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噪声环境下基于蒲丰距离的依概率多峰优化算法

王霞 王耀民 施心陵 高莲 李鹏

王霞, 王耀民, 施心陵, 高莲, 李鹏. 噪声环境下基于蒲丰距离的依概率多峰优化算法. 自动化学报, 2019, 45(x): 1−23 doi: 10.16383/j.aas.c190474
引用本文: 王霞, 王耀民, 施心陵, 高莲, 李鹏. 噪声环境下基于蒲丰距离的依概率多峰优化算法. 自动化学报, 2019, 45(x): 1−23 doi: 10.16383/j.aas.c190474
Wang Xia, Wang Yao-Min, Shi Xin-Ling, Gao Lian, Li Peng. Probabilistic multimodal optimization algorithm based on buffon distance in noisy environment. Acta Automatica Sinica, 2019, 45(x): 1−23 doi: 10.16383/j.aas.c190474
Citation: Wang Xia, Wang Yao-Min, Shi Xin-Ling, Gao Lian, Li Peng. Probabilistic multimodal optimization algorithm based on buffon distance in noisy environment. Acta Automatica Sinica, 2019, 45(x): 1−23 doi: 10.16383/j.aas.c190474

噪声环境下基于蒲丰距离的依概率多峰优化算法

doi: 10.16383/j.aas.c190474
基金项目: 国家自然科学基金(61763046, 61762093, 61662089, 61761048), 云南省第十七批中青年学术和技术带头人资助项目(2014HB019), 云南省高校科技创新团队支持计划资助, 云南省教育厅科学研究基金(2017ZZX229), 云南省应用基础研究重点项目(2018FA036)资助
详细信息
    作者简介:

    王霞:云南大学博士研究生. 主要研究方向为智能优化算法. E-mail: wangxiacsu@163.com

    王耀民:云南大学博士研究生. 主要研究方向为SDN、数据中心和智能优化算法. E-mail: 18988081898@189.cn

    施心陵:云南大学信息学院教授. 研究方向为智能优化算法, 自适应信号处理与信息系统, 医学电子学. 本文通信作者. E-mail: xlshi@ynu.edu.cn

    高莲:云南大学信息学院讲师. 研究方向为生物医学信号处理. E-mail: ylbgl123@sina.com

    李鹏:云南大学信息学院副教授. 研究方向为输变电系统安全诊断、预警与维护决策研究. E-mail: lipeng@ynu.edu.cn

Probabilistic Multimodal Optimization Algorithm based on Buffon Distance in Noisy Environment

Funds: Supported by National Natural Science Foundation of China (61763046, 61762093, 61662089, and 61761048), The17th batches of Young and Middle-aged Leaders in Academic and Technical Reserved Talents Project of Yunnan Province under Grant (2014HB019), The Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province. Yunnan Provincial Department of Education Science Research Fund Project (2017ZZX229), and Key Projects of Applied Basic Research of Yunnan Province (2018FA036)
  • 摘要: 针对噪声环境下求解多个极值点的问题, 本文提出了噪声环境下基于蒲丰距离的依概率多峰优化算法(PMB). 算法依据蒲丰投针原理提出噪声下的蒲丰距离和极值分辨度概念, 理论推导证明了二者与算法峰值检测率符合依概率关系. 在全局范围内依据蒲丰距离划分搜索空间, 可以使PMB算法保持较好的搜索多样性. 在局部范围内利用改进的斐波那契法进行探索, 减少了算法陷入噪声引起的局部最优的概率. 基于34个测试函数, 从依概率特性验证、寻优结果影响因素分析、多极值点寻优和多维函数寻优四个角度进行实验. 证明了蒲丰距离与算法的峰值检测率符合所推导的依概率关系. 对比噪声环境下的改进蝙蝠算法和粒子群算法(Particle Swarm Optimization, PSO), PMB算法在噪声环境中可以依定概率更精确地定位多峰函数的更多极值点, 从而证明了PMB算法原理的正确性和噪声条件下全局寻优的依概率性能, 具有理论意义和实用价值.
  • 图  1  极值分辨度

    Fig.  1  Extreme value resolution

    图  2  函数波形与极值点

    Fig.  2  Function waveform and extremum points

    图  3  蒲丰投针实验设计图

    Fig.  3  Experimental design of buffon needles

    图  4  式(9)证明示意图

    Fig.  4  Diagram to prove (9)

    图  5  蒲丰距离与针长的关系示意图

    Fig.  5  Diagram of the relationship between buffon distance and needle length

    图  6  改进的斐波那契法的区域伸缩过程

    Fig.  6  The process of regional scaling of improved fibonacci method

    图  7  斐波那契法的区域收缩过程

    Fig.  7  The process of regional shrink of fibonacci method

    图  8  PMB算法的多级划分策略

    Fig.  8  The multilevel partition strategy of PMB algorithm

    图  9  $a$ = 0.2时无噪声环境下PMB算法对TF1中的$f_1$的寻优结果

    Fig.  9  Optimization results for $f_1$ of TF1 by PMB algorithm in noiseless environment with $a$ = 0.2

    图  14  $\sigma^2$ = 0.01, $a$ = 0.2时PMB算法对TF1中的$f_1$的寻优结果

    Fig.  14  Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.01 and $a$ = 0.2

    图  11  $\sigma^2$ = 0.05, $a$ = 0.5时PMB算法对TF1中的$f_1$的寻优结果

    Fig.  11  Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.05 and$a$ = 0.5

    图  10  $a$ = 0.5时无噪声环境下PMB算法对TF1中的$f_1$的寻优结果

    Fig.  10  Optimization results for $f_1$ of TF1 by PMB algorithm in noiseless environment with $a$ = 0.5

    图  12  $\sigma^2$ = 0.01,$a$ = 0.5时PMB算法对TF1中的$f_1$的寻优结果

    Fig.  12  Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.01 and $a$ = 0.5

    图  13  $\sigma^2$ = 0.05,$a$ = 0.2时PMB算法对TF1中的$f_1$的寻优结果

    Fig.  13  Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.05 and $a$ = 0.2

    图  15  $a$ = 0.2, PMB算法在不同噪声环境下对TF1中的$f_1$求解所有极值点的定位误差

    Fig.  15  Positioning errors of total-extremum points for $f_1$ of TF1 searched by PMB algorithm with $a$ = 0.2 under different noise environments

    图  16  $a$ = 1, $\sigma^2$ = 0.05, PMB算法对TF1中 $f_2$的寻优结果

    Fig.  16  Optimization results for $f_2$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.05 and $a$ = 1

    表  1  测试函数集1 (TF1)

    Table  1  Test function 1 (TF1)

    函数 名称 定义域 全局极值点 表达式
    $f_1$ [−1, 1] 3.5326($\pm $0.8781, $\pm $0.8781) ${{f}_{1}} = x_{1}^{2}+x_{2}^{2}-\cos(18{{x}_{1}})-\cos (18{{x}_{2}})$
    $f_2$ Levy 05 [−10, 10] −176.1375 (−1.3068, −1.4248) $\begin{aligned} {f}_{2}=\;& \sum\nolimits_{i = 1}^{5}{i\cos [(i-1){{x}_{1}}}+i]\times \sum\nolimits_{j = 1}^{5}{j\cos [(j+1){{x}_{2}}+j]}+ \\&{{({{x}_{1}}+1.42513)}^{2}}+{{({{{x}}_{2}}+0.80032)}^{2}}\end{aligned}$
    $f_3$ Michalewics [0, ${\text{π}}$] −1.8013 (2.2031, 1.5709) ${{f}_{3}} = -\displaystyle\sum\nolimits_{i = 1}^{2}{\sin ({{x}_{i}})}{{\left(\sin \left(i\times x_{i}^{2}/{\text{π}} \right)\right)}^{20}}$
    $f_4$ Periodic [0, ${\text{π}}$] 0.9 (0,0) ${{f}_{4}} = 1+{{\sin }^{2}}({{x}_{1}})+{{\sin }^{2}}({{x}_{2}})-0.1{{ \rm{e}}^{-(x_{1}^{2}+x_{2}^{2})}}$
    $f_5$ Carrom Table [−10, 10] −24.1568($\pm $9.6466, 9.6466) ${{f}_{7}} = -\left[{\left\{\cos ({{x}_{1}})\cos({{x}_{2}}){{ \rm{e}}^{\left| 1-\sqrt{x_{1}^{2}+x_{2}^{2}}/{\text{π}} \right|}}\right\}}^{2}\right]/30$
    $f_6$ Holder Table 2 [−10, 10] -19.2085($\pm $8.0550, $\pm $9.6646) ${{f}_{8}} = -\left| \sin ({{x}_{1}})\cos ({{x}_{2}}){{ \rm{e}}^{\left| 1-\sqrt{{{x}_{1}}+{{x}_{2}}}/{\text{π}} \right|}} \right|$
    下载: 导出CSV

    表  2  测试函数集2(TF2)(维度为5)

    Table  2  Test function 2(TF2)(dimension is 5)

    函数 全局最优值$f_{i}^{*}$ 函数名称
    单峰函数 $f_1$ −1 400 Sphere Function
    $f_2$ −1 300 Rotated High Conditioned Elliptic Function
    $f_3$ −1 200 Rotated Bent Cigar Function
    $f_4$ −1 100 Rotated Discus Function
    $f_5$ −1 000 Different Powers Function
    多峰函数 $f_6$ −900 Rotated Rosenbrock's Function
    $f_7$ −800 Rotated Schaffers F7 Function
    $f_8$ −700 Rotated Ackley's Function
    $f_9$ −600 Rotated Weierstrass Function
    $f_{10}$ −500 Rotated Griewank's Function
    $f_{11}$ −400 Rastrigin's Function
    $f_{12}$ −300 Rotated Rastrigin's Function
    $f_{13}$ −200 Non-Continuous Rotated Rastrigin's Function
    $f_{14}$ −100 Schwefel's Function
    $f_{15}$ 100 Rotated Schwefel's Function
    $f_{16}$ 200 Rotated Katsuura Function
    $f_{17}$ 300 Lunacek Bi_Rastrigin Function
    $f_{18}$ 400 Rotated Lunacek Bi_Rastrigin Function
    $f_{19}$ 500 Expanded Griewank’s plus Rosenbrock's Function
    $f_{20}$ 600 Expanded Scaffer's F6 Function
    组合函数 $f_{21}$ 700 Composition Function 1 (n = 5, Rotated)
    $f_{22}$ 800 Composition Function 2 (n = 3, Unrotated)
    $f_{23}$ 900 Composition Function 3 (n = 3, Rotated)
    $f_{24}$ 1 000 Composition Function 4 (n = 3, Rotated)
    $f_{25}$ 1 100 Composition Function 5 (n = 3, Rotated)
    $f_{26}$ 1 200 Composition Function 6 (n = 5, Rotated)
    $f_{27}$ 1 300 Composition Function 7 (n = 5, Rotated)
    $f_{28}$ 1 400 Composition Function 8 (n = 5, Rotated)
    下载: 导出CSV

    表  5  $a$ = 0.5时PMB算法针对TF1中的 $f_1$在不同噪声环境下的全局极值点优化数值结果

    Table  5  Numerical results of global extremum points for $f_1$ of TF1 searched by PMB algorithm with $a$ = 0.5 under different noise environments

    极值点序号 P-MQHOA[1]无噪声 PMB
    $\sigma^2$ = 0.01 $\sigma^2$ = 0.05
    $x_1$ $x_2$ $f$ $x_1$ $x_2$ $f$ $P_f$ $P_l$ $x_1$ $x_2$ $f$ $P_f$ $P_l$
    1 −0.8781 −0.8781 3.5326 −0.8760 −0.8780 3.6923 1.60E-01 2.07E-03 −0.8805 −0.8775 3.9568 4.24E-01 2.46E-03
    2 −0.8781 0.8781 3.5326 −0.8771 0.8782 3.6953 1.63E-01 1.03E-03 −0.8770 0.8803 3.9510 4.18E-01 2.50E-03
    3 0.8781 −0.8781 3.5326 0.8775 −0.8794 3.6847 1.52E-01 1.05E-03 0.8772 −0.8788 3.9467 4.14E-01 1.14E-03
    4 0.8781 0.8781 3.5326 0.8788 0.8784 3.6917 1.59E-01 8.18E-04 0.8775 0.8756 3.9697 4.37E-01 2.52E-03
    下载: 导出CSV

    表  3  针对TF1中的$f_1$取不同蒲丰距离时PMB算法在不同噪声环境下的优化结果

    Table  3  Optimization results of PMB algorithm for $f_1$ of TF1 in different noise environments with different Buffon distances

    $\sigma^2$ $a$ Num_mean Num_max Num_min $\eta\_{\rm{mean}}$ $\eta\_{\rm{global}}$
    50 次 0.01 0.5 29.62 33 26 82.28 % 100.00 %
    0.2 36.04 37 36 100.11 % 100.00 %
    0.05 0.5 31.32 35 27 87.00 % 100.00 %
    0.2 36.26 38 36 100.72 % 100.00 %
    100 次 0.01 0.5 29.78 33 27 82.72 % 100.00 %
    0.2 36.02 37 36 100.06 % 100.00 %
    0.05 0.5 32.08 36 28 89.11 % 100.00 %
    0.2 36.37 38 36 101.03 % 100.00 %
    200 次 0.01 0.5 29.715 34 23 82.54 % 100.00 %
    0.2 36.02 37 36 100.06 % 100.00 %
    0.05 0.5 32 35 26 88.89 % 100.00 %
    0.2 36.4 40 36 101.11 % 100.00 %
    下载: 导出CSV

    表  4  $a$ = 0.2时PMB算法针对TF1中的 $f_1$在不同噪声环境下的全局极值点优化数值结果

    Table  4  Numerical results of global extremum points for $f_1$ of TF1 searched by PMB algorithm with $a$ = 0.2 under different noise environments

    极值点序号 P-MQHOA[1]无噪声 PMB
    $\sigma^2$ = 0.01 $\sigma^2$ = 0.05
    $x_1$ $x_2$ $f$ $x_1$ $x_2$ $f$ $P_f$ $P_l$ $x_1$ $x_2$ $f$ $P_f$ $P_l$
    1 −0.8781 −0.8781 3.5326 −0.8781 −0.878 3.8057 2.73E-01 1.42E-04 −0.8776 −0.8773 4.1956 6.63E-01 9.37E-04
    2 −0.8781 0.8781 3.5326 −0.8789 0.8784 3.8027 2.70E-01 7.86E-04 −0.8788 0.8782 4.1974 6.65E-01 8.19E-04
    3 0.8781 −0.8781 3.5326 0.8782 −0.8778 3.8086 2.76E-01 3.64E-04 0.8788 −0.8784 4.2048 6.72E-01 7.36E-04
    4 0.8781 0.8781 3.5326 0.8783 0.8775 3.8049 2.72E-01 6.60E-04 0.8774 0.8765 4.2050 6.72E-01 1.79E-03
    下载: 导出CSV

    表  6  P-MQHOA[1]算法与$a$ = 0.2时的PMB算法针对TF1中的 $f_1$在不同环境下的所有极值点求解结果对比

    Table  6  Comparison of total-extremum points for $f_1$ of TF1 searched by P-MQHOA[1]algorithm and PMB algorithm with $a$ = 0.2 under different noise environments

    极值点序号 P-MQHOA[1]无噪声 PMB
    $\sigma^2$ = 0.01 $\sigma^2$ = 0.09
    $x_1$ $x_2$ $f(x_1,x_2)$ $x_1$ $x_2$ $f(x_1,x_2)$ $x_1$ $x_2$ $f(x_1,x_2)$
    1 −0.8781 −0.8781 3.5326 −0.8781 −0.8780 3.8057 −0.8762 −0.8777 4.4627
    2 −0.8781 −0.5269 3.0421 −0.8778 −0.5263 3.3069 −0.8774 −0.5285 3.9294
    3 −0.8781 −0.1756 2.7969 −0.8784 −0.1746 3.0482 −0.8778 −0.1818 3.6818
    4 −0.8781 0.1756 2.7969 −0.8789 0.1763 3.0605 −0.8790 0.1814 3.7123
    5 −0.8781 0.5269 3.0421 −0.8788 0.5277 3.3100 −0.8761 0.5258 3.9341
    6 −0.8781 0.8781 3.5326 −0.8789 0.8784 3.8027 −0.8798 0.8774 4.4562
    7 −0.5269 −0.8781 3.0421 −0.5276 −0.8780 3.3047 −0.5264 −0.8766 3.9330
    8 −0.5269 −0.5269 2.5517 −0.5266 −0.5267 3.4294 −0.52621 −0.5269 2.7918
    9 −0.5269 −0.1756 2.3065 −0.5264 −0.1756 2.5465 −0.5278 −0.1798 3.1998
    10 −0.5269 0.1756 2.3065 −0.5263 0.1760 2.5499 −0.5289 0.1772 3.1915
    11 −0.5269 0.5269 2.5517 −0.5268 0.5256 2.7924 −0.526 0.5278 3.4282
    12 −0.5269 0.8781 3.0421 −0.5268 0.8775 3.3045 −0.5271 0.8770 3.9228
    13 −0.1756 −0.8781 2.7969 −0.1770 −0.8786 3.0600 −0.1745 −0.8800 3.6858
    14 −0.1756 −0.5269 2.3065 −0.1761 −0.5275 2.5472 −0.1792 −0.5261 3.1804
    15 −0.1756 −0.1756 2.0613 −0.1772 −0.1755 2.2932 −0.1775 −0.1789 2.9170
    16 −0.1756 0.1756 2.0613 −0.1765 0.1746 2.2879 −0.1755 0.1774 2.9344
    17 −0.1756 0.5269 2.3065 −0.1763 0.5278 2.5418 −0.1757 0.5250 3.1700
    18 −0.1756 0.8781 2.7969 −0.1756 0.8791 3.0500 −0.1784 0.8817 3.6725
    19 0.1756 −0.8781 2.7969 0.1762 −0.8773 3.0574 0.1769 −0.8757 3.6846
    20 0.1756 −0.5269 2.3065 0.1762 −0.5266 2.5532 0.1794 −0.5270 3.1708
    21 0.1756 −0.1756 2.0613 0.1756 −0.1779 2.2964 0.1802 −0.1758 2.9292
    22 0.1756 0.1756 2.0613 0.1756 0.1752 2.2921 0.1790 0.1766 2.9235
    23 0.1756 0.5269 2.3065 0.1752 0.5269 2.5441 0.1820 0.5250 3.1637
    24 0.1756 0.8781 2.7969 0.1756 0.8800 3.0573 0.1803 0.8784 3.6873
    25 0.5269 −0.8781 3.0421 0.5253 −0.8767 3.2985 0.5250 −0.8808 3.9639
    26 0.5269 −0.5269 2.5517 0.5254 −0.5261 2.8010 0.5256 −0.5244 3.4443
    27 0.5269 −0.1756 2.3065 0.5275 −0.1752 2.5432 0.5266 −0.1785 3.1973
    28 0.5269 0.1756 2.3065 0.5266 0.1775 2.5502 0.5291 0.1789 3.1769
    29 0.5269 0.5269 2.5517 0.5273 0.5259 2.7941 0.5277 0.5257 3.4395
    30 0.5269 0.8781 3.0421 0.5254 0.8776 3.3128 0.5242 0.8817 3.9135
    31 0.8781 −0.8781 3.5326 0.8782 −0.8778 3.8086 0.8793 −0.8759 4.4479
    32 0.8781 −0.5269 3.0421 0.8792 −0.5265 3.3119 0.8771 −0.5288 3.9199
    33 0.8781 −0.1756 2.7969 0.8778 −0.1766 3.0594 0.8774 −0.1782 3.7063
    34 0.8781 0.1756 2.7969 0.8780 0.1760 3.0549 0.8772 0.1784 3.6981
    35 0.8781 0.5269 3.0421 0.8784 0.5265 3.3001 0.8788 0.5279 3.3046
    36 0.8781 0.8781 3.5326 0.8783 0.8775 3.8049 0.8803 0.8792 4.4365
    下载: 导出CSV

    表  7  PMB算法对TF1中的$f_2- f_6$在不同噪声环境下的全局极值点优化数值结果

    Table  7  Numerical results of global extremum points for function $f_2- f_6$ of TF1searched by PMB algorithm under different noise environments

    函数 全局极值 指标 IBA[32] PMB
    $\sigma^2$ = 0.01 $\sigma^2$ = 0.05 $\sigma^2$ = 0.07 $\sigma^2$ = 0.09 $\sigma^2$ = 0.01 $\sigma^2$ = 0.05 $\sigma^2$ = 0.07 $\sigma^2$ = 0.09
    $f_2$ −176.1375 Mean −176.1830 −176.3646 −176.5443 −176.5442 −176.1607 −176.0738 −176.0689 −175.9722
    MAPE 0.0258 0.1289 0.1799 0.2309 0.0132 0.0362 0.0389 0.0938
    $f_3$ −1.8013 Mean −1.8468 −2.0283 −2.1206 −2.2095 −2.0007 −2.3048 −2.4433 −2.5351
    MAPE 2.5281 12.6022 17.7255 22.6617 11.0675 27.9516 35.6393 40.7363
    $f_4$ 0.9 Mean 0.8556 0.6838 0.6053 0.5330 0.6534 0.2620 0.1312 0.0529
    MAPE 4.9308 24.0261 32.7394 46.5648 27.4009 70.8921 85.4263 94.1181
    $f_5$ −24.1568 Mean −24.2016 −24.3806 −24.4665 −24.5567 −24.2903 −24.538 −24.5977 −24.6518
    MAPE 0.1856 0.9266 1.2822 1.6556 0.5526 1.5781 1.8253 2.0490
    $f_6$ −-19.2085 Mean −19.253 −19.4329 −19.5236 −19.6095 −19.3724 −19.6408 −19.7391 −19.8169
    MAPE 0.2319 1.1683 1.6403 2.0875 0.8533 2.2504 2.7625 3.1673
    下载: 导出CSV

    表  8  PMB算法对TF1中$f_2- f_4$在不同噪声环境下得到的全局极值点的定位误差

    Table  8  Positioning errors of global extremum points for function $f_2-f_4$ of TF1 searched by PMB algorithm under different noise environments

    函数 $\sigma^2$ IBA[32] PMB
    $P_l$ $x_1$ $x_2$ $P_l$
    $f_2$ 0.01 0.00082301 −1.3062 −1.4246 0.0007
    0.05 0.0019 −1.3064 −1.4241 0.0009
    0.07 0.0022 −1.3079 −1.4244 0.0011
    0.09 0.0023 −1.3041 −1.4252 0.0027
    $f_3$ 0.01 0.0540 2.2088 1.5727 0.0059
    0.05 0.0132 2.1913 1.5723 0.0119
    0.07 0.0160 2.1882 1.5664 0.0156
    0.09 0.0201 2.1833 1.5710 0.0198
    $f_4$ 0.01 0.0282 0.0611 0.0698 0.0927
    0.05 0.0830 0.0836 0.0623 0.1042
    0.07 1.1067 0.0820 0.0655 0.1049
    0.09 2.2026 0.0579 0.0934 0.1099
    下载: 导出CSV

    表  9  PMB算法对TF1中的$f_5- f_6$在不同噪声环境下的全局极值点优化数值结果

    Table  9  Numerical results of global extremum points for function $f_5- f_6$ of TF1searched by PMB algorithm under different noise environments

    函数 算法 IBA[32] PMB
    $\sigma^2$ 0.01 0.05 0.07 0.09 0.01 0.05 0.07 0.09
    极值点序号 $P_l$ $(x_1, x_2), P_l$
    $f_5$ 1 0.0048 0.0151 0.0179 0.0206 (9.6453, −9.6439), 0.0030 (9.6498, −9.6444), 0.0039 (9.6398, −9.6475), 0.0068 (9.6559, −9.6482), 0.0094
    2 0.0057 0.0162 0.016 0.0158 (−9.6457, 9.6444), 0.0023 (−9.6473, 9.6442), 0.0025 (−9.6520, 9.6450), 0.0057 (−9.6473, 9.6401), 0.0065
    3 0.006 0.0112 0.0149 0.0153 (−9.6454, −9.6455), 0.0016 (−9.6460, −9.6413), 0.0053 (−9.6427, −9.6406), 0.0072 (−9.6407, −9.6419), 0.0076
    4 0.0044 0.0133 0.0151 0.0178 (9.6476, 9.6468), 0.0010 (9.6477, 9.6528), 0.0063 (9.6493, 9.6532), 0.0071 (9.6434, 9.6390), 0.0082
    $f_6$ 1 0.0107 0.0215 0.0238 0.0291 (8.0518, −9.6675), 0.0043 (8.0521, −9.6712), 0.0072 (8.0462, −9.6588), 0.0106 (8.0525, −9.6534), 0.0115
    2 0.0094 0.0207 0.0262 0.0289 (−8.0541, 9.6626), 0.0022 (−8.0532, 9.6629), 0.0025 (−8.0561, 9.6614), 0.0034 (−8.0586, 9.6567), 0.0087
    3 0.0088 0.0199 0.0242 0.0243 (−8.0513, −9.6634), 0.0039 (−8.0541, −9.6590), 0.0057 (−8.0595, −9.6698), 0.0069 (−8.0602, −9.6566), 0.0096
    4 0.0081 0.0241 0.021 0.0238 (8.0529, 9.6611), 0.0041 (8.0503, 9.6688), 0.0063 (8.0547, 9.6757), 0.0111 (8.0642, 9.6743), 0.0134
    下载: 导出CSV

    表  10  PMB算法和PSO算法对TF2在无噪声环境下得到的全局极值点

    Table  10  Global extremum points of TF2 obtained by PMB algorithm and PSO algorithm in noiseless environment

    TF2 Opti
    mum
    PSO PMB
    $f$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $t(s)$ $f$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $t(s)$
    $f_1$ −1 400 −1 400. 5153 −22.003 11.6561 −36.1216 69.3583 −37.5101 1.2932 −1 399.9755 −22.1013 11.5532 −35.9107 69.3869 −37. 6351 4.024
    $f_2$ −1 300 −608.3034 −23.2765 15.9243 −18.1464 70.6333 −35.3268 1.586 −1 284.0072 −22.1909 10.6542 −37.981 69.5946 −37.5306 13.084
    $f_3$ −1 200 −1 191. 9927 −90.9013 11.4835 −35.8577 68.8415 −40.1122 1.3798 −1 199.4865 −32.8088 11.5433 −35.9866 69.289 −38.0021 30.5866
    $f_4$ −1 100 −1 100. 2332 −22.0721 11.3036 −35.9249 69.1684 −37.6371 1.4573 −1 098.4189 −22.0954 11.9348 −36.1631 68.5683 −37.4961 52.0969
    $f_5$ −1 000 −1 000. 4002 −21.9647 11.3823 −36.0354 69.0788 −37.36 1.326 −999.9999 −21.9848 11.5549 −36.0044 69.3447 −37.6114 73.5208
    $f_6$ −900 −900. 1464 −48.0332 16.1924 −31.6003 85.6123 −64.8171 1.2566 −900.0000 −22.0309 11.5686 −36.0034 69.4094 −37.6539 6.7166
    $f_7$ −800 −800. 314 −47.6006 11.5431 −35.9756 69.3023 −37.8813 1.9918 −799.9792 −22.0032 11.5635 −36.004 69.3839 −37.5808 24.2937
    $f_8$ −700 −680.2271 −89.9164 95.3877 55.5365 −63.3428 −20.1823 1.6314 −699.9191 −22.2758 11.5436 −36.0022 69.3631 −37.6026 44.2769
    $f_9$ −600 −600. 3439 75.0307 21.2084 −65.4178 −90.6598 25.5987 10.7053 −599.9286 −22.0921 11.5053 −35.998 69.3653 −37.6013 137.0404
    $f_{10}$ −500 −500. 3287 −22.7446 11.9779 −35.0425 69.3475 −37.3164 1.6048 −499.8497 −21.6232 11.8265 −36.1124 69.0713 −37.5719 164.8766
    $f_{11}$ −400 −400. 4644 −21.9891 11.5276 −36.014 69.3475 −37.5829 1.8479 −399.54 −21.6252 11.9614 −35.9928 69.484 −37.3817 203.6023
    $f_{12}$ −300 −299.4175 −19.0411 5.6211 −50.5124 72.7959 −42.8024 1.9517 −299.2154 −22.224 11.6704 −35.3449 68.6809 −37.0768 244.3073
    $f_{13}$ −200 −199. 3823 −31.2247 9.7199 −34.7315 78.1449 −45.3504 1.8811 −199.7728 −22.0876 11.6491 −35.5865 69.2094 −37.2746 288.9675
    $f_{14}$ −100 −100. 0015 −6.2047 11.5466 −27.1106 69.3496 −32.6014 1.5652 −99.8081 −22.0000 11.4699 −36.0043 69.3659 −37.622 25.5247
    $f_{15}$ 100 156.5509 49.6019 63.3034 −72.268 −3.7613 −79.9879 1.6973 110.4386 −21.8498 11.9445 −35.3707 69.3721 −37.4593 45.1034
    $f_{16}$ 200 199. 8538 −57.0813 58.5504 −31.0707 26.9993 −38.2225 3.9518 200.4084 −22.0509 11.5686 −36.0034 69.4094 −37.6539 31.5485
    $f_{17} $ 300 304. 7026 8.0018 −16.5642 −5.9498 38.2561 −8.1092 1.5401 300.3612 −22.0466 11.5575 −37.5698 69.3452 −37.6059 114.3228
    $f_{18}$ 400 405. 3374 8.8635 −22.2516 −5.391 38.826 −10.5181 1.6256 401.5735 −22.269 12.4883 −38.1444 68.9858 −37.9558 144.0145
    $f_{19} $ 500 499. 8491 −36.1124 −7.1141 −60.5984 57.0958 −63.211 1.4022 500.0023 −22.1995 11.4599 −36.2687 68.9998 −37.9225 175.7896
    $f_{20} $ 600 599. 7283 −43.4837 14.2779 −40.0212 72.9036 −43.5544 1.4423 600.0068 −22.0109 11.5786 −36.0034 69.4014 −37.6519 214.3663
    $f_{21}$ 700 999.5585 74.8671 7.56 60.3875 15.7259 31.6655 1.8884 709.2098 −22.1135 11.7078 −36.2284 69.2862 −37.5341 259.2839
    $f_{22}$ 800 899.5949 −48.5424 53.7715 13.7283 69.8239 −18.62 29 2.6328 821.3446 −22.0219 12.2169 −36.155 69.2986 −37.4481 338.1721
    $f_{23}$ 900 1 040.6959 −47.8633 59.6935 18.944 63.6605 −16.7946 2.5248 926.4912 −22.2323 11.3133 −36.2082 69.665 −37.8375 400.5942
    $f_{24}$ 1 000 1 104.3212 −43.5817 62.9002 24.6363 69.2317 −19.6422 11.4319 1 005.5802 −22.2796 11.6315 −36.0045 69.656 −37.0404 536.6315
    $f_{25}$ 1 100 1 199.5464 −48.5541 53.8377 13.7864 69.8039 −18.5822 11.4093 1 103.9827 −22.2144 11.3766 −35.7601 69.5939 −37.2318 674.0186
    $f_{26}$ 1 200 1 230.9793 −45.3432 36.5543 −61.2965 73.0248 −28.7484 11.8107 1 201.2061 −21.9038 11.8136 −36.0593 69.1101 −37.8049 824.5081
    $f_{27}$ 1 300 1 643.8584 39.9937 64.0666 74.3787 82.5407 88.9271 11.4554 1 316.2217 −22.0002 11.5006 −36.0034 69.4000 −37.6006 977.0531
    $f_{28}$ 1 400 1 699.6997 74.858 7.5684 60.4032 15.7134 31.6678 2.8537 1 404.1964 −21.9492 11.4562 −36.0421 69.5111 −37.5802 1 061.4819
    下载: 导出CSV

    表  11  PSO算法对TF2在$\sigma^2$ = 0.01的噪声环境下得到的全局极值点

    Table  11  Global extremum points of TF2 obtained by PSO algorithm in noise environment of $\sigma^2$ = 0.01

    TF2 Optimum $f$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ t(s) $P_f$ $P_l$
    $f_1$ −1 400 −1 400.4781 −22.0184 11.4865 −35.9794 69.3581 −37.6173 1.404 0.0372 0.2464
    $f_2$ −1 300 −930.7041 −20.245 15.2674 −30.6074 67.6185 −38.8701 1.4961 322.4007 13.6581
    $f_3$ −1 200 −1 199.0062 −50.622 11.5252 −35.947 69.152 −38.649 1.2854 7.0135 40.3071
    $f_4$ −1 100 −1 100.2794 −21.9225 11.64 −36.0456 69.5782 −37.4029 1.4142 0.0462 0.6106
    $f_5$ −1 000 −1 000.3961 −21.994 11.5758 −36.1201 69.1453 −37.9101 1.2586 0.0041 0.5937
    $f_6$ −900 −900.3562 −31.8677 13.59 −34.0754 76.8813 −49.9714 1.218 0.2098 23.8924
    $f_7$ −800 −800.3509 −-22.0771 11.5961 −35.183 69.3788 −37.6198 1.8579 0.0369 25.5374
    $f_8$ −700 −680.3089 −71.3566 −56.4316 57.2977 −28.3447 −56.5965 1.615 0.0817 161.0824
    $f_9$ −600 −599.2936 49.9955 −78.2402 0.1786 32.7815 56.5152 10.4697 1.0503 176.1058
    $f_{10}$ −500 −500.2789 −22.3318 11.3539 −34.8006 70.2354 −36.8967 1.5857 0.0497 1.2581
    $f_{11}$ −400 −400.4319 −21.9215 11.6111 −36.0095 69.3882 −37.6099 1.822 0.0325 0.1181
    $f_{12}$ −300 −300.4235 −21.962 11.6181 −35.9548 69.2314 −37.4736 1.8597 1.006 17.2487
    $f_{13}$ −200 −195.7745 −16.5439 0.7856 −63.1612 78.2956 −49.8104 1.8291 3.6078 33.5188
    $f_{14}$ −100 −76.8796 93.8116 23.4253 −27.1196 69.3665 −37.609 1.5651 23.1219 100.8436
    $f_{15}$ 100 162.9651 63.763 73.9758 −59.6899 6.8698 −69.5651 1.6225 6.4142 26.3496
    $f_{16}$ 200 200.0371 −77.2249 85.4895 −99.6791 9.5552 0.5678 3.677 0.1834 87.4504
    $f_{17}$ 300 305.138 8.0174 −22.2004 −9.134 41.8141 −7.1107 1.5252 0.4354 7.4541
    $f_{18}$ 400 406.2551 10.8862 −24.2858 −0.1323 39.2559 −7.5586 1.5572 0.9177 6.6953
    $f_{19}$ 500 499.7671 −26.9034 2.4016 −45.2641 65.0955 −40.2783 1.3468 0.082 31.6291
    $f_{20}$ 600 599.6336 −71.294 14.1417 −38.0582 69.6662 −29.0219 1.4203 0.0947 31.6063
    $f_{21} $ 700 799.646 −48.5365 53.7646 13.7222 69.8263 −18.6266 1.8519 199.9126 158.1048
    $f_{22}$ 800 1 016.1259 95.0903 −70.2239 −51.7248 71.7396 −52.0955 2.4272 116.5309 203.5029
    $f_{23}$ 900 1 066.8577 −61.6704 86.9732 42.3647 41.522 −18.9232 2.4921 26.1619 44.4746
    $f_{24}$ 1 000 1 111.4644 −61.8977 82.0776 36.3207 76.7673 1.768 11.6881 7.1432 36.8097
    $f_{25}$ 1 100 1 199.6718 −48.5853 53.7148 13.7168 69.7682 −18.5628 11.7015 0.1254 0.1502
    $f_{26}$ 1 200 1 302.6409 −71.6478 58.3538 19.0587 68.5734 −24.1115 12.4385 71.6616 87.5525
    $f_{27}$ 1 300 1 599.7342 74.8682 7.5586 60.3873 15.7236 31.6624 12.1106 44.1241 111.1257
    $f_{28}$ 1 400 1 499.6147 −48.5389 53.7644 13.7202 69.8289 −18.6285 2.8779 200.085 158.1087
    下载: 导出CSV

    表  12  PMB算法对TF2在$\sigma^2$ = 0.01的噪声环境下得到的全局极值点

    Table  12  Global extremum points of TF2 obtained by PMB algorithm in noise environment of $\sigma^2$ = 0.01

    TF2 Optimum f $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $t(s)$ $P_f$ $P_l$
    f1 −1 400 −1 400.0699 −21.8287 11.8038 −36.3299 69.5330 −37.6233 12.7743 0.0943 0.5782
    f2 −1 300 −608.3307 −23.1175 11.4652 −34.0409 70.2874 −36.6119 39.0930 675.6766 4.2854
    f3 −1 200 −1 017.2976 −32.7028 11.5533 −35.9896 69.2990 −38.0010 68.3085 182.1889 0.1070
    f4 −1 100 −1 054.7452 −22.0054 11.9548 −36.1731 68.5083 −37.3461 102.1388 43.6737 0.1863
    f5 −1 000 −1 000.2789 −21.9308 11.4701 −36.3134 69.5171 −37.2096 135.1712 0.2791 0.5448
    f6 −900 −900.0006 −22.1408 11.3658 −36.1004 69.2084 −37.6539 12.8520 0.0006 0.3210
    f7 −800 −800.0919 −21.9418 11.6135 −35.8100 69.3635 −37.6909 40.6402 0.1127 0.2376
    f8 −700 −699.7605 −22.1688 11.4203 −36.5141 69.3235 −37.5792 69.8338 0.1586 0.5393
    f9 −600 −600.1833 −21.8968 12.9478 −32.0218 69.5171 −37.6985 215.7291 0.2547 4.2381
    f10 −500 −499.9168 −21.3850 11.1694 −35.9969 69.3487 −37.4039 251.9397 0.0671 0.7791
    f11 −400 −400.0491 −21.9828 11.4866 −35.7514 69.3762 −37.4913 306.9611 0.5091 0.6597
    f12 −300 −299.9505 −21.8073 11.5243 −35.7096 69.1440 −37.0312 358.0750 0.7351 0.7379
    f13 −200 −200.0731 −21.8323 11.3919 −36.2855 69.2937 −37.4883 412.7622 0.3003 0.8202
    f14 −100 −99.4329 −22.1030 11.5419 −36.1002 69.3419 −37.6012 523.8452 0.3752 0.1612
    f15 100 104.4172 −21.7108 11.3782 −36.1444 69.3134 −37.7273 58.0593 6.0214 1.0070
    f16 200 200.8154 −22.1002 11.4383 −36.1730 69.4102 −37.6045 126.4925 0.4069 0.2250
    f17 300 301.8527 −22.1000 11.6151 −37.4783 69.4012 −37.5859 366.6641 1.4915 0.1345
    f18 400 404.3400 −22.2810 12.4912 −38.0133 68.8766 −37.9101 341.7595 2.7665 0.1771
    f19 500 500.1091 −22.1404 11.2668 −36.2592 68.9192 −37.9133 318.0817 0.1068 0.2178
    f20 600 599.9330 −22.0115 11.5642 −35.9012 69.4102 −37.6413 501.6370 0.0739 0.1041
    f21 700 704.7231 −22.1409 11.5481 −35.9884 69.3233 −37.5984 799.2632 4.4867 0.2989
    f22 800 843.9745 −22.0029 11.7082 −36.0243 69.3881 −38.1587 1 284.7595 22.6299 0.8884
    f23 900 923.2544 −21.9358 11.4092 −36.9182 69.2899 −37.7927 1 405.1109 3.2368 0.8625
    f24 1 000 1 002.7871 −22.0554 11.0213 −36.4846 69.5489 −37.5213 1 645.1116 2.7931 0.9464
    f25 1 100 1 100.8277 −21.8449 11.4803 −35.9915 69.3188 −37.5424 1 886.3396 3.1550 0.6107
    f26 1 200 1 215.0429 −21.8701 11.7852 −37.0089 69.1101 −37.1271 2 148.5756 13.8368 1.1675
    f27 1 300 1 379.5470 −22.2113 11.7141 −36.6134 69.4000 −37.2465 2 412.4953 63.3253 0.7665
    f28 1 400 1 401.3007 −21.9727 11.5578 −36.0055 69.3242 −37.6351 2 555.0831 2.8957 0.2240
    下载: 导出CSV

    表  13  PMB算法对TF2中部分函数在无噪声环境和$\sigma^2$ = 0.01的噪声环境下的多极值点优化结果

    Table  13  Optimization results of multiple extremum points for some functions of TF2 obtained by PMB algorithm in noiseless environment and noise environment of $\sigma^2$ = 0.01

    TF2 Noiseless environment $\sigma^2$ = 0.01
    $f$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $f$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $P_f$ $P_l$
    $f_6$ −900.0000 −22.0309 11.5686 −36.0034 69.4094 −37.6539 −900.0006 −22.1408 11.3658 −36.1004 69.2084 −37.6539 0.0006 0.3210
    −896.0639 −33.8126 44.7393 54.6931 77.1452 −50.0368 −896.0683 −34.4983 44.9367 54.8740 77.5853 −50.9570 0.0044 1.2579
    −899.9794 −28.6913 12.8088 −35.1466 74.6026 −46.3440 −900.2057 −28.8650 12.9076 −35.1465 74.7262 −46.3359 0.2264 0.2351
    −899.9959 −19.0610 10.9852 −36.4042 66.8717 −33.4840 −899.8525 −19.1580 11.2422 −36.3101 66.9289 −32.6954 0.1433 0.8423
    −899.9987 −20.6166 11.3082 −36.2410 68.2617 −35.8258 −899.9635 −20.4386 11.2333 −36.0719 68.0558 −35.3752 0.0353 0.5580
    $f_7$ −799.9792 −22.0032 11.5635 −36.0040 69.3839 −37.5808 −800.0919 −21.9418 11.6135 −35.8100 69.3635 −37.6909 0.1127 0.2376
    −799.8756 −28.3647 11.5956 −35.8878 69.3717 −37.6959 −800.0568 −28.2097 11.5979 −35.9973 69.4689 −37.5283 0.1812 0.2712
    −799.7114 −40.2000 11.5118 −35.9099 69.3766 −37.8352 −799.8792 −40.1102 11.5317 −35.9960 69.3786 −37.8348 0.1678 0.1259
    −799.6562 −56.9976 11.5324 −35.9848 69.1641 −38.1372 −799.8114 −55.1614 11.5224 −35.9948 69.1645 −38.1372 0.1552 1.8363
    −799.5697 −66.1517 11.6099 −35.9001 69.4026 −37.9670 −799.6939 −65.2517 11.6094 −35.9078 69.3432 −37.9666 0.1242 0.9020
    $f_8$ −699.9191 −22.2758 11.5436 −36.0022 69.3631 −37.6026 −699.7605 −22.1688 11.4203 −36.5141 69.3235 −37.5792 0.1586 0.5393
    −699.9042 −24.1426 11.5593 −36.0070 69.3609 −37.3899 −699.7605 −24.1231 11.5323 −35.9885 69.3697 −37.3910 0.1437 0.0391
    −699.6190 −45.4600 11.5691 −36.0014 84.1102 −20.9700 −699.7605 −45.4382 11.5681 −36.0127 84.1234 −20.9690 0.1415 0.0279
    −699.1928 −36.3628 11.5985 −35.9784 69.3447 −36.1163 −699.7605 −36.4531 11.5900 −35.9804 69.3459 −36.0956 0.5676 0.0931
    −699.4293 −30.7647 11.5746 −35.9425 69.3885 −37.6095 −699.7605 −30.7591 11.5895 −35.9425 69.4115 −37.5944 0.3312 0.0318
    $f_16$ 200.4084 −22.0509 11.5686 −36.0034 69.4094 −37.6539 200.8154 −22.1002 11.4383 −36.1730 69.4102 −37.6045 0.4069 0.2250
    200.0320 −52.1005 −61.6883 −32.3610 −46.5936 1.0443 200.3819 −52.1021 −61.7593 −32.3582 −46.5732 1.1053 0.3499 0.0958
    200.0353 −33.1330 −38.1129 2.5053 −47.4949 37.2437 200.1492 −33.1432 −38.1214 2.4901 −47.4899 37.2348 0.1139 0.0226
    200.0370 −25.6295 23.1822 −3.7438 −74.5210 −22.2897 200.1477 −25.6000 23.1812 −3.7542 −74.5312 −22.2982 0.1107 0.0340
    200.0511 31.4036 49.5090 46.7470 −62.4386 −15.8403 200.0922 31.4125 49.4947 46.7173 −62.4472 −15.8504 0.0411 0.0366
    $f_19$ 500.0023 −22.1995 11.4599 −36.2687 68.9998 −37.9225 500.1091 −22.1404 11.2668 −36.2592 68.9192 −37.9133 0.1068 0.2178
    500.0611 −25.0001 6.8588 −39.2834 63.8559 −42.1153 500.1292 −25.0018 6.8592 −39.2723 63.8337 −42.1010 0.0681 0.0287
    500.0907 −46.4426 −3.0700 −49.8269 56.0704 −50.5283 500.0979 −46.4534 −3.0612 −49.8173 56.1213 −50.5448 0.0072 0.0561
    500.1281 −31.3099 4.2889 −42.9418 58.0552 −46.9003 500.1475 −31.3063 4.2901 −42.9317 58.0661 −46.9169 0.0194 0.0226
    500.1030 −34.7069 −10.5994 −60.2413 54.6492 −60.9458 500.1952 −34.7672 −10.5528 −60.2362 54.6518 −60.9347 0.0922 0.0772
    $f_20$ 600.0068 −22.0109 11.5786 −36.0034 69.4014 −37.6519 599.9330 −22.0115 11.5642 −35.9012 69.4102 −37.6413 0.0739 0.1041
    600.0586 −100.0000 11.0005 −37.8204 70.8742 −25.9073 599.9468 −100.0000 11.1015 −37.8113 70.8513 −25.8984 0.1118 0.1043
    600.0591 −41.9112 12.0157 −38.3612 72.3213 −39.5142 600.0655 −41.8397 12.0277 −38.3525 72.3357 −39.5054 0.0063 0.0749
    600.0671 −25.7643 9.0700 −27.3033 69.5256 −36.3012 600.0348 −25.7528 9.0594 −27.3270 69.5188 −36.3140 0.0323 0.0319
    600.1270 −99.9565 11.5712 −36.7345 62.4002 −37.5823 600.2360 −100.0000 11.5834 −36.7231 62.4028 −37.5947 0.1089 0.0483
    下载: 导出CSV
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