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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

文章导航 > 自动化学报  > 2022 >  48(11): 2757-2776
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, 2022, 48(11): 2757−2776 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, 2022, 48(11): 2757−2776 doi: 10.16383/j.aas.c190617
shu
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, 2022, 48(11): 2757−2776 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, 2022, 48(11): 2757−2776 doi: 10.16383/j.aas.c190617
shu

灰狼与郊狼混合优化算法及其聚类优化

doi: 10.16383/j.aas.c190617
  • 1.

    河南师范大学计算机与信息工程学院 新乡 453007

  • 2.

    智慧商务与物联网技术河南省工程实验室 新乡 453007

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. 主要研究方向为智能优化算法, 图像去噪, 图像增强和图像分割. 本文通信作者.E-mail: xinmingzhang@126.com

    姜云：河南师范大学硕士研究生. 主要研究方向为智能优化算法和图像分割.E-mail: jiangyun951120@163.com

    刘尚旺：河南师范大学副教授. 主要研究方向为图像处理和计算机视觉.E-mail: shwl08@126.com

    刘国奇：河南师范大学副教授. 主要研究方向为图像分割和偏微分方程.E-mail: liuguoqi080408@163.com

    窦智：河南师范大学讲师. 主要研究方向为算法及数字图像处理.E-mail: 619534345@163.com

    刘艳：河南师范大学实验师. 主要研究方向为优化算法和图像分割. E-mail: liu_yan122@sina.com

Hybrid Coyote Optimization Algorithm With Grey Wolf Optimizer and Its Application to Clustering Optimization

  • 1.

    College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007

  • 2.

    Engineering Laboratory of Intelligence Business and Internet of Things, Xinxiang 453007

Funds: Supported by National Natural Science Foundation of China (61901160, U1904123) and Key Research Project of Higher Education Institutions of Henan Province (19A520026)
More Information
    Author Bio:

    ZHANG Xin-Ming　Professor at Henan Normal University. His research interest covers intelligence optimization algorithm, image denoising, image enhancement, and image segmentation. Corresponding author of this paper

    JIANG Yun　Master student at Henan Normal University. Her research interest covers intelligence optimization algorithm and image segmentation

    LIU Shang-Wang　Associate professor at Henan Normal University. His research interest covers image processing and computer vision

    LIU Guo-Qi　Associate professor at Henan Normal University. His research interest covers image segmentation and partial differential equation

    DOU Zhi　Lecturer at Henan Normal University. His research interest covers algorithms and digital image processing

    LIU Yan　Laboratory teacher at Henan Normal University. Her research interest covers optimization algorithm and image segmentation

    摘要
  • 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, 进一步获得更好的优化性能. 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.
    Abstract: Coyote optimization algorithm (COA) is a novel swarm intelligence optimization algorithm with great application potential, which was proposed recently. It has a unique search mechanism and the advantages to solve global optimization problems well and so on. But when dealing with the complex optimization problems, it has some defects, such as low search efficiency, poor operability, slow convergence speed and so on. To make up for COA's disadvantages and utilize the advantages of grey wolf optimizer (GWO), a hybrid COA with GWO (HCOAG) is proposed. Firstly, an improved COA (ICOA) is proposed. A Gaussian global-best growing operator replaces the growing operator of the original algorithm to improve the search efficiency and convergence speed, and a dynamic adjustment scheme of coyote number in each group is proposed to enhance the search ability and operability. Secondly, in order to improve the operability and reduce the computational complexity of the algorithm, a simplified GWO (SGWO) is proposed. Finally, ICOA and SGWO are integrated by a sinusoidal crossover strategy to further get better optimization performance. A large number of experimental results on classical benchmark functions and CEC2017 complex functions and K-Means clustering show that, compared with COA, HCOAG has higher search efficiency, stronger operability and faster convergence speed. Compared with other state-of-the-art comparison algorithms, HCOAG has better optimization performance and can solve clustering optimization problems better.
    • HTML全文

  • 图  1  GWO的流程图

    Fig.  1  Flow chart of GWO

    下载: 全尺寸图片 幻灯片

    图  2  组数$N_p $与组内郊狼数$N_c $的分配图

    Fig.  2  Disposition graph of two parameters $N_c $ and $N_p $

    下载: 全尺寸图片 幻灯片

    图  3  GWO与SGWO的等级情况对比

    Fig.  3  Comparison of hierarchies of GWO and SGWO

    下载: 全尺寸图片 幻灯片

    图  4  HCOAG流程图

    Fig.  4  Flow chart of HCOAG

    下载: 全尺寸图片 幻灯片

    图  5  HCOAG与对比算法在4个经典函数上的收敛图

    Fig.  5  Convergence curves of HCOAG and the comparison algorithms on the 4 classical benchmark functions

    下载: 全尺寸图片 幻灯片

    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

    下载: 全尺寸图片 幻灯片

    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

    下载: 全尺寸图片 幻灯片

    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 标准HCOAGCOAGWOHCOAG5HCOAG10ICOASGWO
    F1Mean7.4494×10−41.2099×1031.2813×1094.1072×10−41.9800×10−31.0737×1023.3279×103
    Std1.4801×10−31.2998×1039.6388×1088.5916×10−42.9438×10−31.0569×1024.3271×103
    Rank2571346
    F2Mean1.1941×1012.9013×10213.1831×10324.8580×1031.8078×1018.6764×10153.3582×1014
    Std2.4077×1011.1462×10221.5894×10333.3985×1045.0208×1013.5675×10161.3200×1015
    Rank1673254
    F3Mean9.5410×10−16.0573×1042.8342×1047.6972×10−11.0995×1003.3032×1048.7276×102
    Std1.9288×1001.0177×1049.2323×1039.8032×10−11.6794×1006.8409×1037.2376×102
    Rank2751364
    F4Mean1.8113×1018.4041×1012.0825×1022.0841×1012.8446×1014.8248×1011.0495×102
    Std2.7696×1018.5306×1008.4445×1013.0540×1013.1826×1013.3517×1012.4806×101
    Rank1572346
    F5Mean2.8433×1015.2890×1019.6116×1013.5884×1013.0204×1013.4844×1013.1488×101
    Std6.8886×1001.5025×1013.2690×1011.0115×1018.8983×1001.0983×1019.2242×100
    Rank1675243
    F6Mean1.7483×10−71.6399×10−56.3664×1009.4452×10−71.5005×10−62.8782×10−42.0381×10−2
    Std4.7524×10−79.6428×10−63.1596×1002.4080×10−65.9643×10−61.6254×10−42.7102×10−2
    Rank1472356
    F7Mean6.1055×1017.5148×1011.4460×1026.7082×1015.9300×1016.7675×1016.4025×101
    Std1.0851×1011.3762×1014.6314×1011.1241×1019.4998×1001.1520×1011.1856×101
    Rank2674153
    F8Mean3.2489×1015.6110×1018.4662×1013.6085×1012.9446×1013.6138×1013.1775×101
    Std1.2272×1011.8774×1012.5270×1018.8063×1007.9048×1001.0081×1017.8884×100
    Rank3674152
    F9Mean2.7362×10−15.6225×10−15.5392×1025.2270×10−12.5931×10−18.8559×10−27.4159×100
    Std4.8298×10−11.0209×1003.2695×1028.7374×10−14.6957×10−11.5656×10−16.7112×100
    Rank3574216
    F10Mean2.2671×1032.7575×1033.1862×1032.5574×1032.3435×1032.1380×1032.1424×103
    Std6.1427×1024.6685×1029.7886×1025.3524×1026.0670×1025.6098×1024.0691×102
    Rank3675412
    F11Mean2.1678×1014.1143×1014.9771×1022.9822×1012.6698×1012.2685×1011.0908×102
    Std2.0907×1012.7367×1016.4235×1022.6128×1012.4059×1012.0893×1013.8218×101
    Rank1574326
    F12Mean9.8943×1031.2532×1054.0285×1071.2577×1041.0657×1041.3660×1052.0716×105
    Std6.0932×1031.2555×1057.3849×1076.4424×1036.4606×1039.3092×1041.9967×105
    Rank1473256
    F13Mean1.9749×1032.0357×1042.8073×1063.0265×1033.1829×1033.6293×1021.3271×104
    Std3.8565×1032.6333×1041.6225×1076.2901×1038.0719×1038.9975×1011.5283×104
    Rank2673415
    F14Mean8.6436×1018.0070×1011.3112×1057.7150×1011.0134×1025.6726×1011.4132×104
    Std4.3766×1011.9915×1012.3335×1056.0071×1019.2585×1011.4850×1011.7944×104
    Rank4372516
    F15Mean1.8396×1032.0792×1033.3658×1057.1579×1021.7386×1036.9111×1016.6116×103
    Std2.9044×1037.9984×1037.9125×1051.2272×1032.9477×1031.9083×1018.3961×103
    Rank4572316
    F16Mean3.0243×1027.9869×1028.1416×1023.3715×1023.0883×1024.6416×1025.0252×102
    Std2.0550×1022.8651×1022.6440×1022.1415×1021.8878×1022.6962×1022.4545×102
    Rank1673245
    F17Mean4.7111×1012.2439×1022.7004×1026.4809×1015.3120×1013.7365×1011.3984×102
    Std4.0925×1011.3518×1021.3820×1025.3991×1014.7801×1014.0654×1018.0664×101
    Rank2674315
    F18Mean6.1013×1046.9910×1047.1643×1055.1875×1045.0376×1043.9930×1041.8454×105
    Std5.7031×1041.0210×1058.2799×1053.6270×1043.7592×1042.0034×1041.7045×105
    Rank4573216
    F19Mean3.4042×1014.4886×1034.6400×1053.4163×1013.0105×1022.4678×1015.4815×103
    Std2.0528×1011.3325×1045.4998×1053.9687×1011.1322×1037.1259×1004.9859×103
    Rank2573416
    F20Mean9.6665×1012.4290×1023.6059×1021.0084×1021.1637×1021.0389×1022.0165×102
    Std7.7834×1011.4995×1021.0264×1026.6726×1017.5890×1018.7694×1019.6673×101
    Rank1672435
    F21Mean2.3023×1022.5626×1022.8298×1022.3713×1022.3135×1022.3724×1022.3289×102
    Std8.5095×1001.6800×1012.5684×1011.0815×1019.8453×1001.1261×1019.9428×100
    Rank1674253
    F22Mean1.0010×1021.9999×1038.0434×1021.0005×1021.0005×1021.0005×1021.2974×102
    Std4.8096×10−11.5970×1031.1113×1033.4354×10−13.4444×10−13.4443×10−12.0703×102
    Rank4761325
    F23Mean3.7755×1024.1635×1024.7029×1023.8706×1023.7831×1023.8441×1023.8854×102
    Std1.0911×1011.6898×1012.9324×1011.3271×1018.1817×1001.0543×1011.3692×101
    Rank1674235
    F24Mean4.4827×1025.4044×1025.2489×1024.5484×1024.4530×1024.5862×1024.5671×102
    Std1.0757×1014.5778×1013.3902×1011.1783×1011.1461×1011.2203×1011.1956×101
    Rank2763154
    F25Mean3.8777×1023.8706×1024.7784×1023.8747×1023.8729×1023.8698×1024.0239×102
    Std5.4462×1008.0647×10−12.3819×1011.5625×1001.2735×1005.4095×10−11.5135×101
    Rank5274316
    F26Mean1.2578×1031.6520×1032.0116×1031.3249×1031.2449×1031.3138×1031.5024×103
    Std1.9807×1021.7070×1025.7618×1023.1093×1023.4947×1021.9599×1022.8691×102
    Rank2674135
    F27Mean5.1091×1025.0430×1025.9279×1025.1349×1025.1088×1025.0560×1025.3331×102
    Std7.7116×1008.2707×1003.8462×1018.6401×1008.6837×1007.3827×1001.1175×101
    Rank4175326
    F28Mean3.3558×1024.0555×1025.9941×1023.2930×1023.3793×1023.4828×1024.5710×102
    Std5.3866×1013.6156×1016.9788×1015.1585×1015.1950×1015.3564×1012.3392×101
    Rank2571346
    F29Mean4.5991×1026.6978×1028.5036×1024.8821×1024.6287×1024.5683×1026.2453×102
    Std4.4075×1011.7459×1021.8235×1025.5772×1014.3839×1014.4800×1011.1861×102
    Rank2674315
    F30Mean2.9823×1036.0618×1034.0643×1063.1036×1032.9323×1031.9586×1046.1880×103
    Std6.1135×1024.7022×1033.1688×1068.4665×1025.9332×1027.3086×1032.8250×103
    Rank2473165
    Count101045100
    Ave rank2.205.236.873.102.603.074.93
    Total rank1674235
    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 标准D = 10
    HCOAGCOAGWOHFPSODEBBO
    f1Mean6.0684×10−91.7833×1029.0799×10−153.4157×10−56.7086×10−2
    Std4.8458×10−96.3524×1012.4849×10−142.4485×10−53.1056×10−2
    Rank25134
    f2Mean8.4133×10−62.3737×1001.3222×10−91.3703×10−32.8483×10−2
    Std3.7531×10−64.0964×10−11.1382×10−96.5582×10−47.0993×10−3
    Rank25134
    f3Mean01.6180×1021.0000×10−100
    Std05.0787×1013.0513×10−100
    Rank15411
    f4Mean1.2498×10−104.0253×1001.5325×10−63.5760×10−71.8575×10−3
    Std2.0501×10−101.6104×1009.4192×10−74.1539×10−71.0158×10−3
    Rank15324
    f5Mean2.0046×10−87.1619×1022.4093×10−54.6770×10−62.6132×10−2
    Std5.9686×10−81.5819×1031.4121×10−55.6661×10−61.0613×10−2
    Rank15324
    f6Mean4.1921×10−101.7228×1001.2593×10−24.9377×10−73.5149×10−3
    Std4.3501×10−105.1829×10−16.8779×10−24.8665×10−71.5826×10−3
    Rank15423
    D = 30
    f1Mean1.3966×10−173.2554×1015.4432×10−417.2595×10−92.7076×10−4
    Std3.2255×10−175.9567×1007.1605×10−417.3446×10−91.1010×10−4
    Rank25134
    f2Mean2.8862×10−101.3998×1006.0158×10−245.3463×10−51.3264×10−3
    Std4.6435×10−101.9835×10−16.2049×10−243.9178×10−52.3103×10−4
    Rank25134
    f3Mean03.3700×1013.3333×10−200
    Std07.9877×1001.8257×10−100
    Rank15411
    f4Mean1.0451×10−173.8002×1001.5129×10−21.7278×10−26.4886×10−5
    Std3.0738×10−171.3163×1001.0471×10−23.9296×10−22.7878×10−5
    Rank15342
    f5Mean5.3309×10−171.8376×1011.6587×10−13.2962×10−35.0150×10−4
    Std1.5184×10−165.8919×1001.1940×10−15.1211×10−32.1237×10−4
    Rank15432
    f6Mean1.2484×10−181.8049×1007.9738×10−18.2480×10−38.5930×10−5
    Std1.7135×10−184.9152×10−17.4565×10−14.5176×10−23.9292×10−5
    Rank15432
    Count80422
    Ave rank1.335.002.752.502.92
    Total rank15324
    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${f_1}(x) = \displaystyle\sum_{i = 1}^D {x_i^2}$ [−100, 100]0
    Schwefel 2.22${f_2}(x) = \displaystyle\sum_{i = 1}^D {\left| { {x_i} } \right|} + \prod_{i = 1}^D {\left| { {x_i} } \right|}$ [−10, 10]0
    Step${f_3}(x) = \displaystyle\sum_{i = 1}^D { { {\left( {\left\lfloor { {x_i} + 0.5} \right\rfloor } \right)}^2} }$ [−100, 100]0
    多峰函数Penalized 1${f_4}(x) = \dfrac{\pi}{D}\bigg\{ {10{ {\sin }^2}\left( {\pi {y_i} } \right)} +$
    　　　　$\displaystyle\sum_{i = 1}^{D - 1} { { {\left( { {y_i} - 1} \right)}^2}\left[ {1 + 10{ {\sin }^2}\left( {\pi {y_{i + 1} } } \right)} \right]} { + { {\left( { {y_D} - 1} \right)}^2} } \bigg\} +$
    　　　　$\displaystyle\sum_{i = 1}^D {u\left( { {x_i},10,100,4} \right)}$
    　　　　${y_i} = 1 + \dfrac{1}{4}\left( { {x_i} + 1} \right)$
    　　　　$u\left( { {x_i},a,k,m} \right) = $$\left\{ \begin{aligned}&k{\left( { {x_i} - a} \right)^m},\quad\;\; {x_i} > a\\&0, \quad \quad \quad \quad \quad\quad\;\; - a \le {x_i} \le a\\&k{\left( { - {x_i} - a} \right)^m},\quad {x_i} < - a \end{aligned} \right.$     		[−50, 50]0
    Penalized 2${f_5}(x) = 0.1\bigg\{ { { {\sin }^2} } \left( {\pi {x_1} } \right) + \displaystyle\sum_{i = 1}^{D - 1} { { {\left( { {x_i} - 1} \right)}^2} } \left[ {1 + { {\sin }^2}\left( {3\pi {x_{i + 1} } } \right)} \right] +$
    　　　　$\left( { {x_D} - 1} \right) {\Big[ {1 + { {\sin }^2}\left( {2\pi {x_D} } \right)} \Big]} \bigg\} + \displaystyle\sum_{i = 1}^D {u\left( { {x_i},5,100,4} \right)}$     		[−50, 50]0
    Levy${f_6}(x) = \displaystyle\sum_{i = 1}^{D - 1} { { {\left( { {x_i} - 1} \right)}^2} } \left[ {1 + { {\sin }^2}\left( {3\pi {x_{i + 1} } } \right)} \right] +$
    　　　　${\sin ^2}\left( {3\pi {x_1} } \right) + \left| { {x_D} - 1} \right|\Big[ {1 + { {\sin }^2}\left( {3\pi {x_D} } \right)} \Big]$     		[−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 标准HCOAGCOAGWOMEGWOHFPSODEBBOSaDESE04FWATLBO
    F1Mean7.4494×10−41.2099×1031.2813×1094.5517×1033.9338×1032.7849×1033.0714×1033.2930×1034.3987×1062.9846×103
    Std1.4801×10−31.2998×1039.6388×1081.0677×1035.3689×1034.0364×1033.5072×1034.2328×1031.4055×1063.1471×103
    Rank12108735694
    F2Mean1.1941×1012.9013×10213.1831×10322.8884×1085.3485×1041.5057×10178.6275×10−13.0802×10134.1397×10151.0448×1016
    Std2.4077×1011.1462×10221.5894×10338.0571×1083.2847×1053.5179×10174.9357×1001.1694×10141.5680×10164.7082×1016
    Rank29104381567
    F3Mean9.5410×10−16.0573×1042.8342×1042.2633×1021.5595×10−73.6772×1043.0045×1029.7974×1032.4748×1044.0488×10−4
    Std1.9288×1001.0177×1049.2323×1031.7031×1022.3334×10−75.8394×1037.3017×1023.4377×1036.3467×1031.6647×10−3
    Rank31084195672
    F4Mean1.8113×1018.4041×1012.0825×1022.4815×1016.9386×1018.4851×1016.0423×1018.5881×1011.1370×1025.9054×101
    Std2.7696×1018.5306×1008.4445×1012.8995×1012.1364×1012.2848×10−12.9825×1011.1251×1011.7315×1013.0429×101
    Rank16102574893
    F5Mean2.8433×1015.2890×1019.6116×1015.6912×1018.5624×1015.8216×1015.6192×1014.1688×1011.8456×1028.5717×101
    Std6.8886×1001.5025×1013.2690×1011.0725×1011.7427×1016.5957×1001.4216×1018.1545×1003.3933×1011.8601×101
    Rank13957642108
    F6Mean1.7483×10−71.6399×10−56.3664×1002.4470×10−11.0170×1001.1369×10−138.9317×10−27.5481×10−65.1770×1007.2903×100
    Std4.7524×10−79.6428×10−63.1596×1008.1620×10−22.3644×10001.3955×10−13.9880×10−53.1351×1004.4538×100
    Rank24967153810
    F7Mean6.1055×1017.5148×1011.4460×1028.9106×1011.0407×1029.9725×1019.4945×1017.2448×1012.0998×1021.3661×102
    Std1.0851×1011.3762×1014.6314×1011.0935×1011.9791×1016.4285×1001.9879×1017.3495×1004.4817×1012.4846×101
    Rank13947652108
    F8Mean3.2489×1015.6110×1018.4662×1015.9398×1017.2842×1015.9299×1015.3942×1014.4194×1011.4508×1027.1339×101
    Std1.2272×1011.8774×1012.5270×1011.0663×1011.7967×1016.0788×1001.2792×1016.5834×1002.1470×1011.4852×101
    Rank14968532107
    F9Mean2.7362×10−15.6225×10−15.5392×1027.8267×1003.2733×1014.0125×10−148.3556×1013.0839×10−13.5295×1032.4197×102
    Std4.8298×10−11.0209×1003.2695×1021.1815×1011.3249×1025.4870×10−146.2643×1018.4139×10−19.5511×1021.4491×102
    Rank24956173108
    F10Mean2.2671×1032.7575×1033.1862×1032.4369×1032.9908×1033.2911×1032.3253×1032.3267×1033.7800×1036.0667×103
    Std6.1427×1024.6685×1029.7886×1024.4542×1025.9210×1022.7284×1024.9247×1022.8457×1025.9660×1021.0625×103
    Rank15746823910
    F11Mean2.1678×1014.1143×1014.9771×1022.9612×1011.1553×1023.7430×1011.0032×1024.1343×1011.6164×1021.2672×102
    Std2.0907×1012.7367×1016.4235×1021.0347×1013.9628×1012.3672×1014.3101×1012.7994×1014.5263×1014.5717×101
    Rank14102736598
    F12Mean9.8943×1031.2532×1054.0285×1071.5983×1049.9670×1041.3866×1056.8629×1041.1143×1064.6351×1063.3042×104
    Std6.0932×1031.2555×1057.3849×1074.0434×1031.0658×1059.2097×1043.8252×1048.1422×1053.1121×1062.8646×104
    Rank16102574893
    F13Mean1.9749×1032.0357×1042.8073×1062.0450×1023.0927×1048.1265×1031.1211×1044.6063×1033.7320×1041.4857×104
    Std3.8565×1032.6333×1041.6225×1072.7028×1012.7301×1047.8066×1031.0535×1044.8590×1032.6480×1041.7072×104
    Rank27101845396
    F14Mean8.6436×1018.0070×1011.3112×1056.1985×1016.7377×1034.9240×1034.3238×1037.1204×1042.6955×1053.5454×103
    Std4.3766×1011.9915×1012.3335×1058.6647×1005.5695×1033.2902×1035.7159×1035.9323×1042.4525×1054.1276×103
    Rank32917658104
    F15Mean1.8396×1032.0792×1033.3658×1055.1634×1019.7487×1034.9944×1032.1676×1032.2013×1033.2784×1033.9091×103
    Std2.9044×1037.9984×1037.9125×1051.0713×1011.2114×1046.6468×1033.0178×1031.9756×1031.9819×1034.3347×103
    Rank23101984567
    F16Mean3.0243×1027.9869×1028.1416×1024.4823×1027.7229×1023.9643×1025.6072×1024.9392×1021.2266×1035.0039×102
    Std2.0550×1022.8651×1022.6440×1021.3443×1022.2590×1021.1932×1022.0850×1021.7309×1023.0034×1022.7575×102
    Rank18937264105
    F17Mean4.7111×1012.2439×1022.7004×1026.9544×1012.5591×1028.1642×1018.7684×1011.4116×1025.5825×1022.3994×102
    Std4.0925×1011.3518×1021.3820×1021.7296×1011.2971×1022.2037×1019.1289×1018.5026×1012.3401×1028.8703×101
    Rank16928345107
    F18Mean6.1013×1046.9910×1047.1643×1052.0505×1021.1409×1053.2225×1051.0034×1052.1361×1059.8409×1052.0609×105
    Std5.7031×1041.0210×1058.2799×1054.7536×1011.1535×1051.2197×1051.1019×1051.3261×1051.1184×1061.5131×105
    Rank23915847106
    F19Mean3.4042×1014.4886×1034.6400×1052.9977×1018.6631×1038.3686×1035.9612×1032.0723×1035.2207×1036.3203×103
    Std2.0528×1011.3325×1045.4998×1053.3897×1001.9974×1049.2795×1037.1112×1032.1685×1033.9175×1031.0793×104
    Rank24101986357
    F20Mean9.6665×1012.4290×1023.6059×1021.1363×1022.6516×1025.5205×1011.2989×1021.7303×1024.6345×1022.4392×102
    Std7.7834×1011.4995×1021.0264×1025.2411×1011.1737×1023.5413×1017.0970×1017.2015×1011.7129×1028.4432×101
    Rank26938145107
    F21Mean2.3023×1022.5626×1022.8298×1022.5458×1022.7446×1022.5950×1022.4896×1022.5047×1023.7768×1022.6988×102
    Std8.5095×1001.6800×1012.5686×1013.3247×1011.9517×1017.6690×1001.3195×1018.4442×1007.9462×1011.9589×101
    Rank15948623107
    F22Mean1.0010×1021.9999×1038.0434×1021.0022×1021.4532×1031.0000×1021.0228×1021.0211×1032.1380×1031.0232×102
    Std4.8096×10−11.5970×1031.1113×1034.3917×10−21.8286×1032.3100×10−133.2279×1001.2872×1032.2149×1034.0114×100
    Rank29638147105
    F23Mean3.7755×1024.1635×1024.7029×1023.8959×1024.8447×1024.0323×1024.1472×1024.0247×1025.8963×1024.5003×102
    Std1.0911×1011.6898×1012.9324×1016.8787×1014.4709×1015.6348×1001.8742×1018.1687×1008.8792×1013.0546×101
    Rank16829453107
    F24Mean4.4827×1025.4044×1025.2489×1024.8972×1025.6079×1024.7430×1024.8169×1024.9840×1027.9489×1025.0152×102
    Std1.0757×1014.5778×1013.3902×1011.6597×1015.7847×1016.0055×1002.0610×1011.3899×1018.6391×1012.3700×101
    Rank18749235106
    F25Mean3.8777×1023.8706×1024.7784×1023.8374×1023.8818×1023.8691×1024.0124×1023.8779×1024.1099×1024.0877×102
    Std5.4462×1008.0647×10−12.3819×1011.8246×10−13.4076×1007.5524×10−21.9489×1011.1319×1002.1657×1012.2401×101
    Rank43101627598
    F26Mean1.2578×1031.6520×1032.0116×1032.5051×1021.4922×1031.4821×1031.7344×1031.5337×1032.2418×1031.8645×103
    Std1.9807×1021.7070×1025.7618×1024.1112×1019.6940×1027.2015×1017.1347×1021.9051×1021.7373×1031.0276×103
    Rank26914375108
    F27Mean5.1091×1025.0430×1025.9279×1025.1286×1025.3523×1024.9807×1025.4289×1025.0744×1025.7919×1025.3827×102
    Std7.7116×1008.2707×1003.8462×1016.1632×1002.1095×1014.7270×1001.7086×1013.6242×1003.5317×1011.6907×101
    Rank42105618397
    F28Mean3.3558×1024.0555×1025.9941×1023.6492×1023.5331×1023.2281×1023.3257×1024.1364×1024.6256×1023.6806×102
    Std5.3866×1013.6156×1016.9788×1013.2477×1015.9179×1013.7880×1015.2165×1012.5577×1012.3601×1015.3388×101
    Rank37105412896
    F29Mean4.5991×1026.6978×1028.5036×1025.4385×1026.7006×1025.1851×1025.5826×1025.4778×1021.0120×1037.8922×102
    Std4.4075×1011.7459×1021.8235×1025.4241×1011.4475×1023.4859×1011.0040×1028.1960×1012.1872×1021.3560×102
    Rank16937254108
    F30Mean2.9823×1036.0618×1034.0643×1063.6855×1031.8733×1045.9405×1035.0147×1034.9671×1031.5965×1045.9572×103
    Std6.1135×1024.7022×1033.1688×1063.3042×1023.4470×1042.3158×1031.9712×1032.0934×1038.7877×1033.9139×103
    Rank17102954386
    Count15007161000
    Ave rank1.735.279.103.176.674.374.534.639.036.50
    Total rank16102834597
    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HCOAGCOA/DEBBO
    下界上界下界上界
    F13.8942×10-97.8898×10-33.5001×1005.0055×103
    F51.2935×1014.1788×1012.6865×1019.2083×101
    F114.0954×1007.5899×1011.3819×1018.8108×101
    F293.7200×1026.0977×1024.4500×1026.2345×102
    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 COA1.3039×10−7YES4531230/27/0/3
    HCOAG vs GWO1.8626×10−9YES465030/30/0/0
    HCOAG vs MEGWO2.7741×10−2YES33912630/23/0/7
    HCOAG vs HFPSO5.5879×10−9YES463230/29/0/1
    HCOAG vs DEBBO9.0000×10−6YES4293630/23/0/7
    HCOAG vs SaDE3.5390×10−8YES458730/28/0/2
    HCOAG vs SE041.3039×10−8YES461430/29/0/1
    HCOAG vs FWA1.8626×10−9YES465030/30/0/0
    HCOAG vs TLBO3.7253×10−9YES464130/29/0/1
    下载: 导出CSV

    表  7  Friedman检验结果

    Table  7  Friedman test results

    DpHCOAGCOAGWOMEGWOHFPSODEBBOSaDESE04FWATLBO
    306.3128×10-311.735.279.103.176.674.374.534.639.036.50
    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HCOAGCOAMEGWOHFPSOIPSOIGA
    Wine (178, 13, 3)Mean88.6271116.730791.591693.62289.861789.564
    Std3.4479×10−22.9398×1002.5237×1006.7353×1003.9148×1002.0321×100
    Rank164532
    Heart (270, 13, 2)Mean283.7680295.3786284.5731284.7653285.0072284.4112
    Std3.9989×10−32.3404×1002.3896×10−12.3804×1005.2425×1002.1715×100
    Rank163452
    Iris (150, 4, 3)Mean29.205331.051129.265929.357829.357829.2607
    Std8.8033×10−25.7768×10−11.3448×10−11.0048×1001.0048×1009.2414×10−2
    Rank163442
    Glass (214, 9, 6)Mean55.025575.462168.881662.711457.310260.8651
    Std2.2242×1002.1402×1002.8975×1003.7975×1003.5444×1003.5855×100
    Rank165423
    Newthyroid (215, 5, 3)Mean40.053842.003340.473641.808740.821341.9155
    Std9.8154×10−37.6493×10−14.0104×10−12.8501×1002.0086×1002.8122×100
    Rank162435
    Liver disorders (345, 6, 2)Mean90.344393.534490.384991.024690.336590.3698
    Std3.8530×10−41.0160×1002.8148×10−22.6310×1002.1424×10−22.0447×10−2
    Rank264513
    Balance (625, 4, 3)Mean356.1247357.6041356.502356.0165356.0802356.4092
    Std2.3618×10−14.0943×10−13.3785×10−11.5930×10−12.0303×10−11.4966×10−1
    Rank365124
    Count500110
    Ave rank1.436.003.713.862.863.00
    Total rank164523
    下载: 导出CSV
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出版历程
  • 收稿日期:  2019-09-02
  • 录用日期:  2020-03-11
  • 网络出版日期:  2022-10-20
  • 刊出日期:  2022-11-22

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