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多策略融合改进的自适应蜉蝣算法

蒋宇飞 许贤泽 徐逢秋 高波

蒋宇飞,许贤泽,徐逢秋,等. 多策略融合改进的自适应蜉蝣算法[J]. 北京航空航天大学学报,2024,50(4):1416-1426 doi: 10.13700/j.bh.1001-5965.2022.0492
引用本文: 蒋宇飞,许贤泽,徐逢秋,等. 多策略融合改进的自适应蜉蝣算法[J]. 北京航空航天大学学报,2024,50(4):1416-1426 doi: 10.13700/j.bh.1001-5965.2022.0492
JIANG Y F,XU X Z,XU F Q,et al. Multi-strategy fusion improved adaptive mayfly algorithm[J]. Journal of Beijing University of Aeronautics and Astronautics,2024,50(4):1416-1426 (in Chinese) doi: 10.13700/j.bh.1001-5965.2022.0492
Citation: JIANG Y F,XU X Z,XU F Q,et al. Multi-strategy fusion improved adaptive mayfly algorithm[J]. Journal of Beijing University of Aeronautics and Astronautics,2024,50(4):1416-1426 (in Chinese) doi: 10.13700/j.bh.1001-5965.2022.0492

多策略融合改进的自适应蜉蝣算法

doi: 10.13700/j.bh.1001-5965.2022.0492
基金项目: 国家自然科学基金(51975422)
详细信息
    通讯作者:

    E-mail:xxz@whu.edu.cn

  • 中图分类号: TP301.6

Multi-strategy fusion improved adaptive mayfly algorithm

Funds: National Natural Science Foundation of China (51975422)
More Information
  • 摘要:

    为改进蜉蝣算法全局搜索能力较差、种群多样性较小和自适应能力弱等问题,提出一种多策略融合改进的自适应蜉蝣算法 (MIMA)。采用Sin混沌映射初始化蜉蝣种群,使种群能够均匀分布在解空间中,提高初始种群质量,增强全局搜索能力;引入Tent混沌映射和高斯变异对种群个体进行调节,增加种群多样性的同时调控种群密度,增强局部最优逃逸能力;引入不完全伽马函数,重构自适应动态调节的重力系数,建立全局搜索和局部开发能力之间更好的平衡,进而提升算法收敛精度,有利于提高全局搜索能力;采用随机反向学习 (ROBL) 策略,增强全局搜索能力,提高收敛速度并增强稳定性。利用经典测试函数集进行算法对比,并利用Wilcoxon秩和检验分析算法的优化效果,证明改进的有效性和可靠性。实验结果表明:所提算法与其他算法相比,寻优精度、收敛速度、稳定性都取得了较大提升。

     

  • 图 1  MIMA算法流程

    Figure 1.  MIMA Algorithm Flow Chart

    图 2  测试函数平均收敛曲线

    Figure 2.  Average convergence curves of test functions

    表  1  不同$\alpha $值的搜索结果

    Table  1.   Search results with different ${\boldsymbol{\alpha}} $ values

    $\alpha $ 平均值 最优值 标准差
    1 1.4275×10−40 5.1651×10−48 4.4247×10−40
    1.5 2.9239×10−40 3.7552×10−47 1.3173×10−39
    2 1.0435×10−39 5.1667×10−47 4.2855×10−39
    2.5 3.8230×10−39 8.3485×10−50 2.3260×10−38
    3 1.7396×10−38 2.5809×10−48 6.7601×10−38
    下载: 导出CSV

    表  2  基准测试函数

    Table  2.   Benchmark test function

    函数公式 维度 搜索空间 $ f_{\min } $
    ${f_1}\left( x \right) =\textstyle\sum \limits_{i = 1}^n x_i^2$ 20 [−100,100] 0
    ${f_2}\left( x \right) = \textstyle\sum \limits_{i = 1}^n \left| {{x_i}} \right| + \mathop \prod \limits_{i = 1}^n \left| {{x_i}} \right|$ 20 [−10,10] 0
    ${f_3}\left( x \right) = {\text{max}}\left\{ {\left| {{x_i}} \right|} \right\}$ 20 [−100,100] 0
    ${f_4}\left( x \right) = \mathop \sum \limits_{i = 1}^n {\left( {\left| {{x_i} + 0.5} \right|} \right)^2}$ 20 [−100,100] 0
    ${f_5}\left( x \right) = \mathop \sum \limits_{i = 1}^n x_i^4 + {\rm{random}}\left[ {0,1} \right)$ 20 [−1.28.1.28] 0
    ${f_6}\left( x \right) = \mathop \sum \limits_{i = 1}^n {\left| {{x_i}} \right|^{i + 1}}$ 20 [−1,1] 0
    ${f_7}\left( x \right) = - \mathop \sum \limits_{i = 1}^n \left[ {{x_i}\sin \left( {\sqrt {{x_i}} } \right)} \right]$ 20 [−500,500] −8379.66
    ${f_8}\left( x \right) = \mathop \sum \limits_{i = 1}^n \left( {x_i^2 - 10\cos \left( {2 \text{π} {x_i}} \right) + 10} \right)$ 20 [−5.12,5.12] 0
    ${f_9}\left( x \right) = - 20\exp \left( { - 0.2\sqrt {\dfrac{1}{n}\mathop \sum \limits_{i = 1}^n x_i^2} } \right) - {\mathrm{exp}}\left(\dfrac{1}{n}\mathop \sum \limits_{i = 1}^n \cos \left( {2 \text{π} {x_i}} \right)\right) + 20 + e$ 20 [−32,32] 0
    ${f_{10}}\left( x \right) = \dfrac{1}{{4\;000}}\mathop \sum \limits_{i = 1}^n x_i^2 - \mathop \prod \limits_{i = 1}^n \cos \left( {\dfrac{{{x_i}}}{{\sqrt i }}} \right) + 1$ 20 [−600,600] 0
    $ \begin{gathered} {f_{11}}\left( x \right) = \dfrac{ \text{π} }{n}\Biggr[ 10{\mathrm{si}}{{\mathrm{n}}^2}\left( { \text{π} {y_i}} \right) + \textstyle\sum \limits_{i = 1}^{n - 1} {\left( {{y_i} - 1} \right)^2} \Bigr( {1 + 10{\mathrm{si}}{{\mathrm{n}}^2}\left( { \text{π} {y_{i + 1}}} \right)}\Bigr) +\\ {\left( {{y_n} - 1} \right)^2}\Biggr] + \textstyle\sum \limits_{i = 1}^n u\left( {{x_i},10,100,4} \right) \\ \end{gathered} $ 20 [−50,50] 0
    ${f_{12}}\left( x \right) = {\left[0.002 + \sum \limits_{j = 1}^{25} \dfrac{1}{{j + \sum \limits_{i = 1}^2 {{\left( {{x_i} - {a_{ij}}} \right)}^6}}}\right]^{ - 1}}$ 2 [−65.53,65.53] 1
    ${f_{13}}\left( x \right) = \mathop \sum \limits_{i = 1}^{11} {\left[ {{a_i} - \dfrac{{{x_i}\left( {b_i^2 + {b_i}{x_2}} \right)}}{{b_i^2 + {b_i}{x_3} + {x_4}}}} \right]^2}$ 4 [−5,5] 3.075 × 10−4
    ${f_{14}}\left( x \right) = {\left( {{x_2} - \dfrac{{5.1}}{{4{ \text{π} ^2}}}x_1^2 + \dfrac{5}{ \text{π} }{x_1} - 11} \right)^2} + 10\left( {1 - \dfrac{1}{{8 \text{π} }}} \right){\mathrm{cos}}\;{x_1} + 10$ 2 [−5,10] 0.398
    下载: 导出CSV

    表  3  基准测试函数结果对比

    Table  3.   Comparison of benchmark test function results

    测试函数 平均值
    MIMA IMA PSO GWO SCA IMA2 TCMA
    ${f_1}$ 1.486×10−192 6.023×10−39 2.007 1.945×10−65 2.717×10−8 1.229×10−44 2.310×10−44
    ${f_2}$ 7.613×10−98 2.106×10−16 8.188×10−1 2.540×10−38 1.844×10−9 1.576×10−24 6.570×10−21
    ${f_3}$ 9.818×10−64 2.106×10−3 8.653×10−1 3.919×10−16 5.631×10−1 4.224×10−4 2.976×10−4
    ${f_4}$ 4.930×10−34 1.972×10−33 2.283 4.616×10−1 2.048 0 2.465×10−33
    ${f_5}$ 1.254×10−4 5.163×10−3 5.900×10−3 6.322×10−4 6.508×10−3 3.237×10−3 4.164×10−3
    ${f_6}$ 1.864×10−294 7.675×10−74 4.774×10−8 2.650×10−221 4.803×10−9 2.972×10−72 2.413×10−73
    ${f_7}$ −7.445×103 −6.881×103 −5.169×102 −6.130×103 −3.234×103 −8.380×103 −6.557×103
    ${f_8}$ 0 2.815 1.909×101 4.887×10−1 2.379 3.317×10−1 4.345
    ${f_9}$ 8.882×10−16 1.182 1.840 1.489×10−14 8.627 2.599×10−14 1.614
    ${f_{10}}$ 0 3.780×10−2 1.764×10−1 1.776×10−3 6.454×10−2 2.254×10−2 2.146×10−16
    ${f_{11}}$ 2.382×10−32 3.421×10−2 4.423×10−2 3.328×10−2 2.622×10−1 2.408×10−32 5.183×10−3
    ${f_{12}}$ 9.980×10−1 9.980×10−1 1.267×101 4.131 9.982×10−1 9.980×10−1 9.980×10−1
    ${f_{13}}$ 3.075×10−4 3.075×10−4 6.628×10−4 4.376×10−3 9.713×10−4 1.645×10−3 3.075×10−4
    ${f_{14}}$ 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1 3.987×10−1 3.979×10−1 3.979×10−1
    测试函数 最优值
    MIMA IMA PSO GWO SCA IMA2 TCMA
    ${f_{1}}$ 5.722×10−193 8.786×10−49 6.672×10−1 8.155×10−68 1.571×10−14 1.468×10−51 2.017×10−51
    ${f_{2}}$ 1.450×10−99 1.962×10−26 5.817×10−1 3.137×10−39 6.787×10−13 2.663×10−30 5.440×10−27
    ${f_{3}}$ 1.010×10−97 5.682×10−5 4.711×10−1 8.026×10−18 6.846×10−3 4.710×10−5 1.529×10−5
    ${f_{4}}$ 0 0 1.175 1.410×10−5 1.579 0 0
    ${f_{5}}$ 1.227×10−7 1.762×10−3 9.874×10−4 1.539×10−4 4.763×10−4 1.334×10−3 1.468×10−3
    ${f_{6}}$ 0 3.927×10−93 4.576×10−9 9.344×10−231 9.462×10−22 1.576×10−93 1.037×10−86
    ${f_{7}}$ −7.926×103 −7.787×103 −1.506×103 −8.067×103 −3.824×103 −8.380×103 −7.432×103
    ${f_{8}}$ 0 5.684×10−14 8.689 0 5.684×10−14 0 9.950×10−1
    ${f_{9}}$ 8.882×10−16 2.931×10−14 1.015 7.994×10−15 1.877×10−8 7.994×10−15 6.401×10−6
    ${f_{10}}$ 0 0 4.317×10−2 0 8.378×10−13 0 0
    ${f_{11}}$ 2.356×10−32 2.356×10−32 7.751×10−3 6.514×10−3 1.546×10−1 2.356×10−32 2.356×10−32
    ${f_{12}}$ 9.980×10−1 9.980×10−1 1.267×101 9.980×10−1 9.980×10−1 9.980×10−1 9.980×10−1
    ${f_{13}}$ 3.075×10−4 3.075×10−4 3.166×10−4 3.075×10−4 3.436×10−4 3.075×10−4 3.075×10−4
    ${f_{14}}$ 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1
    测试函数 标准差
    MIMA IMA PSO GWO SCA IMA2 TCMA
    ${f_{1}}$ 0 1.887×10−37 7.375×10−1 3.560×10−65 1.025×10−7 2.807×10−44 7.722×10−44
    ${f_{2}}$ 1.161×10−97 1.043×10−14 1.153×10−1 2.434×10−38 4.370×10−9 6.994×10−24 2.832×10−20
    ${f_{3}}$ 4.279×10−63 6.188×10−3 1.758×10−1 6.989×10−16 1.159 7.688×10−4 1.555×10−4
    ${f_{4}}$ 2.415×10−33 7.542×10−33 7.023×10−1 2.647×10−1 2.190×10−1 0 1.121×10−32
    ${f_{5}}$ 1.318×10−4 2.830×10−3 3.497×10−3 3.159×10−4 5.658×10−3 1.292×10−3 2.459×10−3
    ${f_{6}}$ 0 4.000×10−73 3.824×10−8 0 1.413×10−8 1.590×10−71 8.527×10−73
    ${f_{7}}$ 2.102×102 3.168×102 2.053×102 6.476×102 2.169×102 1.012×10−11 3.688×102
    ${f_{8}}$ 0 1.888 5.034 1.946 5.766 6.957×10−1 1.441
    ${f_{9}}$ 0 7.940×10−1 5.516×10−1 2.602×10−15 9.000 1.917×10−14 7.649×10−1
    ${f_{10}}$ 0 5.098×10−2 6.514×10−2 5.066×10−3 1.382×10−1 2.578×10−2 3.280×10−16
    ${f_{11}}$ 5.825×10−34 7.800×10−2 3.926×10−2 1.549×10−2 4.723×10−2 1.187×10−33 2.791×10−2
    ${f_{12}}$ 7.022×10−17 1.088×10−16 2.139×10−10 4.107 4.398×10−4 9.065×10−17 1.147×10−16
    ${f_{13}}$ 3.048×10−19 1.870×10−19 2.497×10−4 7.997×10−3 4.014×10−4 5.003×10−3 3.278×10−19
    ${f_{14}}$ 3.331×10−16 3.331×10−16 2.714×10−5 4.969×10−5 8.580×10−4 3.331×10−16 3.331×10−16
    下载: 导出CSV

    表  4  Wilcoxon秩和检验$p$值

    Table  4.   $p$-value for Wilcoxon’s rank-sum results

    测试函数 IMA PSO GWO SCA IMA2 TCMA
    ${f_1}$ 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11
    ${f_2}$ 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11
    ${f_3}$ 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11
    ${f_4}$ 7.4577×10−9 2.3638×10−12 2.3638×10−12 2.3638×10−12 1.6074×10−1 3.0199×10−11
    ${f_5}$ 3.0199×10−11 3.0199×10−11 7.3891×10−11 3.0199×10−11 3.0199×10−11 3.0199×10−11
    ${f_6}$ 6.4789×10−12 6.4789×10−12 6.4789×10−12 6.4789×10−12 6.4789×10−12 6.4789×10−12
    ${f_7}$ 2.5306×10−4 3.0199×10−11 9.0632×10−8 3.0199×10−11 2.4306×10−11 2.1947×10−8
    ${f_8}$ 2.5306×10−4 3.0199×10−11 9.0632×10−8 3.0199×10−11 4.5270×10−12 1.2118×10−12
    ${f_9}$ 1.2108×10−12 1.2118×10−12 2.0341×10−13 1.2118×10−12 1.0609×10−12 1.2118×10−12
    ${f_{10}}$ 5.7720×10−11 1.2118×10−12 8.1523×10−2 1.2118×10−12 1.6560×10−11 2.9112×10−2
    ${f_{11}}$ 1.1529×10−9 2.9729×10−11 2.9729×10−11 2.9729×10−11 3.1958×10−7 1.4735×10−7
    ${f_{12}}$ 3.0490×10−1 3.1578×10−12 1.9881×10−11 4.3909×10−12 7.8056×10−1 6.3241×10−1
    ${f_{13}}$ NaN 1.2118×10−12 1.2118×10−12 1.2118×10−12 NaN 8.1404×10−2
    ${f_{14}}$ NaN 1.2118×10−12 1.2118×10−12 1.2118×10−12 NaN NaN
     注:NaN表示算法性能相当。
    下载: 导出CSV

    表  5  消融实验结果

    Table  5.   Experimental results of ablation

    测试函数 平均值
    IMA CMA GMA MIMA
    ${f_1}$ 6.023×10−39 4.459×10−160 8.516×10−156 1.486×10−192
    ${f_2}$ 2.106×10−16 1.159×10−94 1.604×10−82 7.613×10−98
    ${f_3}$ 2.106×10−3 2.058×10−6 2.844×10−51 9.818×10−64
    ${f_4}$ 1.972×10−33 7.396×10−34 1.761×10−31 4.930×10−34
    ${f_5}$ 5.163×10−3 2.098×10−3 1.379×10−4 1.254×10−4
    ${f_6}$ 7.675×10−74 6.998×10−222 7.204×10−217 1.864×10−294
    ${f_7}$ −6.881×103 −7.233×103 −7.105×103 −7.445×103
    ${f_8}$ 2.815 1.678 0 0
    ${f_9}$ 1.182 4.370×10−15 8.882×10−16 8.882×10−16
    ${f_{10}}$ 3.780×10−2 0 0 0
    ${f_{11}}$ 3.421×10−2 1.662×10−2 1.960×10−31 2.382×10−32
    ${f_{12}}$ 9.980×10−1 9.980×10−1 9.980×10−1 9.980×10−1
    ${f_{13}}$ 3.075×10−4 1.310×10−3 3.075×10−4 3.075×10−4
    ${f_{14}}$ 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1
    测试函数 最优值
    IMA CMA GMA MIMA
    ${f_1}$ 8.786×10−49 4.162×10−173 4.062×10−196 5.722×10−193
    ${f_2}$ 1.962×10−26 3.206×10−95 4.226×10−98 1.450×10−99
    ${f_3}$ 5.682×10−5 1.443×10−24 9.776×10−98 1.010×10−97
    ${f_4}$ 0 0 0 0
    ${f_5}$ 1.762×10−3 3.955×10−4 5.288×10−6 1.227×10−7
    ${f_6}$ 3.927×10−93 1.580×10−232 7.363×10−229 0
    ${f_7}$ −7.787×103 −7.906×103 −7.906×103 −7.926×103
    ${f_8}$ 5.684×10−14 0 0 0
    ${f_9}$ 2.931×10−14 8.882×10−16 8.882×10−16 8.882×10−16
    ${f_{10}}$ 0 0 0 0
    ${f_{11}}$ 2.356×10−32 2.356×10−32 2.404×10−32 2.356×10−32
    ${f_{12}}$ 9.980×10−1 9.980×10−1 9.980×10−1 9.980×10−1
    ${f_{13}}$ 3.075×10−4 3.075×10−4 3.075×10−4 3.075×10−4
    ${f_{14}}$ 3.979×10−1 3.979×10−1 3.979×10−1 3.979×10−1
    测试函数 标准差
    IMA CMA GMA MIMA
    ${f_1}$ 1.887×10−37 2.425×10−159 5.945×10−155 0
    ${f_2}$ 1.043×10−14 1.268×10−94 1.123×10−81 1.161×10−97
    ${f_3}$ 6.188×10−3 9.700×10−6 1.240×10−50 4.279×10−63
    ${f_4}$ 7.542×10−33 2.927×10−33 5.428×10−31 2.415×10−33
    ${f_5}$ 2.830×10−3 1.350×10−3 1.045×10−4 1.318×10−4
    ${f_6}$ 4.000×10−73 0 0 0
    ${f_7}$ 3.168×102 2.789×102 4.038×102 2.102×102
    ${f_8}$ 1.888 2.873 0 0
    ${f_9}$ 7.940×10−1 4.974×10−16 0 0
    ${f_{10}}$ 5.098×10−2 0 0 0
    ${f_{11}}$ 7.800×10−2 5.627×10−2 5.236×10−31 5.825×10−34
    ${f_{12}}$ 1.088×10−16 5.439×10−17 2.259×10−10 7.022×10−17
    ${f_{13}}$ 1.870×10−19 4.371×10−3 2.341×10−19 3.048×10−19
    ${f_{14}}$ 3.331×10−16 3.331×10−16 0 3.331×10−16
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-06-16
  • 录用日期:  2022-09-23
  • 网络出版日期:  2022-10-10
  • 整期出版日期:  2024-04-29

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