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多模函数优化的改进花朵授粉算法

郭庆 惠晓滨 张贾奎 李正欣

郭庆, 惠晓滨, 张贾奎, 等 . 多模函数优化的改进花朵授粉算法[J]. 北京航空航天大学学报, 2018, 44(4): 828-840. doi: 10.13700/j.bh.1001-5965.2017.0240
引用本文: 郭庆, 惠晓滨, 张贾奎, 等 . 多模函数优化的改进花朵授粉算法[J]. 北京航空航天大学学报, 2018, 44(4): 828-840. doi: 10.13700/j.bh.1001-5965.2017.0240
GUO Qing, HUI Xiaobin, ZHANG Jiakui, et al. Improved flower pollination algorithm for multimodal function optimization[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(4): 828-840. doi: 10.13700/j.bh.1001-5965.2017.0240(in Chinese)
Citation: GUO Qing, HUI Xiaobin, ZHANG Jiakui, et al. Improved flower pollination algorithm for multimodal function optimization[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(4): 828-840. doi: 10.13700/j.bh.1001-5965.2017.0240(in Chinese)

多模函数优化的改进花朵授粉算法

doi: 10.13700/j.bh.1001-5965.2017.0240
基金项目: 

国家自然科学基金 61502521

详细信息
    作者简介:

    郭庆  男,硕士研究生。主要研究方向:信息系统工程与智能计算等

    惠晓滨  男, 博士, 教授, 博士生导师。主要研究方向:复杂系统建模与仿真、数据分析与智能决策

    张贾奎  男, 硕士研究生。主要研究方向:信息系统工程与智能计算等

    李正欣  男, 博士, 讲师。主要研究方向:复杂系统建模与仿真、数据分析与智能决策

    通讯作者:

    惠晓滨, E-mail: zibai4991@qq.com

  • 中图分类号: TP181

Improved flower pollination algorithm for multimodal function optimization

Funds: 

National Natural Science Foundation of China 61502521

More Information
  • 摘要:

    为了探讨花朵授粉算法(FPA)在解算多模函数优化问题中存在的不足,通过定义种群多样性及差异性指标,定性分析了FPA在多模复杂函数优化中的寻优缺点。基于模拟退火思想优化全局授粉过程,并利用Nelder-Mead单纯形搜索技术对花朵局部授粉进行重构,提出一种新的花朵授粉寻优架构。仿真结果表明,相对于基本的FPA、布谷鸟算法、萤火虫算法,改进花朵授粉算法能够有效避免陷入局部最优,具备优异的全局勘探和局部开采能力,对多模优化问题具有一定优势。

     

  • 图 1  Sphere函数3D图

    Figure 1.  3D figure of Sphere function

    图 2  Sphere函数FPA寻优参数变化曲线

    Figure 2.  Parameter variation curves of Sphere function using FPA optimization

    图 3  Sphere函数FPA寻优花粉分布

    Figure 3.  Pollen distribution of Sphere function using FPA optimization

    图 4  图 3第100代的局部放大

    Figure 4.  Partial enlargement of Fig. 3 the 100th generation

    图 5  Bridge函数3D图

    Figure 5.  3D figure of Bridge function

    图 6  Bridge函数FPA寻优参数变化曲线

    Figure 6.  Parameter variation curves of Bridge function using FPA optimization

    图 7  Bridge函数FPA寻优花粉分布

    Figure 7.  Pollen distribution of Bridge function using FPA optimization

    图 8  Nelder-Mead单纯形法

    Figure 8.  Nelder-Mead simplex method

    图 9  NS-FPA流程

    Figure 9.  Flowchart of NS-FPA

    图 10  Sphere函数NS-FPA寻优参数变化曲线

    Figure 10.  Parameter variation curves of Sphere function using NS-FPA optimization

    图 11  Sphere函数NS-FPA寻优花粉分布

    Figure 11.  Pollen distribution of Sphere function using NS-FPA optimization

    图 12  图 11第100代的局部放大

    Figure 12.  Partial enlargement of Fig. 11 the 100th generation

    图 13  Bridge函数NS-FPA寻优参数变化曲线

    Figure 13.  Parameter variation curves of Bridge function using NS-FPA optimization

    图 14  Bridge函数NS-FPA寻优花粉分布

    Figure 14.  Pollen distribution of Bridge function using NS-FPA optimization

    图 15  二维函数3D图

    Figure 15.  3D figures of 2D functions

    图 16  F11函数50次寻优结果对比

    Figure 16.  Comparison of 50 times' optimization results of F11 function

    图 17  F11函数寻优进化曲线

    Figure 17.  Optimization evolution curves of F11 function

    图 18  F13函数寻优进化曲线

    Figure 18.  Optimization evolution curves of F13 function

    表  1  4个算法的参数设置

    Table  1.   Parameter setup for four algorithms

    算法 参数设置
    FPA 转换概率p=0.8;参数λ=1.5
    CS 发现概率pa=0.25;步长调节量α=0.01;
    参数β=1.5
    FA 步长因子α=0.25;吸引度因子β=0.2;
    光强吸收强度γ=1
    NS-FPA 转换概率p=0.8;温度衰减系数γ=0.7;
    初温概率p0=0.8;参数λ=1.5
    注:公共参数:种群规模N=50;最大迭代次数Kmax,F1~F13取100,F14~F16取10 000。
    下载: 导出CSV

    表  2  16个标准测试函数

    Table  2.   16 standard test functions

    代号 函数 表达式 解空间 最小值
    F1 Sphere [-100, 100]2 0
    F2 Brid [-2π, 2π]2 -106.764 5
    F3 Roots [-2, 2]2 -1
    F4 Bohachevsky [-100, 100]2 0
    F5 Eggcrate [-10, 10]2 0
    F6 Guichi-f4 [-2, 2]2 -3.253 9
    F7 Cross-in-Tray [-10, 10]2 -2.062 61
    F8 Holder Table [-10, 10]2 -19.208 5
    F9 Drop-Wave [-5.12, 5.12]2 -1
    F10 Levy [-10, 10]2 0
    F11 Bridge [-10, 10]2 -3.005 4
    F12 Shubert [-50, 50]2 -186.730 9
    F13 Rastrigin [-20, 20]4 0
    F14 Ackley [-32, 32]30 0
    F15 Griewank [-600, 600]40 0
    F16 Vincent [0.25, 10]50 -50
    下载: 导出CSV

    表  3  16个标准测试函数优化结果对比

    Table  3.   Comparison of optimization results of 16 standard test functions

    函数代号 FPA CS
    最优值 平均值 标准差 最优值 平均值 标准差
    F1 2.82 5.42 1.63 9.85×10-7 2.37×10-3 5.49×10-4
    F2 -106.764 5 -106.762 7 2.24×10-3 -106.764 5 -106.757 0 1.64×10-3
    F3 -0.999 8 -0.997 0 1.88×10-3 -0.999 7 -0.994 5 1.21×10-3
    F4 6.46×10-5 2.46×10-2 3.49×10-2 2.56×10-1 1.81 3.97×10-1
    F5 2.09×10-6 2.41×10-3 5.07×10-3 1.88×10-3 0.53 1.24×10-1
    F6 -3.253 9 -3.253 7 3.20×10-4 -3.253 9 -3.253 3 1.29×10-4
    F7 -2.062 6 -2.062 3 7.46×10-5 -2.062 6 -2.062 5 2.11×10-5
    F8 -19.208 5 -19.208 3 1.73×10-4 -19.208 5 -19.208 2 7.18×10-5
    F9 -0.993 3 -0.947 3 1.77×10-2 -0.986 5 -0.936 2 1.77×10-2
    F10 1.24×10-5 2.93×10-2 3.70×10-2 4.06×10-6 5.60×10-3 1.40×10-3
    F11 -3.004 1 -2.877 1 8.56×10-2 -2.928 0 -2.651 2 7.64×10-2
    F12 -186.730 7 -186.712 5 2.68×10-2 -186.730 8 -186.709 8 6.13×10-3
    F13 6.60 15.36 5.01 2.93 8.05 1.46
    F14 3.87 6.34 1.17 4.66 8.35 0.92
    F15 4.64 11.29 2.66 2.73 6.97 1.21
    F16 -49.61 -49.22 0.16 -49.84 -49.74 9.33×10-2
    F1 8.63×10-11 5.79×10-9 5.66×10-9 3.88×10-9 1.82×10-5 2.89×10-5
    F2 -106.764 5 -104.939 3 5.23 -106.764 5 -106.764 5 4.76×10-6
    F3 -1.000 0 -0.999 8 1.06×10-4 -1.000 0 -0.999 9 7.81×10-5
    F4 6.70×10-7 4.88×10-4 8.66×10-4 8.32×10-9 5.00×10-5 3.94×10-5
    F5 8.34×10-8 3.59×10-6 5.29×10-6 5.15×10-9 1.88×10-7 1.70×10-7
    F6 -3.253 9 -3.253 1 8.20×10-4 -3.253 9 -3.253 9 5.93×10-6
    F7 -2.062 6 -2.062 6 1.77×10-4 -2.062 6 -2.062 6 1.30×10-8
    F8 -19.208 5 -19.207 9 9.73×10-4 -19.208 5 -19.208 5 4.50×10-10
    F9 -1.000 0 -0.994 9 1.75×10-2 -1.000 0 -0.999 9 1.99×10-4
    F10 4.68×10-9 1.41×10-7 2.10×10-7 1.26×10-10 6.79×10-9 7.70×10-9
    F11 -3.005 4 -2.933 6 0.20 -3.005 4 -3.005 1 3.18×10-4
    F12 -186.730 9 -177.764 8 13.30 -186.730 9 -186.730 8 5.50×10-4
    F13 2.38×10-6 1.33 1.41 0 1.18×10-7 7.42×10-7
    F14 1.80×10-3 3.83×10-3 1.12×10-3 6.46×10-12 1.62×10-7 1.09×10-6
    F15 3.00×10-4 7.47×10-2 0.19 2.97×10-4 1.32×10-3 7.53×10-4
    F16 -49.87 -9.62 9.83×10-2 -50.00 -49.99 2.51×10-4
    下载: 导出CSV

    表  4  不同p0γ参数下的NS-FPA寻优结果

    Table  4.   Optimization results of NS-FPA with different p0 and γ

    p0 平均值 标准差 γ 平均值 标准差
    0.1 -0.998 08 9.20×10-3 0.1 -0.996 70 1.28×10-2
    0.2 -0.998 44 9.19×10-3 0.2 -0.999 11 4.04×10-3
    0.3 -0.998 61 8.57×10-3 0.3 -0.999 15 3.27×10-3
    0.4 -0.999 14 4.14×10-3 0.4 -0.999 65 1.40×10-3
    0.5 -0.999 35 2.73×10-3 0.5 -0.999 72 1.21×10-3
    0.6 -0.999 81 9.14×10-4 0.6 -0.999 78 1.14×10-3
    0.7 -0.999 99 6.52×10-5 0.7 -0.999 83 9.39×10-4
    0.8 -0.999 93 9.59×10-5 0.8 -0.999 98 7.47×10-5
    0.9 -0.999 86 7.12×10-4 0.9 -0.999 89 6.15×10-4
    下载: 导出CSV

    表  5  NS-FPA与FPA的平均消耗时间对比

    Table  5.   Comparison of mean consumption time between NS-FPA and FPA

    函数代号 平均耗时/s NS-FPA与FPA平均
    耗时的百分比
    NS-FPA FPA
    F5 0.28 0.20 142.56
    F6 0.28 0.20 142.76
    F11 0.31 0.22 137.15
    F13 0.28 0.22 126.89
    F14 33.56 27.22 123.31
    F15 34.99 28.63 122.24
    下载: 导出CSV
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出版历程
  • 收稿日期:  2017-04-18
  • 录用日期:  2017-05-19
  • 刊出日期:  2018-04-20

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