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基于状态估计反馈的策略自适应差分进化算法

王柳静 张贵军 周晓根

王柳静, 张贵军, 周晓根. 基于状态估计反馈的策略自适应差分进化算法. 自动化学报, 2020, 46(4): 752-766. doi: 10.16383/j.aas.2018.c170338
引用本文: 王柳静, 张贵军, 周晓根. 基于状态估计反馈的策略自适应差分进化算法. 自动化学报, 2020, 46(4): 752-766. doi: 10.16383/j.aas.2018.c170338
WANG Liu-Jing, ZHANG Gui-Jun, ZHOU Xiao-Gen. Strategy Self-adaptive Differential Evolution Algorithm Based on State Estimation Feedback. ACTA AUTOMATICA SINICA, 2020, 46(4): 752-766. doi: 10.16383/j.aas.2018.c170338
Citation: WANG Liu-Jing, ZHANG Gui-Jun, ZHOU Xiao-Gen. Strategy Self-adaptive Differential Evolution Algorithm Based on State Estimation Feedback. ACTA AUTOMATICA SINICA, 2020, 46(4): 752-766. doi: 10.16383/j.aas.2018.c170338

基于状态估计反馈的策略自适应差分进化算法

doi: 10.16383/j.aas.2018.c170338
基金项目: 

国家自然科学基金 61773346

国家自然科学基金 61573317

浙江省自然科学基金重点项目 LZ20F030002

详细信息
    作者简介:

    王柳静  浙江工业大学信息工程学院博士研究生.主要研究方向为智能信息处理, 优化理论及算法设计.E-mail:wlj@zjut.edu.cn

    周晓根   浙江工业大学信息工程学院博士研究生.主要研究方向为智能信息处理, 优化理论及算法设计.E-mail:zhouxiaogen53@126.com

    通讯作者:

    张贵军  浙江工业大学信息工程学院教授.主要研究方向为智能信息处理, 优化理论及算法设计, 生物信息学.本文通信作者. E-mail: zgj@zjut.edu.cn

  • 本文责任编委 刘艳军

Strategy Self-adaptive Differential Evolution Algorithm Based on State Estimation Feedback

Funds: 

National Natural Science Foundation of China 61773346

National Natural Science Foundation of China 61573317

Key Program of Natural Science Foundation of Zhejiang Province of China LZ20F030002

More Information
    Author Bio:

    WANG Liu-jing Ph.D. candidate at the College of Information Engineering, Zhejiang University of Technology. Her research interest covers intelligent information processing, optimization theory, and algorithm design

    ZHOU Xiao-Gen Ph. D. candidate at the College of Information Engineering, Zhejiang University of Technology. His research interest covers intelligent information processing, optimization theory, and algorithm design

    Corresponding author: ZHANG Gui-Jun Professor at the College of Information Engineering, Zhejiang University of Technology. His research interest covers intelligent information processing, optimization theory and algorithm design, and bioinformatics. Corresponding author of this paper
  • Recommended by Associate Editor LIU Yan-Jun
  • 摘要: 借鉴闭环控制思想, 提出基于状态估计反馈的策略自适应差分进化(Differential evolution, DE)算法, 通过设计状态评价因子自适应判定种群个体所处于的阶段, 实现变异策略的反馈调节, 达到平衡算法全局探测和局部搜索的目的.首先, 基于抽象凸理论对种群个体建立进化状态估计模型, 提取下界估计信息并结合进化知识设计状态评价因子, 以判定当前种群的进化状态; 其次, 利用状态评价因子的反馈信息, 实现不同进化状态下策略的自适应调整以指导种群进化, 达到提高算法搜索效率的目的.另外, 20个典型测试函数与CEC2013测试集的实验结果表明, 所提算法在计算代价、收敛速度和解的质量方面优于主流改进差分进化算法和非差分进化算法.
    Recommended by Associate Editor LIU Yan-Jun
    1)  本文责任编委 刘艳军
  • 图  1  SEFDE算法思想

    Fig.  1  Main idea of SEFDE

    图  2  下界估计模型$h_k({\bm x})$

    Fig.  2  Underestimation model $h_k({\bm x})$

    图  3  进化状态估计模型$H({\bm x})$

    Fig.  3  Estimation model of evolution state $H({\bm x})$

    图  4  不同$K$值下的进化状态估计模型

    Fig.  4  Model of evolution state under different $K$

    图  5  不同$g$值下Rg的进化状态估计模型

    Fig.  5  Rg model of evolution state under different $g$

    图  6  Lm2状态评价因子变化曲线

    Fig.  6  The curve graph of Lm2 state judgment factor

    图  7  状态评价因子变化曲线

    Fig.  7  The curve graph of state judgment factor

    图  8  DELU、SHADE、EPSDE、CoDE、SEFDE的平均收敛速度曲线图

    Fig.  8  The curve graph of mean convergence speed on DELU、SHADE、EPSDE、CoDE、SEFDE

    表  1  标准测试函数的参数

    Table  1  The parameters of benchmark functions

    函数名 表达式 维数($N$) 取值范围 全局最小值
    Sphere $f_1({\bm x})=\sum\limits_{i=1}^{N}x_i^2$ 30, 50 $(-100, 100)^N$ 0
    SumSquares $f_2({\bm x})=\sum\limits_{i=1}^{N}ix_i^2$ 30, 50 $(-10, 10)^N$ 0
    Schwefel 2.22 $f_3({\bm x})=\sum\limits_{i=1}^{N}\left|x_i\right|+\prod\limits_{i=1}^{N}\left|x_i\right|$ 30, 50 $(-10, 10)^N$ 0
    Exponential $f_4({\bm x})=-\exp(-0.5$$\sum\limits_{i=1}^{N}$$x_i^2)$ 30, 50 $(-1, 1)^N$ $-1$
    Tablet $f_5({\bm x})=10^6x_1^2+\sum\limits_{i=2}^{N}x_i^2$ 30, 50 $(-100, 100)^N$ 0
    Step $f_6({\bm x})=\sum\limits_{i=1}^{N}\left\lfloor x_i+0.5\right\rfloor^2$ 30, 50 $(-100, 100)^N$ 0
    Zakharov $f_7({\bm x})=\sum\limits_{i=1}^{N}x_i^2+\sum\limits_{i=1}^{N}(0.5ix_i)^2+\sum\limits_{i=1}^{N}(0.5ix_i)^4$ 30, 50 $(-5, 10)^N$ 0
    Rosenbrock $f_8({\bm x})=\sum\limits_{i=1}^{N-1}[100(x_{i+1}-x_i^2)^2+(x_i-1)^2]$ 30, 50 $(-30, 30)^N$ 0
    Griewank $f_9({\bm x})=1+\frac{1}{4000}\sum\limits_{i=1}^{N}x_i^2-\prod\limits_{i=1}^{N}\cos(\frac {x_i}{\sqrt{i}})$ 30, 50 $(-600, 600)^N$ 0
    Schaffer 2 $f_{10}({\bm x})=\sum\limits_{i=1}^{N-1}(x_i^2+x_{i+1}^2)^{0.25}(\sin^2(50(x_i^2+x_{i+1}^2)^{0.1})+1)$ 30, 50 $(-100, 100)^N$ 0
    Schwefel 2.26 $f_{11}({\bm x})=-\sum\limits_{i=1}^{N}x_i\sin(\sqrt{\left\|x_i\right\|})$ 30, 50 $(-500, 500)^N$ $-12 569.18$
    Himmelblau $f_{12}({\bm x})=N^{-1}\sum\limits_{i=1}^{N}(x_i^4-16x_i^2+5x_i)$ 30, 50 $(-5, 5)^N$ -78.3323
    Levy and Montalvo 1 $f_{13}({\bm x})=\frac{\pi}{N}(10\sin^2(\pi y_1)+\sum\limits_{i=1}^{N-1}(y_i-1)^{2} [1+\\ 10\sin^2(\pi y_{i+1})]+(y_N-1)^2)$, $y_i=1+\frac{1}{4}(x_i+1)$ 30, 50 $(-10, 10)^N$ 0
    Levy and Montalvo 2 $f_{14}({\bm x})=0.1(\sin^2(3\pi x_i)+\sum\limits_{i=1}^{N-1}(x_i-1)^{2} [1+\sin^2(3\pi x_{i+1})]+\\ (x_N-1)^2[1+\sin^2(2\pi x_N)])$ 30, 50 $(-5, 5)^N$ 0
    Ackley $f_{15}({\bm x})=-20\exp(-0.02\sqrt{N^{-1}\sum\limits_{i=1}^{N}x_i^2}) -\exp(N^{-1}$ 30, 50 $(-30, 30)^N$ 0
    $\sum\limits_{i=1}^{N}\cos(2\pi x_i))+20+e$
    Rastrigin $f_{16}({\bm x})=10N+\sum\limits_{i=1}^{N}(x_i^2-10\cos(2\pi x_i))$ 30, 50 $(-5, 5)^N$ 0
    Penalized 1 $f_{17}(x)=\frac{\pi}{N}\{\sum\nolimits_{i=1}^{N-1}(y_i-1)^2[1+ \sin(\pi y_{i+1})]+(y_N-1)^2+$
    $(10\sin^2(\pi y_1))\}+\sum\nolimits_{i=1}^Nu(x_i, 10, 100, 4)$,
    $y_i=1+\frac{x_i+1}{4}$
    $u(x_i, a, k, m) = \left\{ \begin{array}{ll} k(x_i-a)^m, ~~~~ x_i> a \\ 0, ~~~~-a\leq x_i\leq a \\ k(-x_i-a)^m, ~~~~x_i < -a \end{array} \right.$ 30, 50 $(-50, 50)^N$ 0
    Penalized 2 $f_{18}(x)=0.1\{\sin^2(3\pi x_1)+\sum\nolimits_{i=1}^{N-1} (x_i-1)^2[1+\sin^2(3\pi x_{i+1})]+$
    $(x_N-1)^2[1+\sin^2(2\pi x_N)]\}+\sum\nolimits_{i=1}^Nu(x_i, 5, 100, 4)$ 30, 50 $-(50, 50)^N$ 0
    Neumaier $f_{19}({\bm x})=\sum\limits_{i=1}^{N}(x_i-1)^2- \sum\limits_{i=2}^{N}x_ix_{i-1}+\frac{N(N+4)(N-1)}{6}$ 30, 50 $(-900, 900)^N$ 0
    Alpine $f_{20}({\bm x})=\sum\limits_{i=1}^{N}\left|x_i\sin x_i+0.1x_i\right|$ 30, 50 $(-10, 10)^N$ 0
    下载: 导出CSV

    表  2  SEFDE中参数$K$设置下的平均函数评价次数和成功率

    Table  2  Average numbers of function evaluations and success rates of parameter settings $K$ in SEFDE

    Fun $N$ $K=2$ $K=3$ $K=4$ $K=5$ $K=6$
    FEs SR FEs SR FEs SR FEs SR FEs SR
    $f_{1}$ 30 1.51E+04 1.00 1.19E+04 1.00 1.26E+04 1.00 1.21E+04 1.00 1.18E+04 1.00
    $f_{2}$ 30 1.17E+04 1.00 1.14E+04 1.00 1.19E+04 1.00 1.06E+04 1.00 1.14E+04 1.00
    $f_{3}$ 30 1.85E+04 1.00 1.76E+04 1.00 1.73E+04 1.00 1.75E+04 1.00 1.83E+04 1.00
    $f_{4}$ 30 7.88E+03 1.00 7.89E+03 1.00 7.85E+03 1.00 7.83E+03 1.00 7.87E+03 1.00
    $f_{5}$ 30 2.94E+04 1.00 2.97E+04 1.00 2.93E+04 1.00 2.91E+04 1.00 2.90E+04 1.00
    $f_{6}$ 30 8.94E+03 1.00 8.61E+03 1.00 8.56E+03 1.00 8.28E+03 1.00 8.45E+03 1.00
    $f_{7}$ 30 1.46E+05 1.00 1.45E+05 1.00 1.46E+05 1.00 1.46E+05 1.00 1.44E+05 1.00
    $f_{8}$ 30 1.24E+05 1.00 1.20E+05 1.00 1.21E+05 1.00 1.22E+05 1.00 1.21E+05 1.00
    $f_{9}$ 30 1.28E+04 1.00 1.26E+04 1.00 1.25E+04 1.00 1.27E+04 1.00 1.24E+04 1.00
    $f_{10}$ 30 1.00E+05 1.00 1.09E+05 1.00 1.10E+05 1.00 1.07E+05 1.00 1.09E+05 1.00
    $f_{11}$ 30 4.43E+04 1.00 4.34E+04 1.00 4.57E+04 1.00 4.40E+04 1.00 4.48E+04 1.00
    $f_{12}$ 30 1.24E+04 1.00 1.71E+04 1.00 1.74E+04 1.00 1.64E+04 1.00 1.70E+04 1.00
    $f_{13}$ 30 1.26E+04 1.00 1.24E+04 1.00 1.26E+04 1.00 1.27E+04 1.00 1.26E+04 1.00
    $f_{14}$ 30 1.18E+04 1.00 1.19E+04 1.00 1.20E+04 1.00 1.22E+04 1.00 1.19E+04 1.00
    $f_{15}$ 30 2.02E+04 1.00 1.95E+04 1.00 2.01E+04 1.00 2.02E+04 1.00 1.98E+04 1.00
    $f_{16}$ 30 5.60E+04 1.00 5.44E+04 1.00 5.62E+04 1.00 5.34E+04 1.00 5.66E+04 1.00
    $f_{17}$ 30 1.90E+04 1.00 1.96E+04 1.00 1.91E+04 1.00 1.92E+04 1.00 1.98E+04 1.00
    $f_{18}$ 30 2.32E+04 1.00 2.29E+04 1.00 2.32E+04 1.00 2.30E+04 1.00 2.29E+04 1.00
    $f_{19}$ 30 1.72E+05 1.00 1.70E+05 1.00 1.67E+05 1.00 1.70E+05 1.00 1.68E+05 1.00
    $f_{20}$ 30 3.62E+04 1.00 3.75E+04 1.00 3.13E+04 1.00 3.48E+04 1.00 3.47E+04 1.00
    AVE 4.41E+04 1.000 4.42E+04 1.000 4.41E+04 1.000 4.40E+04 1.000 4.41E+04 1.000
    下载: 导出CSV

    表  3  函数评价次数和成功率对比数据

    Table  3  Compared data on function evaluations and success rates

    Fun $N$ DELU SHADE EPSDE CoDE SEFDE
    FEs SR FEs SR FEs SR FEs SR FEs SR
    $f_{1}$ 30 1.22E+04 1.00 2.35E+04 1.00 1.67E+04 1.00 4.02E+04 1.00 ${\bf1.21E+04}$ 1.00
    $f_{2}$ 30 1.07E+04 1.00 2.16E+04 1.00 1.51E+04 1.00 3.63E+04 1.00 ${\bf1.06E+04}$ 1.00
    $f_{3}$ 30 3.15E+04 1.00 3.42E+04 1.00 2.32E+04 1.00 5.64E+04 1.00 ${\bf1.75E+04}$ 1.00
    $f_{4}$ 30 7.98E+03 1.00 1.67E+04 1.00 9.36E+03 1.00 2.24E+04 1.00 ${\bf7.83E+03}$ 1.00
    $f_{5}$ 30 2.55E+04 1.00 2.67E+04 1.00 ${\bf1.83E+04}$ 1.00 4.13E+04 1.00 2.91E+04 1.00
    $f_{6}$ 30 1.16E+04 1.00 1.19E+04 1.00 ${\bf8.33E+03}$ 1.00 2.01E+04 1.00 ${\bf8.28E+03}$ 1.00
    $f_{7}$ 30 7.71E+04 1.00 ${\bf5.98E+04}$ 1.00 1.33E+05 1.00 7.80E+04 1.00 1.46E+05 1.00
    $f_{8}$ 30 ${\bf7.09E+04}$ 1.00 1.09E+05 1.00 1.16E+05 0.87 2.37E+05 1.00 1.22E+05 1.00
    $f_{9}$ 30 1.54E+04 1.00 2.56E+04 1.00 1.76E+04 0.83 4.39E+04 1.00 ${\bf1.27E+04}$ 1.00
    $f_{10}$ 30 ${\bf9.78E+04}$ 1.00 1.08E+05 0.97 1.13E+05 1.00 1.73E+05 1.00 1.07E+05 1.00
    $f_{11}$ 30 ${\bf1.34E+04}$ 1.00 9.67E+04 1.00 3.71E+04 1.00 6.00E+04 1.00 4.40E+04 1.00
    $f_{12}$ 30 1.98E+04 1.00 3.35E+04 0.97 2.33E+04 1.00 3.56E+04 1.00 ${\bf1.64E+04}$ 1.00
    $f_{13}$ 30 ${\bf1.05E+04}$ 1.00 1.67E+04 1.00 1.30E+04 1.00 2.63E+04 1.00 1.27E+04 1.00
    $f_{14}$ 30 ${\bf8.71E+03}$ 1.00 1.71E+04 1.00 1.15E+04 1.00 2.70E+04 1.00 1.22E+04 1.00
    $f_{15}$ 30 2.65E+04 1.00 3.21E+04 1.00 2.32E+04 1.00 5.68E+04 1.00 ${\bf2.02E+04}$ 1.00
    $f_{16}$ 30 5.94E+04 1.00 1.24E+05 1.00 6.40E+04 1.00 1.30E+05 0.97 ${\bf5.34E+04}$ 1.00
    $f_{17}$ 30 ${\bf9.17E+03}$ 1.00 1.92E+04 1.00 1.29E+04 1.00 3.03E+04 1.00 1.92E+04 1.00
    $f_{18}$ 30 1.52E+04 1.00 ${\bf1.50E+04}$ 1.00 1.59E+04 0.97 3.54E+04 1.00 2.30E+04 1.00
    $f_{19}$ 30 1.68E+05 0.97 ${\bf1.09E+05}$ 1.00 NA 0.00 2.69E+05 1.00 1.70E+05 1.00
    $f_{20}$ 30 NA 0.00 NA 0.00 1.00E+05 1.00 NA 0.00 ${\bf3.48E+04}$ 1.00
    AVE 4.96E+04 0.95 6.00E+04 0.947 5.36E+04 0.934 8.59E+04 0.949 $\textbf{4.40E+04}$ $\textbf{1.000}$
    下载: 导出CSV

    表  4  DELU、SHADE、EPSDE、CoDE、SEFDE 对30维测试函数的优化结果, 平均值(标准差)

    Table  4  Compared data of 30 D optimization results on DELU、SHADE、EPSDE、CoDE、SEFDE, Mean (Std)

    Fun $N$ DELU SHADE EPSDE CoDE SEFDE
    Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev)
    $f_{1}$ 30 9.25E-19(2.99E-36)$^+$ 2.87E-19(2.31E-19)$^+$ 3.87E-31(9.46E-31)$^+$ 2.16E-10(1.73E-10)$^+$ ${\bf4.01E-38}({\bf5.05E-38})$
    $f_{2}$ 30 2.56E-32(4.00E-32)$^+$ 4.40E-20(3.25E-20)$^+$ 5.52E-31(1.94E-30)$^+$ 2.83E-11(2.61E-11)$^+$ ${\bf5.83E-42}({\bf9.48E-42})$
    $f_{3}$ 30 1.46E-11(1.23E-11)$^+$ 4.98E-10(2.39E-10)$^+$ 1.29E-16(2.15E-16)$^+$ 3.86E-06(1.32E-06)$^+$ ${\bf2.03E-23}({\bf1.62E-23})$
    $f_{4}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 4.07E-17(5.44E-17)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 1.30E-14(1.13E-14)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{5}$ 30 1.65E-20(4.70E-20)$^-$ 1.97E-18(2.27E-18)$^-$ ${\bf8.85E-30}({\bf1.75E-29})$$^-$ 4.04E-10(3.04E-10)$^+$ 1.06E-15(3.87E-16)
    $f_{6}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{7}$ 30 ${\bf1.82E-04}({\bf3.39E-04})$$^-$ 3.57E-04(1.00E-03)$^-$ 1.42E+01(2.25E+01)$^-$ 5.69E-04(7.53E-04)$^-$ 3.59E+01(1.12E+01)
    $f_{8}$ 30 ${\bf1.28E-04}({\bf1.36E-04})$$^-$ 1.81E+01(1.11E+00)$^-$ 8.61E+00(2.09E+00)$^-$ 1.97E+01(5.80E-01)$^-$ 2.54E+01(8.26E-01)
    $f_{9}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 2.47E-04(1.35E-03)$^+$ 6.57E-04(2.50E-03)$^+$ 2.47E-07(7.34E-07)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{10}$ 30 ${\bf2.03E-02}({\bf6.76E-03})$$^-$ 3.56E-01(5.96E-02)$^+$ 2.35E-01(1.31E-01)$^+$ 1.97E+00(4.24E-01)$^+$ 1.84E-01(7.45E-02)
    $f_{11}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 1.75E+02(5.85E+01)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 1.66E-02(3.13E-02)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{12}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 3.14E-05(1.02E-07)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{13}$ 30 2.19E-30(2.11E-30)$^+$ 5.53E-21(5.00E-21)$^+$ 2.91E-29(1.14E-28)$^+$ 1.80E-13(3.36E-13)$^+$ ${\bf1.57E-32}({\bf0.00E+00})$
    $f_{14}$ 30 1.36E-32(3.13E-34)$^+$ 4.07E-21(6.03E-21)$^+$ 1.75E-32(9.04E-33)$^+$ 1.31E-13(1.51E-13)$^+$ ${\bf1.35E-32}({\bf0.00E+00})$
    $f_{15}$ 30 2.42E-10(1.35E-10)$^+$ 1.19E-10(6.88E-11)$^+$ 6.04E-15(1.66E-15)$^+$ 3.75E-06(1.88E-06)$^+$ ${\bf4.00E-15}({\bf1.02E-15})$
    $f_{16}$ 30 1.99E-06(2.50E-06)$^+$ 2.30E+01(2.38E+00)$^+$ 5.48E-02(1.39E-01)$^+$ 3.30E+01(5.81E+00)$^+$ ${\bf5.18E-07}({\bf1.12E-06})$
    $f_{17}$ 30 4.57E-26(7.30E-26)$^+$ 2.85E-20(4.26E-20)$^+$ ${\bf6.02E-32}({\bf1.48E-31})$$^-$ 1.44E-12(1.26E-12)$^+$ 2.09E-26(1.95E-26)
    $f_{18}$ 30 3.80E-21(4.03E-21)$^+$ 3.51E-19(3.01E-19)$^+$ 3.66E-04(2.01E-03)$^+$ 2.45E-11(2.36E-11)$^+$ ${\bf2.48E-21}({\bf4.41E-21})$
    $f_{19}$ 30 ${\bf3.67E+02}({\bf4.24E+02})$$^-$ 6.53E+02(6.50E+02)$^+$ 9.64E+02(4.81E+02)$^+$ 6.92E+02(8.00E+02)$^+$ 6.23E+02(5.05E+02)
    $f_{20}$ 30 2.20E-01(1.67E-05)$^+$ 1.60E-02(1.92E-03)$^+$ 1.38E-03(1.06E-03)$^+$ 1.75E+00(1.47E+00)$^+$ ${\bf2.07E-13}({\bf2.86E-13})$
    $+/\approx/-$ 10/5/3 16/1/3 12/4/4 16/2/2 -/-/-
    下载: 导出CSV

    表  5  DELU、SHADE、EPSDE、CoDE、SEFDE对50维测试函数的优化结果, 平均值(标准差)

    Table  5  Compared data of 50 D optimization results on DELU、SHADE、EPSDE、CoDE、SEFDE, Mean (Std)

    Fun $N$ DELU SHADE EPSDE CoDE SEFDE
    Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev)
    $f_{1}$ 50 4.24E-42(4.13E-42)$^+$ 2.05E-53(3.34E-53)$^+$ 2.52E-50(7.95E-50)$^+$ 4.94E-21(5.74E-21)$^+$ ${\bf2.30E-57}({\bf2.91E-57})$
    $f_{2}$ 50 1.92E-32(4.01E-32)$^+$ ${\bf1.35E-53}({\bf3.43E-53})$$^-$ 1.80E-51(3.75E-51)$^-$ 4.46E-22(6.24E-22)$^+$ ${\bf1.59E-42}({\bf2.13E-42})$
    $f_{3}$ 50 7.87E-29(9.66E-29)$^+$ 2.22E-27(1.67E-27)$^+$ 5.63E-29(1.68E-28)$^+$ 6.05E-12(2.77E-12)$^+$ ${\bf5.52E-29}({\bf4.63E-29})$
    $f_{4}$ 50 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 9.99E-17(3.51E-17)$^+$ 1.33E-16(4.68E-17)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{5}$ 50 7.64E-21(8.52E-21)$^+$ ${\bf1.03E-52}({\bf1.07E-52})$$^-$ 3.04E-50(7.79E-50)$^-$ 1.01E-20(1.21E-20)$^+$ 1.50E-27(1.66E-27)
    $f_{6}$ 50 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 5.00E-01(7.07E-01)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{7}$ 50 1.77E-04(9.11E-05)$^-$ ${\bf8.22E-07}({\bf1.80E-06})$$^-$ 2.22E+02(6.11E+01)$^+$ 2.80E-04(3.97E-04)$^-$ 9.43E+01(9.34E+00)
    $f_{8}$ 50 ${\bf2.59E-06}({\bf1.29E-06})$$^-$ 1.34E+01(1.83E+00)$^-$ 2.72E+01(1.81E+01)$^-$ 5.16E+01(2.64E+01)$^+$ 3.73E+01(3.74E+00)
    $f_{9}$ 50 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 2.46E-03(5.69E-03)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{10}$ 50 2.28E-02(1.14E-02)$^-$ 5.45E-02(3.68E-02)$^-$ ${\bf3.12E-03}({\bf4.16E-03})$$^-$ 1.11E-01(7.80E-02)$^-$ 2.19E-01(2.67E-01)
    $f_{11}$ 50 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 1.19E+01(3.79E+01)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{12}$ 50 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 3.39E-01(3.95E-01)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{13}$ 50 9.42E-33(1.44E-48)$^+$ 9.42E-33(1.44E-48)$^+$ 9.46E-33(6.47E-35)$^+$ 4.76E-25(4.46E-25)$^+$ ${\bf1.44E-33}({\bf1.11E-33})$
    $f_{14}$ 50 1.35E-32(2.88E-48)$^+$ 1.35E-32(2.88E-48)$^+$ 1.40E-32(8.62E-34)$^+$ 1.06E-24(8.10E-25)$^+$ ${\bf1.33E-32}({\bf0.00E+00})$
    $f_{15}$ 50 6.13E-15(1.95E-15)$^+$ 7.11E-15(0.00E+00)$^+$ 1.14E-14(4.97E-15)$^+$ 8.59E-12(4.34E-12)$^+$ ${\bf1.04E-15}({\bf3.89E-15})$
    $f_{16}$ 50 2.45E-07(1.58E-07)$^+$ 1.84E-01(5.31E-02)$^+$ 4.42E+00(1.09E+01)$^+$ 4.27E+01(7.39E+00)$^+$ ${\bf1.96E-07}({\bf1.10E-07})$
    $f_{17}$ 50 3.25E-30(1.48E-30)$^-$ ${\bf9.42E-33}({\bf1.44E-48})$$^-$ 1.00E-32(1.96E-33)$^-$ 9.53E-24(4.52E-25)$^+$ 1.68E-26(3.74E-26)
    $f_{18}$ 50 3.08E-32(5.23E-33)$^-$ ${\bf1.36E-32}({\bf3.90E-34})$$^-$ 1.10E-03(3.47E-03)$^+$ 1.29E-22(1.22E-22)$^+$ 1.71E-24(3.73E-24)
    $f_{19}$ 50 8.90E+03(6.05E+03)$^-$ ${\bf1.04E+03}({\bf6.24E+02})$$^-$ 1.26E+04(1.65E+03)$^-$ 1.09E+04(1.44E+03)$^-$ 1.66E+04(2.24E+03)
    $f_{20}$ 50 2.70E-01(4.67E-02)$^+$ 2.60E-01(5.16E-02)$^+$ 2.80E-01(4.22E-02)$^+$ 3.20E-01(4.22E-02)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $+/\approx/-$ 9/5/6 10/2/7 12/2/6 12/5/3 -/-/-
    下载: 导出CSV

    表  6  CLPSO、CMA-ES、GL-25和SEFDE的优化结果性能对比数据, 平均值(标准差)

    Table  6  Compared data of optimization results on CLPSO、CMA-ES、GL-25 and SEFDE, Mean (Std)

    Fun $N$ CLPSO CMA-ES GL-25 SEFDE
    Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev)
    $f_{1}$ 30 9.13E-14(3.33E-14)$^+$ 1.97E-29(2.07E-30)$^+$ 4.78E-87(1.69E-86)$^+$ ${\bf1.26E-103}({\bf2.39E-103})$
    $f_{2}$ 30 9.79E-15(5.44E-15)$^+$ 3.75E-28(4.61E-29)$^+$ 2.66E-78(1.25E-77)$^+$ ${\bf2.40E-112}({\bf3.36E-112})$
    $f_{3}$ 30 4.48E-09(1.48E-09)$^+$ 2.03E-14(9.76E-16)$^+$ 2.98E-28(1.12E-27)$^+$ ${\bf3.68E-60}({\bf3.36E-112})$
    $f_{4}$ 30 3.37E-16(7.98E-17)$^+$ 4.81E-17(5.60E-17)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{5}$ 30 9.33E-14(6.40E-14)$^+$ 1.50E-24(2.15E-25)$^+$ ${\bf4.38E-85}({\bf2.40E-84})$$^-$ 2.42E-45(2.29E-45)
    $f_{6}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{7}$ 30 4.73E+00(1.20E+00)$^+$ ${\bf2.95E-27}({\bf4.82E-28})$$^-$ 3.14E-02(8.77E-02)$^-$ 2.66E-01(2.28E-01)
    $f_{8}$ 30 1.95E+01(2.62E+00)$^-$ ${\bf2.66E-01}({\bf1.01E+00})$$^-$ 2.21E+01(6.19E-01)$^+$ 2.19E+01(5.02E-01)
    $f_{9}$ 30 2.40E-09(3.67E-09)$^+$ 1.40E-03(3.70E-03)$^+$ 1.61E-15(4.55E-15)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{10}$ 30 1.54E-01(2.09E-02)$^-$ 2.50E+02(1.23E+01)$^+$ 2.95E+00(1.00E+00)$^+$ ${\bf1.30E-01}({\bf6.42E-02})$
    $f_{11}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 5.19E+03(6.07E+02)$^+$ 4.46E+03(1.36E+03)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{12}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 1.30E+01(2.50E+00)$^+$ 3.14E-05(3.97E-14)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{13}$ 30 1.13E-17(6.31E-18)$^+$ 9.13E-01(1.06E+00)$^+$ 8.82E-31(4.10E-30)$^+$ ${\bf1.57E-32}({\bf0.00E+00})$
    $f_{14}$ 30 5.94E-17(3.31E-17)$^+$ 1.10E-03(3.35E-03)$^+$ 1.28E-30(5.05E-30)$^+$ ${\bf1.35E-32}({\bf0.00E+00})$
    $f_{15}$ 30 9.76E-08(2.38E-08)$^+$ 1.93E+01(1.97E-01)$^+$ 1.09E-13(1.91E-13)$^+$ ${\bf7.55E-15}({\bf0.00E+00})$
    $f_{16}$ 30 9.41E-07(6.48E-07)$^+$ 2.20E+02(5.64E+01)$^+$ 3.07E+01(2.71E+01)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $f_{17}$ 30 3.49E-16(1.59E-16)$^+$ 1.73E-02(3.93E-02)$^+$ 1.26E+02(1.09E+01)$^+$ ${\bf1.57E-32}({\bf0.00E+00})$
    $f_{18}$ 30 2.52E-14(1.22E-14)$^+$ 2.20E-03(4.47E-03)$^+$ 1.97E+03(1.93E+02)$^+$ ${\bf1.35E-32}({\bf0.00E+00})$
    $f_{19}$ 30 6.26E+03(9.93E+02)$^+$ ${\bf5.59E-10}({\bf7.36E-11})$$^-$ 2.49E+03(4.07E+02)$^+$ 5.60E+02(1.36E+02)
    $f_{20}$ 30 2.33E-04(9.49E-05)$^+$ 8.96E-02(1.46E-01)$^+$ 3.55E-04(9.33E-04)$^+$ ${\bf1.23E-10}({\bf2.47E-10})$
    $+/\approx/-$ 15/3/2 16/1/3 16/2/2 -/-/-
    下载: 导出CSV

    表  7  SHADE、IDE、ZPEDE、SinDE、SEFDE的优化结果性能对比数据, 平均值(标准差)

    Table  7  Compared data of optimization results on SHADE、IDE、ZEPDE、SinDE、SEFDE, Mean (Std)

    Fun N SHADE IDE ZEPDE SinDE SEFDE
    Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev) Mean error (Std Dev)
    $F_{1}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 2.27E-13(1.53E-28)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $F_{2}$ 30 ${\bf2.66E+04}({\bf1.13E+04})$$^-$ 1.68E+06(4.23E+05)$^-$ 1.97E+05(7.53E+04)$^-$ 2.66E+06(8.33E+05)$^-$ 2.85E+07(4.18E+06)
    $F_{3}$ 30 8.80E+05(1.96E+06)$^+$ 1.38E+05(1.85E+05)$^+$ 1.50E+06(2.07E+06)$^+$ 1.01E+05(3.77E+05)$^+$ ${\bf5.80E+04}({\bf3.74E+04})$
    $F_{4}$ 30 ${\bf1.61E-03}(1.41E-03)$$^-$ 6.85E+03(1.10E+03)$^+$ 7.66E-01(4.78E-01)$^-$ 8.28E+03(1.54E+03)$^+$ 3.13E+03(1.12E+03)
    $F_{5}$ 30 ${\bf0.00E+00}(0.00E+00)$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 1.14E-13(7.65E-29)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $F_{6}$ 30 4.28E+01(5.52E+00)$^+$ 4.34E+01(2.62E-04)$^+$ 4.34E+01(3.18E-13)$^+$ 4.34E+01(1.44E-14)$^+$ ${\bf1.52E+01}({\bf1.95E-01})$
    $F_{7}$ 30 2.33E+01(9.32E+00)$^-$ 3.18E+00(1.55E+00)$^-$ 1.37E+01(4.88E+00)$^-$ ${\bf6.10E-01}(5.97E-01)$$^-$ 2.80E+01(7.96E+00)
    $F_{8}$ 30 2.09E+01(1.68E-01)$^+$ 2.11E+01(2.44E-02)$^+$ 2.11E+01(1.17E-01)$^+$ 2.11E+01(3.59E-02)$^+$ ${\bf2.09E+01}({\bf2.78E-02})$
    $F_{9}$ 30 5.54E+01(1.98E+00)$^+$ 3.56E+01(5.54E+00)$^+$ 3.74E+01(5.85E+00)$^+$ 3.48E+01(4.34E+00)$^+$ ${\bf3.04E+01}({\bf6.01E-01})$
    $F_{10}$ 30 7.37E-02(3.67E-02)$^+$ 4.38E-02(2.17E-02)$^+$ 1.37E-01(6.96E-02)$^+$ 7.93E-02(3.57E-02)$^+$ ${\bf2.71E-02}({\bf1.69E-03})$
    $F_{11}$ 30 ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ ${\bf0.00E+00}({\bf0.00E+00})$$^\approx$ 3.65E-01(6.12E-01)$^+$ 5.92E+00(2.86E+00)$^+$ ${\bf0.00E+00}({\bf0.00E+00})$
    $F_{12}$ 30 5.86E+01(1.11E+01)$^-$ 6.89E+01(8.82E+00)$^-$ 6.04E+01(1.76E+01)$^-$ ${\bf5.61E+01}(1.41E+01)$$^-$ 1.21E+02(9.62E+00)
    $F_{13}$ 30 1.45E+02(1.95E+01)$^+$ 1.34E+02(2.28E+01)$^+$ 1.32E+02(3.62E+01)$^+$ 1.39E+02(3.41E+01)$^+$ ${\bf1.24E+02}({\bf1.29E+00})$
    $F_{14}$ 30 ${\bf3.45E-02}({\bf1.93E-02})$$^-$ 1.17E+02(8.38E+01)$^+$ 4.83E+00(2.70E+00)$^-$ 2.34E+02(9.23E+01)$^-$ 5.85E+00(2.05E+00)
    $F_{15}$ 30 6.82E+03(4.41E+02)$^+$ 6.54E+03(5.91E+02)$^+$ 6.59E+03(9.36E+03)$^+$ 6.80E+03(1.00E+03)$^+$ ${\bf5.95E+03}({\bf2.37E+02})$
    $F_{16}$ 30 1.28E+00(2.07E-01)$^-$ 1.59E+00(2.36E-01)$^-$ ${\bf7.82E-01}({\bf6.74E-01})$$^-$ 2.08E+00(3.66E-01)$^-$ 1.02E+02(1.66E-01)
    $F_{17}$ 30 5.08E+01(4.27E-14)$^+$ 5.92E+01(1.41E+00)$^+$ 5.11E+01(1.60E-01)$^+$ 6.52E+01(3.47E+00)$^+$ ${\bf4.21E+01}({\bf6.06E+01})$
    $F_{18}$ 30 1.37E+02(1.29E+01)$^-$ 1.68E+02(1.27E+01)$^-$ ${\bf1.03E+02}({\bf1.19E+01})$$^-$ 1.41E+02(2.27E+01)$^-$ 3.01E+02(9.05E+00)
    $F_{19}$ 30 2.64E+00(2.83E-01)$^-$ ${\bf2.24E+00}({\bf3.66E-01})$$^-$ 3.71E+00(7.55E-01)$^-$ 4.85E+00(8.82E-01)$^-$ 1.03E+02(1.58E-01)
    $F_{20}$ 30 1.93E+01(7.70E-01)$^-$ 1.93E+01(4.47E-01)$^-$ 1.97E+01(7.88E-01)$^-$ ${\bf1.92E+01}(7.52E-01)$$^-$ 1.12E+02(9.46E-02)
    $F_{21}$ 30 8.45E+02(3.63E+02)$^+$ 7.32E+02(3.82E+02)$^+$ 6.33E+02(4.48E+02)$^+$ 5.84E+02(4.22E+02)$^+$ ${\bf5.03E+02}({\bf4.03E+01})$
    $F_{22}$ 30 ${\bf1.33E+01}({\bf7.12E+00})$$^-$ 6.88E+01(2.03E+01)$^-$ 4.23E+02(5.75E+02)$^+$ 3.51E+02(2.72E+02)$^+$ 2.41E+02(1.24E+01)
    $F_{23}$ 30 7.63E+03(6.58E+02)$^+$ 7.32E+03(6.92E+02)$^+$ 7.02E+03(8.73E+02)$^+$ 6.59E+03(8.47E+02)$^+$ ${\bf6.38E+03}({\bf2.25E+02})$
    $F_{24}$ 30 2.34E+02(1.01E+01)$^+$ 2.02E+02(1.14E+00)$^-$ 2.35E+02(1.09E+01)$^+$ ${\bf2.00E+02}(1.34E-01)$$^-$ 2.32E+02(1.18E+00)
    $F_{25}$ 30 3.40E+02(3.09E+01)$^+$ 3.03E+02(1.09E+01)$^+$ 3.23E+02(1.31E+01)$^+$ 2.97E+02(1.33E+01)$^+$ ${\bf2.58E+02}({\bf1.05E+01})$
    $F_{26}$ 30 2.58E+02(8.08E+01)$^-$ ${\bf2.23E+02}({\bf4.46E+01})$$^-$ 2.27E+02(6.20E+01)$^-$ 2.76E+02(5.96E+01)$^-$ 3.02E+02(1.90E-01)
    $F_{27}$ 30 9.36E+02(3.07E+02)$^+$ ${\bf3.58E+02}({\bf3.30E+01})$$^-$ 9.38E+02(1.40E+02)$^+$ 4.75E+02(1.55E+02)$^-$ 6.11E+02(1.62E+02)
    $F_{28}$ 30 4.58E+02(4.13E+02)$^+$ ${\bf4.00E+02}({\bf0.00E+00})$$^\approx$ ${\bf4.00E+02}({\bf0.00E+00})$$^\approx$ ${\bf4.00E+02}(0.00E+00)$$^\approx$ ${\bf4.00E+02}({\bf0.00E+00})$
    $+/\approx/-$ 14/3/11 13/4/11 15/3/10 16/1/11 -/-/-
    下载: 导出CSV
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出版历程
  • 收稿日期:  2017-06-20
  • 录用日期:  2017-12-06
  • 刊出日期:  2020-04-24

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