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