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方波触发勘探与开发的粒子群优化算法

邓志诚 孙辉 赵嘉 王晖 吕莉 谢海华

邓志诚, 孙辉, 赵嘉, 王晖, 吕莉, 谢海华. 方波触发勘探与开发的粒子群优化算法. 自动化学报, 2022, 48(12): 3042−3061 doi: 10.16383/j.aas.c190842
引用本文: 邓志诚, 孙辉, 赵嘉, 王晖, 吕莉, 谢海华. 方波触发勘探与开发的粒子群优化算法. 自动化学报, 2022, 48(12): 3042−3061 doi: 10.16383/j.aas.c190842
Deng Zhi-Cheng, Sun Hui, Zhao Jia, Wang Hui, Lv Li, Xie Hai-Hua. Particle swarm optimization with square wave triggered exploration and exploitation. Acta Automatica Sinica, 2022, 48(12): 3042−3061 doi: 10.16383/j.aas.c190842
Citation: Deng Zhi-Cheng, Sun Hui, Zhao Jia, Wang Hui, Lv Li, Xie Hai-Hua. Particle swarm optimization with square wave triggered exploration and exploitation. Acta Automatica Sinica, 2022, 48(12): 3042−3061 doi: 10.16383/j.aas.c190842

方波触发勘探与开发的粒子群优化算法

doi: 10.16383/j.aas.c190842
基金项目: 国家自然科学基金(52069014, 62066030, 61663029, 51669014, 61663028), 江西省教育厅科技项目(GJJ201915, GJJ180940), 江西省研究生创新专项资金项目(YC2018-S422)资助
详细信息
    作者简介:

    邓志诚:南昌工程学院硕士研究生. 主要研究方向为群智能算法.E-mail: deng_zc215@163.com

    孙辉:南昌工程学院教授. 1988年获清华大学硕士学位, 2002年获南昌大学博士学位. 主要研究方向为群智能算法, 粗糙集和变分不等原理. 本文通信作者.E-mail: sun_hui2006@163.com

    赵嘉:南昌工程学院教授. 主要研究方向为群智能算法与数据挖掘.E-mail: zhaojia925@163.com

    王晖:南昌工程学院教授. 主要研究方向为群智能算法与水资源优化.E-mail: huiwang@nit.edu.cn

    吕莉:南昌工程学院教授. 主要研究方向为群智能算法与目标跟踪.E-mail: lvli623@163.com

    谢海华:南昌工程学院硕士研究生. 主要研究方向为群智能算法.E-mail: pxlh_xhh@163.com

Particle Swarm Optimization With Square Wave Triggered Exploration and Exploitation

Funds: Supported by National Natural Science Foundation of China (52069014, 62066030, 61663029, 51669014, 61663028), Science and Technology Research Project of Education Department of Jiangxi Province (GJJ201915, GJJ180940), and Innovation Fund Designated for Graduate Students of Jiangxi Province (YC2018-S422)
More Information
    Author Bio:

    DENG Zhi-Cheng Master stude-nt at Nanchang Institute of Technology. His main research interest is swarm intelligence algorithm

    SUN Hui Professor at Nanchang Institute of Technology. He received his master degree from Tsinghua University in 1988 and received his Ph.D. degree from Nanchang University in 2002, respectively. His research interest covers swarm intelligent algorithm, rough sets and variational inequality principle. Corresponding author of this paper

    ZHAO Jia Professor at Nanch-ang Institute of Technology. His research interest covers swarm intelligent algorithm and data mining

    WANG Hui Professor at Nanch-ang Institute of Technology. His research interest covers swarm intelligent algorithm and water resources optimization

    LV Li Professor at Nanchang Institute of Technology. Her research interest covers swarm intelligent algorithm and single target track

    XIE Hai-Hua Master student at Nanchang Institute of Technology. His main research interest is swarm intelligence algorithm

  • 摘要: 在粒子群优化算法中, 当勘探时间持续过长, 将可能导致种群在解空间过度徘徊; 种群在开发阶段陷入局部最优后, 难以再次进行全局勘探. 针对上述问题, 提出方波触发勘探与开发的粒子群优化算法. 依据方波的周期特性, 在前半个周期内使用标准粒子群优化算法执行全局勘探, 后半个周期使用改进的更新公式执行局部开发. 经过实验验证, 在方波触发机制下, 通过为粒子提供多变步长, 可达到周期性触发勘探与开发的目的. 使用多类型测试函数, 将该算法与新改进粒子群算法、改进人工蜂群算法、改进差分算法在30、50和100维下比较, 实验结果表明, 该算法在收敛速度和精度上更具优势.
  • 图  1  不同f 值的失败次数

    Fig.  1  Number of failures for different f

    图  2  粒子在10维中的步长变化

    Fig.  2  Step size changes of 10 dimensions of particles

    图  3  PSO和PSO+SW的种群多样性变化

    Fig.  3  The population’s diversity of PSO and PSO + SW

    图  4  PSO和PSO + SW的收敛性

    Fig.  4  The convergence performance of PSO and PSO + SW

    图  5  14个函数进化曲线图

    Fig.  5  Evolution curves of 14 functions

    表  1  Penalized 2函数上使用不同的f值寻找全局最优值的迭代次数

    Table  1  Numbers of iterations for finding the global optimum using different values of f on the Penalized 2 function

    运行次数0.010.0090.0080.0070.0060.0050.0040.0030.002
    1117041227311981116251014699851125211198×
    2222217802223056
    313065119791249911119298471079595002
    4140523210421801228887
    52133362118961109211033×2
    614060210895136421013239472×
    727581499911061107127612133321222
    88652222476229418426
    92122761387311136221007822
    1014962140722108342210544×
    111436210913018123861034127421157129060
    12222222210181488
    1321457713927100192112131085510712
    141334622145713428213974018615
    15163312661113962210691222
    1622212466139392990694388822
    1714325137271511322107564134241319
    182147021220512848930222
    1913871201812345222115659610×
    20350521170110586118158292×
    213314717229782114065039
    22222212163210889601529
    23×14773139781197631087022517
    241431327021368221041311753639
    259272103682117071158785722
    2683012428212196222140839400
    272213887211969105661233922
    28×29531184822297068173
    291481312621221245411169944722
    30297510766124059605332971348214269
    平均479851486028527750214488451049073219
    下载: 导出CSV

    表  2  Levy函数上使用不同的f值寻找全局最优值的迭代次数

    Table  2  Numbers of iterations for finding the global optimum using different values of f on the Levy function

    运行次数0.010.0090.0080.0070.0060.0050.0040.0030.002
    113874142771319810780104211355711056948313787
    2282611832144922222
    322136892113141016713358212695
    4×13732204211042222105502
    5×2268621116011234134941239×
    61483713975299062215785523200
    751542130052118689903222
    82140631244106202210440962413760
    912041221362×10850137622
    102×1250161044696032212647
    11×125502222108369370620
    121292421169511196×9632222
    1321345043117304908149294571354513183
    142233062783222805
    15×1287318611032114139453107739836625
    16137293222270122
    1722×1223512042103592××
    1813234141431254622288281201210905
    19278022210954112686213752
    2021488512626128062776118028737
    2113953222222×2
    22459413092130931470111422×10400902212969
    2324313222138122446
    241366721199211462×14429952114542
    25214440222434223×
    26×75413095214461010210023213288
    27×992956×2959222
    28×100921107214639210721116318908
    29×22263612115222
    30144361292012514114532221033012355
    平均588759545259492340724321449442525146
    下载: 导出CSV

    表  3  26个基准测试函数

    Table  3  Information of 26 benchmark functions

    函数名函数式范围最优解
    Sphere${f_1}(x) = \displaystyle\sum\limits_{i = 1}^D {{x_i}^2} $[−100, 100]0
    Schwefel2.22${f_2}\left( x \right) = \displaystyle\sum\limits_{i = 1}^D {\left| { {x_i} } \right| + \prod_{i = 1}^D {\left| { {x_i} } \right|} }$[−10, 10]0
    Rosebrock${f_3}(x) = \displaystyle\sum\limits_{i = 1}^D {\left[ {100{ {\left( { {x_{i + 1} } - x_i^2} \right)}^2} + { {\left( {1 - x_i^2} \right)}^2} } \right]}$[−5, 10]0
    Quartic${f_4}(x) = \displaystyle\sum\limits_{i = 1}^D {i{x_i^4} + random\left[ {0,1} \right)}$[−1.28, 1.28]0
    Schwefel2.26${f_5}(x) = 418.98288727243380{\cdot }D{\rm{ - } }\displaystyle\sum\limits_{i \;=\; 1}^D { {x_i}\sin \sqrt {\left| { {x_i} } \right|} }$$\overline { {f} } _5(x) = \displaystyle\sum\limits_{i = 1}^D { - {x_i}\sin \sqrt {\left| { {x_i} } \right|} },\; (D = 100)$[−500, 500]0 ~ 418.9826 × D
    Rastrigin${f_6}(x) = \displaystyle\sum\limits_{i = 1}^D {\left[ { {x_i^2} - 10\cos 2\pi {x_i} + 10} \right]}$[−5.12, 5.12]0
    Ackley${f_7}(x) = - 20{\rm{exp} }\left( { - 0.2\sqrt {\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D { {x_i^2} } } } \right) - {\rm{exp} }\left( {\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D {\cos 2\pi x_i} } \right) + 20 + {\rm{e}}$[−50, 50]0
    Griewank${f_8}(x) = \dfrac{1}{{4000}}\displaystyle\sum\limits_{i = 1}^D {{{\left( {{x_i}} \right)}^2}} - \prod _{i = 1}^D\cos \left( {\dfrac{{{x_i}}}{{\sqrt i }}} \right) + 1$[−600, 600]0
    Penalized 1${f_9}(x) = \dfrac{\pi }{D} \bigg\{ {\displaystyle\sum\limits_{i \;=\; 1}^{D - 1} {({y_i} - 1)} ^2}[1 + \sin (\pi {y_{i + 1} })] + {({y_D} - 1)^2} + 10{\sin ^2}(\pi {y_1}) \bigg\} +$$\displaystyle\sum\limits_{i = 1}^D {u({x_i},10,100,4),{y_i} = 1 + \dfrac{ { {x_i} + 1} }{4} }u({x_i},a,k,m) = \left\{ {\begin{aligned} &u({x_i},a,k,m),&{x_i} > a\quad\qquad \\ &0,& - a \le {x_i} \le a\;\\ &k{ {( - {x_i} - a)}^m},&{x_i} < - a \qquad\end{aligned} } \right.$[−100, 100]0
    Penalized 2${f_{10} }\left( x \right) = 0.1\bigg\{ { { {\sin }^2}\left( {\pi {x_1} } \right) + \displaystyle\sum\limits_{i \;=\; 1}^{D - 1} { { {\left( { {x_i} - 1} \right)}^2}[1 + { {\sin }^2}\left( {3\pi {x_{i + 1} } } \right)]+} }$${\left( { {x_D} - 1} \right)^2} {\left[ {1 + { {\sin }^2}\left( {2\pi {x_{i + 1} } } \right)} \right]} \bigg\} + \displaystyle\sum\limits_{i = 1}^D {u\left( { {x_i},5,100,4} \right)}$[−100, 100]0
    Rotated Schwefel2.26${f_{11} }(x) = 4.18.928\cdot D - \displaystyle\sum\limits_{i \;=\; 1}^D { {z_i} } , where \; {z_i} = \left\{ \begin{aligned} & - {y_i}\sin (\sqrt {|{y_i}|} ) if|{y_i}| \le 500 \\ & 0, \quad {\rm{否则} } \end{aligned} \right.$$ {y_i} = y_i^` + 420.96, y_i^` = M \cdot(x - 420.96), M \;is \; an \; orthogonal \; matrix $[−500, 500]0
    Rotated Rastrigin${f_{12} }(x) = \displaystyle\sum\limits_{i \;=\; 1}^D {[y_i^2 - 10\cos (2\pi {y_i}) + 10], where\;y = M \cdot x,\;} M \;is\; an\; orthogonal\; matrix$[−5.12, 5.12]0
    Rotated Ackly${f_{13} }(x) = - 20\exp \left[ - 0.2\sqrt {\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D {y_i^2} }\right ] - \exp \left [\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D {\cos (2\pi {y_i})} \right ] + (20 + {\rm{e} })$$where\;y = M \cdot x , M \;is\; an\; orthogonal\; matrix$[−32, 32]0
    Rotated Griewank${f_{14} }(x) = \dfrac{1}{ {4000} }\displaystyle\sum\limits_{i = 1}^D {y_i^2 - \prod\limits_{i = 1}^D {\cos (\dfrac{ { {y_i} } }{ {\sqrt i } })} + 1, where\; y = M \cdot x,\;} M \;is\; an \; orthogonal \; matrix$[−600, 600]0
    Elliptic${f_{15}}\left( x \right) = {\displaystyle\sum\limits_{i = 1}^D {\left( {{{10}^6}} \right)} ^{\dfrac{{i - 1}}{{D - 1}}}}x_{_i}^2$[−100, 100]0
    SumSquare${f_{16}}\left( x \right) = \displaystyle\sum\limits_{i = 1}^D {ix_{_i}^2} $[−10, 10]0
    SumPower${f_{17} }\left( x \right) = {\displaystyle\sum\limits_{i \;=\; 1}^D {\left| { {x_i} } \right|} ^{\left( {i + 1} \right)} }$[−1, 1]0
    Schwefel2.21${f_{18} }(x) ={\rm max}\left\{ {\left| { {x_i} } \right|,1 \le i \le D} \right\}$[−100, 100]0
    Step${f_{19} }(x) = \displaystyle\sum\limits_{i \;=\; 1}^D {\left\lfloor { {x_i} + 0.5} \right\rfloor }$[−100, 100]0
    Exponential${f_{20} }\left( x \right) = \exp \left( {0.5\displaystyle\sum\limits_{i\; =\; 1}^D { {x_i} } } \right)$[−10, 10]0
    NCRastrigin${f_{21} }\left( x \right) = \displaystyle\sum\limits_{i\; =\; 1}^D {\left[ {y_i^2 - 10\cos \left( {2\pi {y_i} } \right) + 10} \right]}$[−5.12, 5.12]0
    Alpine${f_{22} }\left( x \right) = \displaystyle\sum\limits_{i\; = \;1}^{D - 1} {\left| { {x_i}\sin \left( { {x_i} } \right) + 0.1{x_i} } \right|}$[−10, 10]0
    Levy${f_{23}}\left( x \right) = \displaystyle\sum\limits_{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]$[−10, 10]0
    Weierstrass${f_{24} }\left( x \right) = \displaystyle\sum\limits_{i \;=\; 1}^D {\left( {\displaystyle\sum\limits_{k \;=\; 0}^{k_{max} } {\left[ { {a^k}\cos \left( {2\pi {b^k}\left( { {x_i} + 0.5} \right)} \right)} \right]} } \right)} - D\displaystyle\sum\limits_{k = 0}^{k_{max} } {\left[ { {a^k}\cos \left( {2\pi {b^k}0.5} \right)} \right]}$$a = 0.5;b = 3;k_{max} = 20$[−1, 1]0
    Himmelballa${f_{25}}\left( x \right) = \dfrac{1}{D}\displaystyle\sum\limits_1^D {\left( {x_{_i}^4 - 16x_i^2 + 5{x_i}} \right)} $[−5, 5]−78.332
    Michalewice${f_{26} }\left( x \right) = - \displaystyle\sum\limits_{i \;=\; 1}^D {\sin \left( { {x_i} } \right){ {\sin }^{20} }\left( {\dfrac{ {ix_i^2} }{\pi } } \right)}$$\left[ {0,\pi } \right]$−50, $D=50 $
    下载: 导出CSV

    表  4  30维实验结果对比

    Table  4  Comparison of 30-dimensional test results

    函数指标CLPSOHPSO-TVACDMS-PSOLFPSOPSOLFRPSOLFH-PSO-SCACSWTPSO
    f1 MBF 8.06 × 10−96 2.83 × 10−33 1.53 × 10−113 4.69 × 10−31 0 0 0 0
    SD 3.53 × 10−95 3.19 × 10−33 5.14 × 10−113 2.50 × 10−30 0 0 0 0
    f2 MBF 5.50 × 10−57 9.03 × 10−20 2.18 × 102 2.64 × 10−17 0 0 4.09 × 10−217 0
    SD 1.92 × 10−56 9.58 × 10−20 1.83 × 102 6.92 × 10−17 0 0 0 0
    f3 MBF 4.18 × 101 2.39 × 101 3.49 × 101 2.38 × 101 2.68 × 101 1.01 × 101 2.36 × 101 3.99
    SD 3.35 × 101 2.65 × 101 2.76 × 101 3.17 × 10−1 1.03 9.69 × 10−1 1.51 × 10−1 2.22 × 101
    f4 MBF 1.74 × 10−3 9.82 × 10−2 6.09 × 10−4 2.41 × 10−3 2.60 × 10−5 6.50 × 10−3 3.94 × 10−237 8.89 × 10−4
    SD 7.83 × 10−4 3.26 × 10−2 4.18 × 10−4 8.07 × 10−4 2.10 × 10−5 5.50 × 10−3 0 1.58 × 10−3
    f5 MBF 3.82 × 10−4 1.59 × 103 3.21 × 103 1.37 × 103 2.00 × 103 2.85 × 103 6.64 × 103 3.51 × 102
    SD 1.28 × 10−7 3.26 × 102 6.51 × 102 6.36 × 102 6.08 × 102 3.79 × 102 2.39 × 103 1.14 × 103
    f6 MBF 1.27 × 101 9.43 1.49 × 101 4.54 0 0 0 0
    SD 4.22 3.48 3.62 1.03 × 101 0 0 0 0
    f7 MBF 1.42 × 10−14 7.29 × 10−14 6.06 × 10−12 1.68 × 10−14 8.88 × 10−16 8.88 × 10−16 8.88 × 10−16 5.88 × 10−16
    SD 7.46 × 10−15 3.00 × 10−14 3.90 × 10−13 4.84 × 10−15 0 0 0 0
    f8 MBF 3.20 × 10−3 9.75 × 10−3 1.85 × 10−3 8.14 × 10−17 0 0 0 0
    SD 4.93 × 10−3 8.33 × 10−3 4.07 × 10−3 4.46 × 10−16 0 0 0 0
    f9 MBF 1.36 × 10−33 2.71 × 10−29 0 4.67 × 10−31 1.66 × 10−2 1.98 × 10−32 2.10 × 10−1 1.58 × 10−32
    SD 2.82 × 10−33 1.88 × 10−29 0 9.01 × 10−31 1.27 × 10−2 9.23 × 10−33 9.67 × 10−1 5.07 × 10−33
    f10 MBF 1.65 × 10−33 2.79 × 10−28 6.16 × 10−35 1.51 × 10−28 9.01 × 10−9 1.65 × 10−32 1.09 1.36 × 10−32
    SD 4.03 × 10−33 2.18 × 10−28 2.76 × 10−34 8.00 × 10−28 1.11 × 10−8 4.36 × 10−33 1.42 4.84 × 10−33
    f11 MBF 4.39 × 103 5.32 × 103 4.04 × 103 5.51 × 103 1.54 × 103 9.98 × 103 4.48 × 103 6.63 × 103
    SD 3.51 × 102 7.00 × 102 5.68 × 102 5.64 × 102 4.28 × 102 7.58 × 10−12 1.73 × 102 1.43 × 103
    f12 MBF 8.17 × 101 5.29 × 103 4.20 × 101 1.79 0 0 0 0
    SD 1.08 × 101 1.25 × 101 9.74 9.81 0 0 0 0
    f13 MBF 5.91 × 10−5 9.29 2.42 × 10−14 1.65 × 10−14 0 0 0 0
    SD 6.46 × 10−5 2.07 1.52 × 10−14 5.40 × 10−15 0 0 0 0
    f14 MBF 7.69 × 10−5 9.26 × 10−3 1.02 × 10−2 1.48 × 10−3 0 0 0 0
    SD 7.66 × 10−5 8.80 × 10−3 1.24 × 10−2 6.17 × 10−3 0 0 0 0
    下载: 导出CSV

    表  6  100维实验结果对比

    Table  6  Comparison of 100-dimensional test results

    函数指标CLPSOHPSO-TVACDMS-PSOLFPSOPSOLFRPSOLFH-PSO-SCACSWTPSO
    f1 MBF 4.16 × 10−75 5.48 × 10−26 4.89 × 10−77 2.65 × 10−10 0 0 0 0
    SD 1.80 × 10−74 2.59 × 10−25 1.40 × 10−76 3.89 × 10−8 0 0 0 0
    f2 MBF 3.45 × 10−45 3.77 × 10−15 5.83 × 102 1.02 0 1.22 × 10−252 0 0
    SD 1.52 × 10−44 1.20 × 10−14 3.54 × 101 3.21 0 0 0 0
    f3 MBF 1.46 × 102 1.34 × 102 1.20 × 102 2.56 × 101 9.04 × 101 9.15 × 101 9.27 × 101 1.03 × 101
    SD 4.78 × 101 2.28 × 102 3.62 × 101 5.23 × 10−1 2.57 × 101 1.50 4.31 2.55 × 101
    f4 MBF 7.00 × 10−3 1.17 × 10−48 6.40 × 10−3 3.21 × 10−3 2.35 × 10−5 8.68 × 10−3 3.25 × 10−125 6.28 × 10−4
    SD 1.53 × 10−3 1.27 × 10−47 2.56 × 10−3 3.24 × 10−3 1.92 × 10−5 5.47 × 10−3 0 1.60 × 10−3
    f5 MBF 5.36 2.30 × 103 6.58 × 104 3.56 × 104 2.46 × 104 1.29 × 103 2.66 × 104 6.16 × 102
    SD 2.35 × 101 2.88 × 103 5.21 × 101 3.56 × 103 5.75 × 103 2.05 × 103 4.11 × 103 2.08 × 103
    f6 MBF 7.02 4.90 × 101 1.95 × 101 3.25 × 101 0 0 0 0
    SD 1.00 × 102 4.21 × 101 2.59 × 101 5.68 × 101 0 0 0 0
    f7 MBF 2.74 × 10−14 3.21 × 10−12 9.91 × 10−1 3.87 × 10−9 5.89 × 10−16 5.89 × 10−16 5.89 × 10−16 5.88 × 10−16
    SD 5.17 × 10−15 4.29 × 10−11 4.43 5.89 × 10−9 0 0 0 0
    f8 MBF 3.33 × 10−17 2.71 × 10−3 7.40 × 10−4 3.89 × 10−2 0 0 0 0
    SD 7.29 × 10−17 2.32 × 10−2 2.28 × 10−3 5.78 × 10−2 0 0 0 0
    f9 MBF 9.33 × 10−3 2.39 × 10−24 1.56 × 10−2 9.56 × 10−2 2.15 × 10−2 1.27 × 10−12 5.47 × 10−1 1.90 × 10−31
    SD 2.87 × 10−2 7.19 × 10−24 4.23 × 10−2 3.46 × 10−2 3.26 × 10−2 1.24 × 10−11 7.08 × 10−1 5.09 × 10−30
    f10 MBF 1.10 × 10−3 1.74 × 10−21 5.49 × 10−3 8.97 × 10−2 6.32 × 10−3 3.25 × 10−3 4.50 2.80 × 10−30
    SD 3.38 × 10−3 7.05 × 10−21 1.26 × 10−2 2.65 × 10−2 2.53 × 10−2 9.57 × 10−2 6.71 3.77 × 10−29
    f11 MBF 6.32 × 103 2.85 × 104 1.12 × 104 3.56 × 104 4.18 × 104 3.43 × 104 2.07 × 104 2.99 × 104
    SD 2.32 × 102 5.14 × 103 3.65 × 103 3.25 × 103 1.67 × 102 3.57 × 103 8.63 × 102 7.92 × 103
    f12 MBF 2.56 × 103 5.47 × 102 9.32 × 102 1.25 × 102 0 0 0 0
    SD 5.32 × 102 1.03 × 102 1.35 × 102 6.32 × 101 0 0 0 0
    f13 MBF 6.87 × 10−3 1.05 × 10−6 3.56 × 10−6 6.89 × 10−6 0 0 0 0
    SD 3.98 × 10−3 2.54 × 10−6 3.89 × 10−5 1.58 × 10−5 0 0 0 0
    f14 MBF 2.32 × 10−2 3.89 × 10−3 8.25 × 10−1 9.58 × 10−2 0 0 0 0
    SD 1.25 × 10−2 5.89 × 10−3 9.23 × 10−1 2.56 × 10−2 0 0 0 0
    下载: 导出CSV

    表  5  50维实验结果对比

    Table  5  Comparison of 50-dimensional test results

    函数 指标CLPSOHPSO-TVACDMS-PSOLFPSOPSOLFRPSOLFH-PSO-SCACSWTPSO
    f1 MBF 2.28 × 10−85 5.50 × 10−11 8.62 × 10−103 9.17 × 10−17 0 6.57 × 10−201 0 0
    SD 3.87 × 10−85 1.42 × 10−10 2.82 × 10−102 3.20 × 10−16 0 0 0 0
    f2 MBF 4.47 × 10−52 1.12 × 10−7 3.09 × 102 8.00 × 10−1 0 8.01 × 10−124 4.69 × 10−245 0
    SD 1.22 × 10−51 1.76 × 10−7 2.87 × 101 4.44 0 2.36 × 10−122 0 0
    f3 MBF 6.97 × 101 1.36 × 102 7.03 × 101 4.41 × 101 4.74 × 101 4.41 × 101 4.71 × 101 7.24
    SD 4.45 × 101 5.90 × 102 4.49 × 101 2.82 × 10−1 9.84 × 10−1 4.74 × 101 6.75 5.33 × 101
    f4 MBF 3.31 × 10−3 3.78 × 10−23 1.15 × 10−3 2.37 × 10−3 1.60 × 10−5 3.26 × 10−3 8.85 × 10−176 9.08 × 10−4
    SD 9.05 × 10−4 1.39 × 10−22 3.92 × 10−4 1.01 × 10−3 1.17 × 10−5 5.36 × 10−3 0 2.68 × 10−3
    f5 MBF 9.53 9.83 × 102 9.76 × 103 4.36 × 103 5.07 × 103 5.96 × 102 1.20 × 104 5.38 × 102
    SD 4.76 × 101 1.76 × 103 5.62 × 102 1.27 × 103 9.74 × 102 1.43 × 103 2.50 × 103 2.01 × 103
    f6 MBF 2.42 × 101 2.31 × 101 2.99 × 101 1.26 × 101 0 0 0 0
    SD 6.40 3.06 × 101 6.30 1.61 × 101 0 0 0 0
    f7 MBF 1.01 × 10−14 1.53 × 10−4 7.75 × 10−8 6.53 × 10−10 8.88 × 10−16 5.89 × 10−16 8.88×10−16 5.89 × 10−16
    SD 3.32 × 10−15 1.91 × 10−3 2.79 × 10−7 2.46 × 10−9 0 0 0 0
    f8 MBF 4.93 × 10−4 4.50 × 10−4 7.40 × 10−4 4.86 × 10−3 0 0 0 0
    SD 2.20 × 10−3 2.80 × 10−2 2.28 × 10−3 1.36 × 10−2 0 0 0 0
    f9 MBF 3.11 × 10−3 2.88 × 10−7 3.11 × 10−3 1.94 × 10−17 3.15 × 10−2 2.26 × 10−14 2.96 × 10−1 1.15 × 10−32
    SD 1.39 × 10−2 7.95 × 10−6 1.39 × 10−2 7.21 × 10−17 1.66 × 10−2 3.46 × 10−13 5.03 × 10−1 1.79 × 10−32
    f10 MBF 5.49 × 10−4 3.49 × 10−8 8.63 × 10−34 2.39 × 10−4 4.23 × 10−8 9.67 × 10−13 2.21 1.56 × 10−32
    SD 2.46 × 10−3 4.85 × 10−7 2.30 × 10−33 1.69 × 103 5.84 × 10−8 1.12 × 10−11 3.38 2.39 × 10−32
    f11 MBF 5.96 × 103 1.07 × 104 1.78 × 103 5.72 × 103 1.79 × 103 1.32 × 104 8.71 × 103 1.14 × 104
    SD 6.22 × 102 2.85 × 103 6.03 × 102 1.64 × 103 4.44 × 102 2.93 × 103 5.30 × 102 3.27 × 103
    f12 MBF 1.41 × 103 2.61 × 102 4.91 × 102 2.56 × 102 0 0 0 0
    SD 2.35 × 102 6.73 × 101 1.20 × 102 3.44 × 101 0 0 0 0
    f13 MBF 2.07 × 10−4 7.31 × 10−9 1.81 × 10−8 7.97 × 10−9 0 0 0 0
    SD 5.52 × 10−3 9.51 × 10−9 4.09 × 10−7 2.99 × 10−8 0 0 0 0
    f14 MBF 5.22 × 10−3 4.58 × 10−3 4.11 × 10−1 3.74 × 10−3 0 0 0 0
    SD 1.19 × 10−2 6.37 × 10−3 3.72 × 10−1 7.97 × 10−3 0 0 0 0
    下载: 导出CSV

    表  7  各PSO算法的Friedman 检验结果

    Table  7  Friedman test results of each POS algorithm

    算法D = 30 (排名)D = 50 (排名)D = 100 (排名)
    SWTPSO2.86 (1)2.46 (1)2.61 (1)
    H-PSO-SCAC3.86 (4)3.93 (4)3.71 (4)
    RPSOLF3.79 (3)3.46 (3)3.46 (2)
    PSOLF3.57 (2)3.39 (2)3.68 (3)
    LFPSO5.43 7)5.25 (5)6.50 (8)
    DMS-PSO5.32 (6)5.96 (7)6.04 (7)
    HPSO-TVAC6.50 (8)5.57 (6)5.04 (6)
    CLPSO4.68 (5)5.96 (7)4.96 (5)
    下载: 导出CSV

    表  8  SWTPSO与9种相关算法实验结果对比

    Table  8  Comparison of experimental results of SWTPSO and nine related algorithms

    函数指标BABCCoDEJADEjDEscopSaDEAABCLSCMAESABCVSSMPGABCSWTPSO
    f1 MBF 1.01 × 10−14 1.16 × 10−37 2.50 × 10−87 1.15 × 10−36 1.48 × 10−57 4.60 × 10−35 1.21 × 10−28 6.68 × 10−23 2.47 × 10−51 0
    SD 5.07 × 10−14 2.69 × 10−37 8.62 × 10−87 5.75 × 10−36 6.02 × 10−57 1.67 × 10−35 2.09 × 10−29 3.16 × 10−32 9.52 × 10−51 0
    f2 MBF 3.14 × 10−29 1.25 × 10−34 4.43 × 10−78 4.77 × 10−36 8.97 × 10−55 4.48 × 10−33 4.88 × 10−23 1.11 × 10−23 4.54 × 10−49 0
    SD 4.95 × 10−29 1.50 × 10−34 2.21 × 10−77 1.85 × 10−35 2.38 × 10−54 2.16 × 10−33 9.20 × 10−24 4.17 × 10−23 1.32 × 10−48 0
    f3 MBF 1.23 × 10−13 1.94 × 10−38 2.27 × 10−85 1.80 × 10−41 5.22 × 10−58 2.54 × 10−35 4.22 × 10−27 1.77 × 10−33 6.05 × 10−53 0
    SD 6.16 × 10−13 4.08 × 10−38 1.14 × 10−82 1.47 × 10−16 1.49 × 10−57 1.24 × 10−35 9.12 × 10−28 7.88 × 10−33 1.39 × 10−52 0
    f4 MBF 2.96 × 10−92 6.20 × 10−143 3.01 × 10−100 3.58 × 10−124 3.38 × 10−80 1.52 × 10−50 6.82 × 10−13 7.21 × 10−41 8.51 × 10−103 0
    SD 0 3.15 × 10−142 1.00 × 10−99 1.79 × 10−123 1.59 × 10−79 4.88 × 10−50 5.39 × 10−13 3.56 × 10−40 3.61 × 10−102 0
    f5 MBF 2.18 × 10−6 1.21 × 10−20 1.89 × 10−41 1.71 × 10−22 1.71 × 10−38 4.98 × 10−18 3.69 × 10−4 2.89 × 10−18 2.95 × 10−28 1.13 × 10−300
    SD 1.06 × 10−5 8.94 × 10−21 8.32 × 10−41 8.46 × 10−22 7.47 × 10−38 1.34 × 10−18 1.84 × 10−3 6.52 × 10−18 4.04 × 10−28 0
    f6 MBF 7.20 6.00 × 10−5 8.20 × 10−10 2.52 3.42 × 10−1 1.16 × 10−1 6.00 × 10−15 2.06 2.12 × 101 0
    SD 2.78 9.51 × 10−5 9.51 × 10−10 5.61 × 10−1 4.67 × 10−1 1.24 × 10−1 7.45 × 10−16 3.77 × 10−1 4.04 0
    f7 MBF 0 0 0 0 0 0 1.60 × 10−1 0 0 0
    SD 0 0 0 0 0 0 4.73 × 10−1 0 0 0
    f8 MBF 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 5.40 × 10−66 2.67 × 10−109 2.67 × 10−109 2.96 × 10−109
    SD 2.62 × 10−116 9.67 × 10−125 9.60 × 10−125 9.65 × 10−125 3.71 × 10−122 2.77 × 10−119 1.99 × 10−65 3.08 × 10−120 3.06 × 10−122 1.14 × 10−111
    f9 MBF 5.74 × 10−2 8.17 × 10−3 2.50 × 10−3 3.91 × 10−3 1.33 × 10−2 1.63 × 10−2 2.80 × 10−1 6.14 × 10−2 5.02 × 10−2 6.61 × 10−4
    SD 1.11 × 10−2 2.79 × 10−3 1.54 × 10−3 1.47 × 10−3 3.33 × 10−3 4.68 × 10−3 6.70 × 10−2 1.38 × 10−2 7.98 × 10−3 1.37 × 10−2
    f10 MBF 6.29 × 10−2 3.32 × 101 3.19 × 10−1 2.58 × 101 9.45 3.09 1.75 × 10−25 1.09 × 10−1 4.32 × 10−1 8.19
    SD 1.24 × 10−1 2.26 × 101 1.10 2.68 × 101 2.08 × 101 1.36 × 101 4.31 × 10−26 2.52 × 10−1 7.44 × 10−1 5.68 × 101
    f15 MBF 0 7.34 × 10−1 1.78 × 10−11 1.03 × 10−14 6.77 × 10−1 0 3.89 × 102 0 0 0
    SD 0 8.82 × 10−1 1.74 × 10−11 1.31 × 10−14 6.87 × 10−1 0 7.11 × 101 0 0 0
    f16 MBF 0 2.28 × 101 2.74 × 10−8 8.02 × 10−2 4.40 × 10−1 8.00 × 10−2 3.78 × 102 0 0 0
    SD 0 4.68 2.07 × 10−8 2.77 × 10−1 6.51 × 10−1 2.77 × 10−1 4.90 × 101 0 0 0
    f17 MBF 0 2.96 × 10−4 7.88 × 10−4 1.47 × 10−16 6.88 × 10−3 4.44 × 10−18 1.38 × 10−3 3.67 × 10−14 0 0
    SD 0 1.48 × 10−3 2.82 × 10−3 2.60 × 10−16 1.22 × 10−2 2.22 × 10−17 3.34 × 10−3 1.84 × 10−13 0 0
    f18 MBF 3.78 × 10−12 4.74 4.75 6.18 × 101 3.64 × 10−12 1.41 × 10−11 9.01 × 103 4.66 × 10−12 1.05 × 10−11 5.38 × 102
    SD 7.28 × 10−13 2.37 × 101 2.32 × 101 1.69 × 102 0 3.53 × 10−12 1.13 × 103 2.47 × 10−12 3.03 × 10−12 2.01 × 103
    f19 MBF 7.50 × 10−15 2.81 × 10−15 6.22 × 10−15 1.43 × 101 1.32 2.65 × 10−14 1.99 × 101 1.70 × 10−14 2.51 × 10−14 5.89 × 10−16
    SD 2.49 × 10−15 7.11 × 10−16 0 6.50 4.90 × 10−1 3.48 × 10−15 2.63 × 10−2 5.75 × 10−33 4.55 × 10−15 0
    f20 MBF 1.22 × 10−13 9.42 × 10−33 2.49 × 10−3 1.91 × 10−31 3.24 × 10−2 9.42 × 10−33 4.98 × 10−3 1.07 × 10−32 9.42 × 10−33 1.15 × 10−32
    SD 6.09 × 10−13 1.40 × 10−48 1.24 × 10−2 5.28 × 10−31 9.01 × 10−2 1.40 × 10−48 1.72 × 10−2 5.74 × 10−33 1.40 × 10−48 1.79 × 10−32
    f21 MBF 2.50 × 10−15 1.55 × 10−33 1.01 × 10−4 5.02 × 10−32 1.09 × 10−2 1.50 × 10−33 8.01 × 103 1.80 × 10−33 1.50 × 10−33 1.56 × 10−32
    SD 1.25 × 10−14 2.47 × 10−34 8.05 × 10−5 5.95 × 10−32 3.00 × 10−2 0 6.25 × 103 1.08 × 10−33 0 2.39 × 10−32
    f22 MBF 2.07 × 10−16 4.39 × 10−3 1.01 × 10−4 2.95 × 10−5 1.75 × 10−16 5.32 × 10−11 8.59 × 10−1 2.14 × 10−16 3.21 × 10−7 3.10 × 10−290
    SD 7.79 × 10−16 6.44 × 10−3 8.05 × 10−5 3.09 × 10−5 2.63 × 10−16 9.68 × 10−11 8.49 × 10−1 7.43 × 10−16 4.58 × 10−7 0
    f23 MBF 1.35 × 10−31 1.35 × 10−31 1.35 × 10−31 1.30 × 10−30 7.39 × 10−2 1.35 × 10−31 3.77 × 10−1 1.35 × 10−31 1.35 × 10−31 1.78 × 10−31
    SD 2.23 × 10−47 2.47 × 10−33 2.23 × 10−47 3.69 × 10−30 1.07 × 10−1 2.23 × 10−47 1.22 2.23 × 10−47 2.23 × 10−47 2.83 × 10−31
    f24 MBF 0 3.45 3.33 × 10−1 0 4.17 × 10−4 2.56 × 10−6 9.41 0 1.56 × 10−2 0
    SD 0 2.94 × 10−1 4.45 × 10−2 0 5.94 × 10−4 7.39 × 10−6 4.19 0 2.04 × 10−2 0
    f25 MBF −7.83 × 101 −7.83 × 101 −7.83 × 101 −7.83 × 101 −7.83 × 101 −7.83 × 101 −6.43 × 101 −7.83 × 101 −7.83 × 101 −6.26 × 101
    SD 1.16 × 10−14 4.10 × 10−15 2.26 × 10−1 1.12 × 10−13 1.13 × 10−1 4.10 × 10−15 2.63 1.00 × 10−14 1.12 × 10−14 3.31
    f26 MBF −5.00 × 101 −4.86 × 101 −4.98 × 101 −5.00 × 101 −4.83 × 101 −5.00 × 101 −4.10 × 101 −5.00 × 101 −5.00 × 101 −3.99 × 101
    SD 6.79 × 10−6 4.45 × 10−1 3.86 × 10−2 9.23 × 10−3 2.70 × 10−1 9.13 × 10−4 2.65 3.82 × 10−7 4.26 × 10−4 2.13 × 101
    下载: 导出CSV

    表  9  8种新相关算法和SWTPSO的Friedman测试结果

    Table  9  Friedman test results of 8 new correlation algorithms and SWTPSO

    算法SWTPSOMPGABCAABCLSJADEABCVSSBABCCoDEjDEscopSaDECMAES
    平均值 (排名)3.86 (1)4.41 (2)5.11 (3)5.05 (4)5.07 (5)5.11 (6)5.59 (7)5.91 (8)6.07 (9)8.82 (10)
    下载: 导出CSV

    表  10  6种算法在15个函数上的比较结果

    Table  10  Results of the fifteen functions for six algorithms

    函数指标GDHSODEIGHSGABCIGPSOSWTPSO
    f1 MBF 8.07 × 10−3 5.67 × 10−81 2.14 × 10−10 8.58 × 102 4.45 × 10−156 0
    SD 8.87 × 10−4 9.55 × 10−81 1.22 × 10−11 1.52 × 103 9.29 × 10−156 0
    f2 MBF 7.20 × 10−1 3.60 × 10−6 3.96 × 10−1 8.47 × 101 0 0
    SD 4.16 × 10−2 1.13 × 10−5 3.07 × 10−1 3.06 × 101 0 0
    f3 MBF 6.17 1.81 × 10−5 2.72 × 10−6 8.49 × 104 0 0
    SD 7.70 × 10−1 4.69 × 10−5 9.49 × 10−7 3.51 × 104 0 0
    f4 MBF 6.71 × 10−2 9.27 × 10−2 3.53 5.34 × 101 0 0
    SD 1.05 × 10−2 2.33 × 10−2 2.67 × 10−1 1.93 × 101 0 0
    f5 MBF 1.07 × 102 9.69 × 10−12 1.32 × 102 6.07 × 105 4.18 × 10−2 1.03 × 101
    SD 2.39 × 101 3.06 × 10−11 4.49 × 101 8.23 × 105 2.86 × 10−2 2.55 × 101
    f6 MBF 0 0 0 2.35 × 103 0 0
    SD 0 0 0 2.90 × 103 0 0
    f7 MBF 2.87 × 10−3 1.83 × 10−3 1.80 × 10−2 8.57 × 10−1 1.40 × 10−4 6.28 × 10−4
    SD 5.74 × 10−4 2.02 × 10−3 2.17 × 10−3 6.27 × 10−1 1.45 × 10−4 1.60 × 10−3
    f8 MBF −4.19 × 104 −4.17 × 104 −3.99 × 104 −4.20 × 104 −4.19 × 104 −3.34 × 104
    SD 4.57 × 10−2 4.35 × 10−2 9.76 × 10−1 4.57 × 10−2 1.33 × 10−2 1.28 × 104
    f9 MBF 1.16 × 10−2 4.52 × 10−8 4.41 × 10−8 4.64 × 101 0 0
    SD 1.14 × 10−3 1.41 × 10−7 3.47 × 10−9 1.63 × 101 0 0
    f10 MBF 1.44 × 10−2 1.72 × 10−4 5.91 × 10−6 7.03 3.55 × 10−15 5.88 × 10−16
    SD 6.71 × 10−4 5.44 × 10−4 1.63 × 10−7 2.47 0 0
    f11 MBF 4.50 × 10−3 7.40 × 10−4 2.28 × 10−3 2.14 × 101 0 0
    SD 5.18 × 10−4 2.28 × 10−3 4.27 × 10−3 3.00 × 101 0 0
    f12 MBF 6.83 × 10−6 3.49 × 10−25 5.26 × 10−13 2.93 × 10−1 1.50 × 10−7 1.90 × 10−31
    SD 6.46 × 10−7 8.66 × 10−25 4.25 × 10−14 9.27 × 10−1 3.70 × 10−8 5.09 × 10−30
    f13 MBF 3.33 × 10−4 1.07 × 10−15 1.86 × 10−6 2.04 × 105 1.75 × 10−5 2.80 × 10−30
    SD 4.66 × 10−5 5.60 × 10−6 5.89 × 10−6 4.44 × 105 5.60 × 10−6 3.77 × 10−29
    f14 MBF 1.94 × 101 1.23 × 101 4.06 × 101 3.17 × 101 0 0
    SD 1.20 7.70 7.70 5.15 0 0
    f15 MBF 1.33 × 10−1 4.29 × 102 3.15 × 101 2.18 × 105 0 0
    SD 1.61 × 10−2 8.19 × 102 6.83 × 101 2.42 × 104 0 0
    下载: 导出CSV

    表  11  6种算法的Friedman检验结果

    Table  11  Friedman test results of 6 algorithms

    算法平均值 (排名)
    SWTPSO1.83 (1)
    IGPSO1.97 (2)
    ODE3.13 (3)
    IGHS3.93 (4)
    GDHS4.20 (5)
    GABC5.93 (6)
    下载: 导出CSV

    表  12  CEC2015函数集

    Table  12  CEC2015 test suite

    函数序号函数类型描述最优值
    1单峰函数Bent Cigar 旋转函数100
    2Discus 旋转函数200
    3简单多峰函数Schwefel 偏移旋转函数300
    4Schwefel偏移旋转函数400
    5Katsuura 偏移旋转函数500
    6HappyCat 偏移旋转函数600
    7HGBat 偏移旋转函数700
    8Griewank + Rosenbrocl 扩展偏移旋转函数800
    9Schffer's F6 扩展偏移旋转函数900
    10混合函数混合函数 1 (N = 3)1000
    11混合函数 2 (N = 4)1100
    12混合函数 3 (N = 5)1200
    13组合函数组合函数 1 (N = 5)1300
    14组合函数 2 (N = 3)1400
    15组合函数 3 (N = 5)1500
    下载: 导出CSV

    表  13  CEC2015实验结果对比

    Table  13  Test results comparison of CEC2015 test suite

    函数符号PSOFAFFPSOHPSOFFHFPSOSWTPSO
    1MBF3.9049 × 1092.8899 × 10109.2383 × 10104.7539 × 1091.1795×1093.0501 × 108
    2MBF9.9760 × 1041.3418 × 1056.9430 × 1069.7376 × 1048.5653 × 1045.6540 × 104
    3MBF3.3113 × 1023.3850 × 1023.4771 × 1023.3059 × 1023.2638 × 1023.2631 × 102
    4MBF7.7928 × 1038.0330 × 1039.6696 × 1036.8199 × 1025.1202×1035.8445×103
    5MBF5.0418 × 1025.0425 × 1025.0586 × 1025.0422 × 1025.0410 × 1025.0403 × 102
    6MBF6.0096 × 1026.0410 × 1026.0755 × 1026.0090 × 1026.0076 × 1026.0053 × 102
    7MBF7.0405 × 1027.6997 × 1028.9401 × 1027.0750 × 1027.0074 × 1027.0046 × 102
    8MBF4.0013 × 1032.3491 × 106 1.6032 × 1081.7191 × 1042.6354 × 1031.5070 × 103
    9MBF9.1358 × 1029.1381 × 1029.1427 × 1029.1365 × 1029.1337 × 1029.1328 × 102
    10MBF7.5602 × 1062.9938 × 1073.9391 × 1081.1337 × 1075.4690 × 1061.1662 × 107
    11MBF1.1614 × 1031.2889 × 1032.1094 × 1031.1551 × 1031.1336 × 1031.1267 × 103
    12MBF2.2417 × 1032.9655 × 1034.9876 × 1052.0617 × 1031.7752 × 1031.7819 × 103
    13MBF1.7719 × 1031.9596 × 1033.6329 × 1031.7390 × 1031.6866×1031.6552 × 103
    14MBF1.6644 × 1031.7522 × 1032.1683 × 1031.6711 × 1031.6469 × 1031.6510 × 103
    15MBF2.5522 × 1032.8256 × 1033.9013 × 1032.6529 × 1032.4467 × 1032.4639 × 103
    下载: 导出CSV

    表  14  Friedman检验结果

    Table  14  Results of Friedman test

    算法秩均值 (排名)
    SWTPSO1.53 (1)
    HFPSO1.73 (2)
    HPSOFF3.33 (3)
    PSO3.40 (4)
    FA5.00 (5)
    FFPSO6.00 (6)
    下载: 导出CSV
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出版历程
  • 收稿日期:  2019-12-11
  • 录用日期:  2020-05-03
  • 网络出版日期:  2022-12-23
  • 刊出日期:  2022-12-23

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