Mean 3.8777×102 3.8706×102 4.7784×102 3.8747×102 3.8729×102 3.8698×102 4.0239×102
    Std 5.4462×100 8.0647×10−1 2.3819×101 1.5625×100 1.2735×100 5.4095×10−1 1.5135×101
    Rank 5 2 7 4 3 1 6
    F26 Mean 1.2578×103 1.6520×103 2.0116×103 1.3249×103 1.2449×103 1.3138×103 1.5024×103
    Std 1.9807×102 1.7070×102 5.7618×102 3.1093×102 3.4947×102 1.9599×102 2.8691×102
    Rank 2 6 7 4 1 3 5
    F27 Mean 5.1091×102 5.0430×102 5.9279×102 5.1349×102 5.1088×102 5.0560×102 5.3331×102
    Std 7.7116×100 8.2707×100 3.8462×101 8.6401×100 8.6837×100 7.3827×100 1.1175×101
    F27 Rank 4 1 7 5 3 2 6
    F28 Mean 3.3558×102 4.0555×102 5.9941×102 3.2930×102 3.3793×102 3.4828×102 4.5710×102
    Std 5.3866×101 3.6156×101 6.9788×101 5.1585×101 5.1950×101 5.3564×101 2.3392×101
    Rank 2 5 7 1 3 4 6
    F29 Mean 4.5991×102 6.6978×102 8.5036×102 4.8821×102 4.6287×102 4.5683×102 6.2453×102
    Std 4.4075×101 1.7459×102 1.8235×102 5.5772×101 4.3839×101 4.4800×101 1.1861×102
    Rank 2 6 7 4 3 1 5
    F30 Mean 2.9823×103 6.0618×103 4.0643×106 3.1036×103 2.9323×103 1.9586×104 6.1880×103
    Std 6.1135×102 4.7022×103 3.1688×106 8.4665×102 5.9332×102 7.3086×103 2.8250×103
    Rank 2 4 7 3 1 6 5
    Count 10 1 0 4 5 10 0
    Ave Rank 2.20 5.23 6.87 3.10 2.60 3.07 4.93
    Total Rank 1 6 7 4 2 3 5
    下载: 导出CSV

    表  2  在6个经典函数上的实验结果对比

    Table  2  comparison results on the 6 classic functions

    函数 D=10
    HCOAG COA GWO HFPSO DEBBO
    f1 Mean 6.0684×10-9 1.7833×102 9.0799×10-15 3.4157×10-5 6.7086×10-2
    Std 4.8458×10-9 6.3524×101 2.4849×10-14 2.4485×10-5 3.1056×10-2
    Rank 2 5 1 3 4
    f2 Mean 8.4133×10-6 2.3737×100 1.3222×10-9 1.3703×10-3 2.8483×10-2
    Std 3.7531×10-6 4.0964×10-1 1.1382×10-9 6.5582×10-4 7.0993×10-3
    Rank 2 5 1 3 4
    f3 Mean 0 1.6180×102 1.0000×10-1 0 0
    Std 0 5.0787×101 3.0513×10-1 0 0
    Rank 1 5 4 1 1
    f4 Mean 1.2498×10-10 4.0253×100 1.5325×10-6 3.5760×10-7 1.8575×10-3
    Std 2.0501×10-10 1.6104×100 9.4192×10-7 4.1539×10-7 1.0158×10-3
    Rank 1 5 3 2 4
    f5 Mean 2.0046×10-8 7.1619×102 2.4093×10-5 4.6770×10-6 2.6132×10-2
    Std 5.9686×10-8 1.5819×103 1.4121×10-5 5.6661×10-6 1.0613×10-2
    Rank 1 5 3 2 4
    f6 Mean 4.1921×10-10 1.7228×100 1.2593×10-2 4.9377×10-7 3.5149×10-3
    Std 4.3501×10-10 5.1829×10-1 6.8779×10-2 4.8665×10-7 1.5826×10-3
    Rank 1 5 4 2 3
    D=30
    f1 Mean 1.3966×10-17 3.2554×101 5.4432×10-41 7.2595×10-9 2.7076×10-4
    Std 3.2255×10-17 5.9567×100 7.1605×10-41 7.3446×10-9 1.1010×10-4
    Rank 2 5 1 3 4
    f2 Mean 2.8862×10-10 1.3998×100 6.0158×10-24 5.3463×10-5 1.3264×10-3
    Std 4.6435×10-10 1.9835×10-1 6.2049×10-24 3.9178×10-5 2.3103×10-4
    Rank 2 5 1 3 4
    f3 Mean 0 3.3700×101 3.3333×10-2 0 0
    Std 0 7.9877×100 1.8257×10-1 0 0
    Rank 1 5 4 1 1
    f4 Mean 1.0451×10-17 3.8002×100 1.5129×10-2 1.7278×10-2 6.4886×10-5
    Std 3.0738×10-17 1.3163×100 1.0471×10-2 3.9296×10-2 2.7878×10-5
    Rank 1 5 3 4 2
    f5 Mean 5.3309×10-17 1.8376×101 1.6587×10-1 3.2962×10-3 5.0150×10-4
    Std 1.5184×10-16 5.8919×100 1.1940×10-1 5.1211×10-3 2.1237×10-4
    Rank 1 5 4 3 2
    f6 Mean 1.2484×10-18 1.8049×100 7.9738×10-1 8.2480×10-3 8.5930×10-5
    Std 1.7135×10-18 4.9152×10-1 7.4565×10-1 4.5176×10-2 3.9292×10-5
    Rank 1 5 4 3 2
    Count 8 0 4 2 2
    Ave Rank 1.33 5.00 2.75 2.50 2.92
    Total Rank 1 5 3 2 4
    下载: 导出CSV

    表  3  6个经典函数的情况

    Table  3  Details of 6 classical benchmark functions

    类型 函数名称 函数表达式 搜索范围 最小值
    单峰函数 Sphere ${f_1}(x) = \sum\nolimits_{i = 1}^D {x_i^2}\hspace{200pt} $ [-100,100] 0
    Schwefel 2.22 ${f_2}(x) = \sum\nolimits_{i = 1}^D {\left| {{x_i}} \right|} + \prod\nolimits_{i = 1}^D {\left| {{x_i}} \right|} \hspace{200pt} $ [-10,10] 0
    Step ${f_3}(x) = \sum\nolimits_{i = 1}^D {{{\left( {\left\lfloor {{x_i} + 0.5} \right\rfloor } \right)}^2}}\hspace{200pt}$ [-100,100] 0
    多峰函数 Penalized 1 $\begin{array}{l} {f_4}(x) = \dfrac{\pi}{D}\left\{ {10{ {\sin }^2}\left( {\pi {y_i} } \right)} \right. + \\ \sum\nolimits_{i = 1}^{D - 1} { { {\left( { {y_i} - 1} \right)}^2}\left[ {1 + 10{ {\sin }^2}\left( {\pi {y_{i + 1} } } \right)} \right]} \left. { + { {\left( { {y_D} - 1} \right)}^2} } \right\} + \\ \sum\nolimits_{i = 1}^D {u\left( { {x_i},10,100,4} \right),} \\ {y_i} = 1 + \dfrac{1}{4}\left( { {x_i} + 1} \right), \;\;\;\;u\left( { {x_i},a,k,m} \right) = \\ \left\{ \begin{array}{l} k{\left( { {x_i} - a} \right)^m},\;\;\;\;\;\;{x_i} > a,\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - a \le {x_i} \le a,\\ k{\left( { - {x_i} - a} \right)^m},\;\;\;{x_i} < - a. \end{array} \right. \end{array}\hspace{200pt}$ [-50,50] 0
    Penalized 2 $\begin{array}{l} {f_5}(x) = 0.1\left\{ {{{\sin }^2}} \right.\left( {\pi {x_1}} \right) + \sum\nolimits_{i = 1}^{D - 1} {{{\left( {{x_i} - 1} \right)}^2}} \left[ {1 + {{\sin }^2}\left( {3\pi {x_{i + 1}}} \right)} \right] + \\ \left( {{x_D} - 1} \right)\left. {\left[ {1 + {{\sin }^2}\left( {2\pi {x_D}} \right)} \right]} \right\} + \sum\nolimits_{i = 1}^D {u\left( {{x_i},5,100,4} \right)} \end{array}\hspace{200pt}$ [-50,50] 0
    Levy $\begin{array}{l} {f_6}(x) = \sum\nolimits_{i = 1}^{D - 1} {{{\left( {{x_i} - 1} \right)}^2}} \left[ {1 + {{\sin }^2}\left( {3\pi {x_{i + 1}}} \right)} \right] + \\ {\sin ^2}\left( {3\pi {x_1}} \right) + \left| {{x_D} - 1} \right|\left[ {1 + {{\sin }^2}\left( {3\pi {x_D}} \right)} \right] \end{array}\hspace{200pt}$ [-10,10] 0
    下载: 导出CSV

    表  4  在30维CEC2017复杂函数上的优化结果对比

    Table  4  Comparison results on the 30-dimensional complex functions from CEC2017

    函数 HCOAG COA GWO MEGWO HFPSO DEBBO SaDE SE04 FWA TLBO
    F1 Mean 7.4494×10−4 1.2099×103 1.2813×109 4.5517×103 3.9338×103 2.7849×103 3.0714×103 3.2930×103 4.3987×106 2.9846×103
    Std 1.4801×10−3 1.2998×103 9.6388×108 1.0677×103 5.3689×103 4.0364×103 3.5072×103 4.2328×103 1.4055×106 3.1471×103
    Rank 1 2 10 8 7 3 5 6 9 4
    F2 Mean 1.1941×101 2.9013×1021 3.1831×1032 2.8884×108 5.3485×104 1.5057×1017 8.6275×10−1 3.0802×1013 4.1397×1015 1.0448×1016
    Std 2.4077×101 1.1462×1022 1.5894×1033 8.0571×108 3.2847×105 3.5179×1017 4.9357×100 1.1694×1014 1.5680×1016 4.7082×1016
    Rank 2 9 10 4 3 8 1 5 6 7
    F3 Mean 9.5410×10−1 6.0573×104 2.8342×104 2.2633×102 1.5595×10−7 3.6772×104 3.0045×102 9.7974×103 2.4748×104 4.0488×10−4
    Std 1.9288×100 1.0177×104 9.2323×103 1.7031×102 2.3334×10−7 5.8394×103 7.3017×102 3.4377×103 6.3467×103 1.6647×10−3
    Rank 3 10 8 4 1 9 5 6 7 2
    F4 Mean 1.8113×101 8.4041×101 2.0825×102 2.4815×101 6.9386×101 8.4851×101 6.0423×101 8.5881×101 1.1370×102 5.9054×101
    Std 2.7696×101 8.5306×100 8.4445×101 2.8995×101 2.1364×101 2.2848×10−1 2.9825×101 1.1251×101 1.7315×101 3.0429×101
    Rank 1 6 10 2 5 7 4 8 9 3
    F5 Mean 2.8433×101 5.2890×101 9.6116×101 5.6912×101 8.5624×101 5.8216×101 5.6192×101 4.1688×101 1.8456×102 8.5717×101
    Std 6.8886×100 1.5025×101 3.2690×101 1.0725×101 1.7427×101 6.5957×100 1.4216×101 8.1545×100 3.3933×101 1.8601×101
    Rank 1 3 9 5 7 6 4 2 10 8
    F6 Mean 1.7483×10−7 1.6399×10−5 6.3664×100 2.4470×10−1 1.0170×100 1.1369×10−13 8.9317×10−2 7.5481×10−6 5.1770×100 7.2903×100
    Std 4.7524×10−7 9.6428×10−6 3.1596×100 8.1620×10−2 2.3644×100 0 1.3955×10−1 3.9880×10−5 3.1351×100 4.4538×100
    Rank 2 4 9 6 7 1 5 3 8 10
    F7 Mean 6.1055×101 7.5148×101 1.4460×102 8.9106×101 1.0407×102 9.9725×101 9.4945×101 7.2448×101 2.0998×102 1.3661×102
    Std 1.0851×101 1.3762×101 4.6314×101 1.0935×101 1.9791×101 6.4285×100 1.9879×101 7.3495×100 4.4817×101 2.4846×101
    Rank 1 3 9 4 7 6 5 2 10 8
    F8 Mean 3.2489×101 5.6110×101 8.4662×101 5.9398×101 7.2842×101 5.9299×101 5.3942×101 4.4194×101 1.4508×102 7.1339×101
    Std 1.2272×101 1.8774×101 2.5270×101 1.0663×101 1.7967×101 6.0788×100 1.2792×101 6.5834×100 2.1470×101 1.4852×101
    Rank 1 4 9 6 8 5 3 2 10 7
    F9 Mean 2.7362×10−1 5.6225×10−1 5.5392×102 7.8267×100 3.2733×101 4.0125×10−14 8.3556×101 3.0839×10−1 3.5295×103 2.4197×102
    Std 4.8298×10−1 1.0209×100 3.2695×102 1.1815×101 1.3249×102 5.4870×10−14 6.2643×101 8.4139×10−1 9.5511×102 1.4491×102
    Rank 2 4 9 5 6 1 7 3 10 8
    F10 Mean 2.2671×103 2.7575×103 3.1862×103 2.4369×103 2.9908×103 3.2911×103 2.3253×103 2.3267×103 3.7800×103 6.0667×103
    Std 6.1427×102 4.6685×102 9.7886×102 4.4542×102 5.9210×102 2.7284×102 4.9247×102 2.8457×102 5.9660×102 1.0625×103
    Rank 1 5 7 4 6 8 2 3 9 10
    F11 Mean 2.1678×101 4.1143×101 4.9771×102 2.9612×101 1.1553×102 3.7430×101 1.0032×102 4.1343×101 1.6164×102 1.2672×102
    Std 2.0907×101 2.7367×101 6.4235×102 1.0347×101 3.9628×101 2.3672×101 4.3101×101 2.7994×101 4.5263×101 4.5717×101
    Rank 1 4 10 2 7 3 6 5 9 8
    F12 Mean 9.8943×103 1.2532×105 4.0285×107 1.5983×104 9.9670×104 1.3866×105 6.8629×104 1.1143×106 4.6351×106 3.3042×104
    Std 6.0932×103 1.2555×105 7.3849×107 4.0434×103 1.0658×105 9.2097×104 3.8252×104 8.1422×105 3.1121×106 2.8646×104
    Rank 1 6 10 2 5 7 4 8 9 3
    F13 Mean 1.9749×103 2.0357×104 2.8073×106 2.0450×102 3.0927×104 8.1265×103 1.1211×104 4.6063×103 3.7320×104 1.4857×104
    Std 3.8565×103 2.6333×104 1.6225×107 2.7028×101 2.7301×104 7.8066×103 1.0535×104 4.8590×103 2.6480×104 1.7072×104
    Rank 2 7 10 1 8 4 5 3 9 6
    F14 Mean 8.6436×101 8.0070×101 1.3112×105 6.1985×101 6.7377×103 4.9240×103 4.3238×103 7.1204×104 2.6955×105 3.5454×103
    Std 4.3766×101 1.9915×101 2.3335×105 8.6647×100 5.5695×103 3.2902×103 5.7159×103 5.9323×104 2.4525×105 4.1276×103
    Rank 3 2 9 1 7 6 5 8 10 4
    F15 Mean 1.8396×103 2.0792×103 3.3658×105 5.1634×101 9.7487×103 4.9944×103 2.1676×103 2.2013×103 3.2784×103 3.9091×103
    Std 2.9044×103 7.9984×103 7.9125×105 1.0713×101 1.2114×104 6.6468×103 3.0178×103 1.9756×103 1.9819×103 4.3347×103
    Rank 2 3 10 1 9 8 4 5 6 7
    F16 Mean 3.0243×102 7.9869×102 8.1416×102 4.4823×102 7.7229×102 3.9643×102 5.6072×102 4.9392×102 1.2266×103 5.0039×102
    Std 2.0550×102 2.8651×102 2.6440×102 1.3443×102 2.2590×102 1.1932×102 2.0850×102 1.7309×102 3.0034×102 2.7575×102
    Rank 1 8 9 3 7 2 6 4 10 5
    F17 Mean 4.7111×101 2.2439×102 2.7004×102 6.9544×101 2.5591×102 8.1642×101 8.7684×101 1.4116×102 5.5825×102 2.3994×102
    Std 4.0925×101 1.3518×102 1.3820×102 1.7296×101 1.2971×102 2.2037×101 9.1289×101 8.5026×101 2.3401×102 8.8703×101
    Rank 1 6 9 2 8 3 4 5 10 7
    F18 Mean 6.1013×104 6.9910×104 7.1643×105 2.0505×102 1.1409×105 3.2225×105 1.0034×105 2.1361×105 9.8409×105 2.0609×105
    Std 5.7031×104 1.0210×105 8.2799×105 4.7536×101 1.1535×105 1.2197×105 1.1019×105 1.3261×105 1.1184×106 1.5131×105
    Rank 2 3 9 1 5 8 4 7 10 6
    F19 Mean 3.4042×101 4.4886×103 4.6400×105 2.9977×101 8.6631×103 8.3686×103 5.9612×103 2.0723×103 5.2207×103 6.3203×103
    Std 2.0528×101 1.3325×104 5.4998×105 3.3897×100 1.9974×104 9.2795×103 7.1112×103 2.1685×103 3.9175×103 1.0793×104
    Rank 2 4 10 1 9 8 6 3 5 7
    F20 Mean 9.6665×101 2.4290×102 3.6059×102 1.1363×102 2.6516×102 5.5205×101 1.2989×102 1.7303×102 4.6345×102 2.4392×102
    Std 7.7834×101 1.4995×102 1.0264×102 5.2411×101 1.1737×102 3.5413×101 7.0970×101 7.2015×101 1.7129×102 8.4432×101
    Rank 2 6 9 3 8 1 4 5 10 7
    F21 Mean 2.3023×102 2.5626×102 2.8298×102 2.5458×102 2.7446×102 2.5950×102 2.4896×102 2.5047×102 3.7768×102 2.6988×102
    Std 8.5095×100 1.6800×101 2.5686×101 3.3247×101 1.9517×101 7.6690×100 1.3195×101 8.4442×100 7.9462×101 1.9589×101
    Rank 1 5 9 4 8 6 2 3 10 7
    F22 Mean 1.0010×102 1.9999×103 8.0434×102 1.0022×102 1.4532×103 1.0000×102 1.0228×102 1.0211×103 2.1380×103 1.0232×102
    Std 4.8096×10−1 1.5970×103 1.1113×103 4.3917×10−2 1.8286×103 2.3100×10−13 3.2279×100 1.2872×103 2.2149×103 4.0114×100
    Rank 2 9 6 3 8 1 4 7 10 5
    F23 Mean 3.7755×102 4.1635×102 4.7029×102 3.8959×102 4.8447×102 4.0323×102 4.1472×102 4.0247×102 5.8963×102 4.5003×102
    Std 1.0911×101 1.6898×101 2.9324×101 6.8787×101 4.4709×101 5.6348×100 1.8742×101 8.1687×100 8.8792×101 3.0546×101
    Rank 1 6 8 2 9 4 5 3 10 7
    F24 Mean 4.4827×102 5.4044×102 5.2489×102 4.8972×102 5.6079×102 4.7430×102 4.8169×102 4.9840×102 7.9489×102 5.0152×102
    Std 1.0757×101 4.5778×101 3.3902×101 1.6597×101 5.7847×101 6.0055×100 2.0610×101 1.3899×101 8.6391×101 2.3700×101
    Rank 1 8 7 4 9 2 3 5 10 6
    F25 Mean 3.8777×102 3.8706×102 4.7784×102 3.8374×102 3.8818×102 3.8691×102 4.0124×102 3.8779×102 4.1099×102 4.0877×102
    Std 5.4462×100 8.0647×10−1 2.3819×101 1.8246×10−1 3.4076×100 7.5524×10−2 1.9489×101 1.1319×100 2.1657×101 2.2401×101
    Rank 4 3 10 1 6 2 7 5 9 8
    F26 Mean 1.2578×103 1.6520×103 2.0116×103 2.5051×102 1.4922×103 1.4821×103 1.7344×103 1.5337×103 2.2418×103 1.8645×103
    Std 1.9807×102 1.7070×102 5.7618×102 4.1112×101 9.6940×102 7.2015×101 7.1347×102 1.9051×102 1.7373×103 1.0276×103
    Rank 2 6 9 1 4 3 7 5 10 8
    F27 Mean 5.1091×102 5.0430×102 5.9279×102 5.1286×102 5.3523×102 4.9807×102 5.4289×102 5.0744×102 5.7919×102 5.3827×102
    Std 7.7116×100 8.2707×100 3.8462×101 6.1632×100 2.1095×101 4.7270×100 1.7086×101 3.6242×100 3.5317×101 1.6907×101
    Rank 4 2 10 5 6 1 8 3 9 7
    F28 Mean 3.3558×102 4.0555×102 5.9941×102 3.6492×102 3.5331×102 3.2281×102 3.3257×102 4.1364×102 4.6256×102 3.6806×102
    Std 5.3866×101 3.6156×101 6.9788×101 3.2477×101 5.9179×101 3.7880×101 5.2165×101 2.5577×101 2.3601×101 5.3388×101
    Rank 3 7 10 5 4 1 2 8 9 6
    F29 Mean 4.5991×102 6.6978×102 8.5036×102 5.4385×102 6.7006×102 5.1851×102 5.5826×102 5.4778×102 1.0120×103 7.8922×102
    Std 4.4075×101 1.7459×102 1.8235×102 5.4241×101 1.4475×102 3.4859×101 1.0040×102 8.1960×101 2.1872×102 1.3560×102
    Rank 1 6 9 3 7 2 5 4 10 8
    F30 Mean 2.9823×103 6.0618×103 4.0643×106 3.6855×103 1.8733×104 5.9405×103 5.0147×103 4.9671×103 1.5965×104 5.9572×103
    Std 6.1135×102 4.7022×103 3.1688×106 3.3042×102 3.4470×104 2.3158×103 1.9712×103 2.0934×103 8.7877×103 3.9139×103
    Rank 1 7 10 2 9 5 4 3 8 6
    Count 15 0 0 7 1 6 1 0 0 0
    Ave Rank 1.73 5.27 9.10 3.17 6.67 4.37 4.53 4.63 9.03 6.50
    Total Rank 1 6 10 2 8 3 4 5 9 7
    下载: 导出CSV

    表  5  在30维CEC2017复杂函数上的上下界结果对比

    Table  5  Comparison of upper and lower bounds on the 30-dimensional complex functions from CEC2017

    函数 HCOAG COA/DEBBO
    下界 上界 下界 上界
    F1 3.8942×10-9 7.8898×10-3 3.5001×100 5.0055×103
    F5 1.2935×101 4.1788×101 2.6865×101 9.2083×101
    F11 4.0954×100 7.5899×101 1.3819×101 8.8108×101
    F29 3.7200×102 6.0977×102 4.4500×102 6,2345×102
    下载: 导出CSV

    表  6  Wilcoxon符号秩检验结果

    Table  6  Wilcoxon sign rank test results

    p a=0.05 R+ R n/w/t/l
    HCOAG vs COA 1.3039×10−7 YES 453 12 30/27/0/3
    HCOAG vs GWO 1.8626×10−9 YES 465 0 30/30/0/0
    HCOAG vs MEGWO 2.7741×10−2 YES 339 126 30/23/0/7
    HCOAG vs HFPSO 5.5879×10−9 YES 463 2 30/29/0/1
    HCOAG vs DEBBO 9.0000×10−6 YES 429 36 30/23/0/7
    HCOAG vs SaDE 3.5390×10−8 YES 458 7 30/28/0/2
    HCOAG vs SE04 1.3039×10−8 YES 461 4 30/29/0/1
    HCOAG vs FWA 1.8626×10−9 YES 465 0 30/30/0/0
    HCOAG vs TLBO 3.7253×10−9 YES 464 1 30/29/0/1
    下载: 导出CSV

    表  7  Friedman检验结果

    Table  7  Friedman test results

    D p HCOAG COA GWO MEGWO HFPSO DEBBO SaDE SE04 FWA TLBO
    30 6.3128×10-31 1.73 5.27 9.10 3.17 6.67 4.37 4.53 4.63 9.03 6.50
    下载: 导出CSV

    表  8  6种算法在K-Means聚类优化上的结果对比

    Table  8  Comparison results of the 6 algorithms on K-Means clustering optimization

    数据集 HCOAG COA MEGWO HFPSO IPSO IGA
    Wine(178,13,3) Mean 88.6271 116.7307 91.5916 93.622 89.8617 89.564
    Std 3.4479×10−2 2.9398×100 2.5237×100 6.7353×100 3.9148×100 2.0321×100
    Rank 1 6 4 5 3 2
    Heart(270,13,2) Mean 283.7680 295.3786 284.5731 284.7653 285.0072 284.4112
    Std 3.9989×10−3 2.3404×100 2.3896×10−1 2.3804×100 5.2425×100 2.1715×100
    Rank 1 6 3 4 5 2
    Iris(150,4,3) Mean 29.2053 31.0511 29.2659 29.3578 29.3578 29.2607
    Std 8.8033×10−2 5.7768×10−1 1.3448×10−1 1.0048×100 1.0048×100 9.2414×10−2
    Rank 1 6 3 4 4 2
    Glass(214,9,6) Mean 55.0255 75.4621 68.8816 62.7114 57.3102 60.8651
    Std 2.2242×100 2.1402×100 2.8975×100 3.7975×100 3.5444×100 3.5855×100
    Rank 1 6 5 4 2 3
    Newthyroid(215,5,3) Mean 40.0538 42.0033 40.4736 41.8087 40.8213 41.9155
    Std 9.8154×10−3 7.6493×10−1 4.0104×10−1 2.8501×100 2.0086×100 2.8122×100
    Rank 1 6 2 4 3 5
    Liver Disorders(345,6,2) Mean 90.3443 93.5344 90.3849 91.0246 90.3365 90.3698
    Std 3.8530×10−4 1.0160×100 2.8148×10−2 2.6310×100 2.1424×10−2 2.0447×10−2
    Rank 2 6 4 5 1 3
    Balance(625,4,3) Mean 356.1247 357.6041 356.502 356.0165 356.0802 356.4092
    Std 2.3618×10−1 4.0943×10−1 3.3785×10−1 1.5930×10−1 2.0303×10−1 1.4966×10−1
    Rank 3 6 5 1 2 4
    Count 5 0 0 1 1 0
    Ave Rank 1.43 6.00 3.71 3.86 2.86 3.00
    Total Rank 1 6 4 5 2 3
    下载: 导出CSV
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