Strategy Self-adaptive Differential Evolution Algorithm Based on State Estimation Feedback
-
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.Abstract: Inspired by the idea of closed-loop control, a strategy self-adaptive differential evolution (DE) algorithm based on state estimation feedback is proposed, the stage of individual can be self-adaptively determined by designing the state judgment factor, and achieve the feedback adjustment of mutation strategies. Consequently, the algorithm can get a trade-off between the exploration and exploitation. Firstly, the estimation model of evolution state is established based on abstract convex theory, from which the underestimation information is extracted combining with the evolutionary information to design the state judgment factor, so that the evolution state of the current population is estimated. Secondly, according to the feedback information of the state judgment factor, the strategy in different evolution state is adaptively selected to guide the evolution of the population. Therefore, the searching efficiency of the algorithm can be improved. Additionally, experimental results of 20 benchmark functions and CEC2013 test set show that the proposed algorithm is superior to the main-stream differential evolution variants and non-differential evolution algorithms mentioned in this paper in terms of computational cost, convergence speed, and solution quality.
-
Key words:
- Differential evolution (DE) /
- state estimation /
- feedback /
- global optimization /
- abstract convex
1) 本文责任编委 刘艳军 -
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
-
[1] Storn R, Price K. Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 1997, 11(4): 341-359 doi: 10.1023/A:1008202821328 [2] Das S, Suganthan P N. Differential evolution: a survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation, 2011, 15(1): 4-31 doi: 10.1109/TEVC.2010.2059031 [3] Raghunathan T, Ghose D. Differential evolution based 3-D guidance law for a realistic interceptor model. Applied Soft Computing, 2014, 16: 20-33 doi: 10.1016/j.asoc.2013.11.017 [4] 胡蓉, 钱斌.一种求解随机有限缓冲区流水线调度的混合差分进化算法.自动化学报, 2009, 35(12): 1580-1586 doi: 10.3724/SP.J.1004.2009.01580Hu Rong, Qian Bin. A hybrid differential evolution algorithm for stochastic flow shop scheduling with limited buffers. Acta Automatica Sinica, 2009, 35(12): 1580-1586 doi: 10.3724/SP.J.1004.2009.01580 [5] Pandit M, Srivastava L, Sharma M. Environmental economic dispatch in multi-area power system employing improved differential evolution with fuzzy selection. Applied Soft Computing, 2015, 28: 498-510 doi: 10.1016/j.asoc.2014.12.027 [6] Zhang G J, Zhou X G, Yu X F, Hao X H, Yu L. Enhancing protein conformational space sampling using distance profile-guided differential evolution. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2017, 14(6): 1288-1301 doi: 10.1109/TCBB.2016.2566617 [7] 朱腾, 王京春, 熊智华.基于改进DE-NMPC的酸碱中和反应pH值控制.自动化学报, 2010, 36(1): 159-163 doi: 10.3724/SP.J.1004.2010.00159Zhu Teng, Wang Jing-Chun, Xiong Zhi-Hua. DE-based nonlinear model predictive control of a pH neutralization process. Acta Automatica Sinica, 2010, 36(1): 159-163 doi: 10.3724/SP.J.1004.2010.00159 [8] Qiao D P, Pang G K H. A modified differential evolution with heuristic algorithm for nonconvex optimization on sensor network localization. IEEE Transactions on Vehicular Technology, 2016, 65(3): 1676-1689 doi: 10.1109/TVT.2015.2409319 [9] Das S, Mullick S S, Suganthan P N. Recent advances in differential evolution - an updated survey. Swarm and Evolutionary Computation, 2016, 27: 1-30 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=abbffecca110c67e0656d3a7220658fc [10] Dragoi E N, Dafinescu V. Parameter control and hybridization techniques in differential evolution: a survey. Artificial Intelligence Review, 2016, 45(4): 447-470 doi: 10.1007/s10462-015-9452-8 [11] Cheng M Y, Tran D H. Two-phase differential evolution for the multiobjective optimization of time - cost tradeoffs in resource-constrained construction projects. IEEE Transactions on Engineering Management, 2014, 61(3): 450-461 doi: 10.1109/TEM.2014.2327512 [12] Liu Z Z, Wang Y, Yang S X, Cai Z X. Differential evolution with a two-stage optimization mechanism for numerical optimization. In: Proceedings of the 2016 IEEE Congress on Evolutionary Computation (CEC). Vancouver, BC, Canada: IEEE, 2016. 3170-3177 [13] Tang L X, Dong Y, Liu J Y. Differential evolution with an individual-dependent mechanism. IEEE Transactions on Evolutionary Computation, 2015, 19(4): 560-574 doi: 10.1109/TEVC.2014.2360890 [14] Yu W J, Shen M E, Chen W N, Zhan Z H, Gong Y J, Lin Y, Liu Q, Zhang J. Differential evolution with two-level parameter adaptation. IEEE Transactions on Cybernetics, 2014, 44(7): 1080-1099 doi: 10.1109/TCYB.2013.2279211 [15] 张贵军, 陈铭, 周晓根.动态小生境半径两阶段多模态差分进化算法.控制与决策, 2016, 31(7): 1185-1191 http://d.old.wanfangdata.com.cn/Periodical/kzyjc201607006Zhang Gui-Jun, Chen Ming, Zhou Xiao-Gen. Two-stage differential evolution algorithm using dynamic niche radius for multimodal optimization. Control and Decision, 2016, 31(7): 1185-1191 http://d.old.wanfangdata.com.cn/Periodical/kzyjc201607006 [16] Price K, Storn R M, Lampinen J A. Differential Evolution - A Practical Approach to Global Optimization. Berlin Heidelberg: Springer, 2005: 1-24 [17] 张贵军, 何洋军, 郭海锋, 冯远静, 徐建明.基于广义凸下界估计的多模态差分进化算法.软件学报, 2013, 24(6): 1177-1195 http://d.old.wanfangdata.com.cn/Periodical/rjxb201306002Zhang Gui-Jun, He Yang-Jun, Guo Hai-Feng, Feng Yuan-Jing, Xu Jian-Ming. Differential evolution algorithm for multimodal optimization based on abstract convex underestimation. Journal of Software, 2013, 24(6): 1177-1195 http://d.old.wanfangdata.com.cn/Periodical/rjxb201306002 [18] 周晓根, 张贵军, 郝小虎.局部抽象凸区域剖分差分进化算法.自动化学报, 2015, 41(7): 1315-1327 doi: 10.16383/j.aas.2015.c140680Zhou Xiao-Gen, Zhang Gui-Jun, Hao Xiao-Hu. Differential evolution algorithm with local abstract convex region partition. Acta Automatica Sinica, 2015, 41(7): 1315-1327 doi: 10.16383/j.aas.2015.c140680 [19] Beliakov G. Extended cutting angle method of global optimization. Pacific Journal of Optimization, 2008, 4(1): 153-176 [20] Beliakov G, Lim K F. Challenges of continuous global optimization in molecular structure prediction. European Journal of Operational Research, 2007, 181(3): 1198-1213 doi: 10.1016/j.ejor.2005.08.033 [21] 周晓根, 张贵军, 郝小虎, 俞立.一种基于局部Lipschitz下界估计支撑面的差分进化算法.计算机学报, 2016, 39(12): 2631-2651 doi: 10.11897/SP.J.1016.2016.02631Zhou Xiao-Gen, Zhang Gui-Jun, Hao Xiao-Hu, Yu Li. Differential evolution algorithm based on local lipschitz underestimate supporting hyperplanes. Chinese Journal of Computers, 2016, 39(12): 2631-2651 doi: 10.11897/SP.J.1016.2016.02631 [22] Ali M M, Khompatraporn C, Zabinsky Z B. A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. Journal of Global Optimization, 2005, 31(4): 635-672 doi: 10.1007/s10898-004-9972-2 [23] Sarker R A, Elsayed S M, Ray T. Differential evolution with dynamic parameters selection for optimization problems. IEEE Transactions on Evolutionary Computation, 2014, 18(5): 689-707 [24] Dragoi E N, Dafinescu V. Parameter control and hybridization techniques in differential evolution: a survey. Artificial Intelligence Review, 2015, 45(4): 447-470 [25] Zhou X G, Zhang G J. Abstract convex underestimation assisted multistage differential evolution. IEEE Transactions on Cybernetics, 2017, 47(9): 2730-2741 doi: 10.1109/TCYB.2017.2710626 [26] Zhou X G, Zhang G J, Hao X H, Yu L. A novel differential evolution algorithm using local abstract convex underestimate strategy for global optimization. Computers & Operations Research, 2016, 75: 132-149 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=dd11fc41fe4a435f970157e38de1f775 [27] Tanabe R, Fukunaga A. Success-history based parameter adaptation for differential evolution. In: Proceedings of the 2013 IEEE Congress on Evolutionary Computation (CEC). Cancun, Mexico: IEEE, 2013. 71-78 [28] Mallipeddi R, Suganthan P N, Pan Q K, Tasgetiren M F. Differential evolution algorithm with ensemble of parameters and mutation strategies. Applied Soft Computing, 2011, 11(2): 1679-1696 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=cface92266298dbf8ec3818bbe607897 [29] Wang Y, Cai Z X, Zhang Q F. Differential evolution with composite trial vector generation strategies and control parameters. IEEE Transactions on Evolutionary Computation, 2011, 15(1): 55-66 doi: 10.1109/TEVC.2010.2087271 [30] Corder G W, Foreman D I. Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach. Hoboken, NJ: Wiley 2009. 38-55 [31] Shang Y W, Qiu Y H. A note on the extended Rosenbrock function. Evolutionary Computation, 2006, 14(1): 119-126 [32] Liang J J, Qin A K, Suganthan P N, Baskar S. Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Transactions on Evolutionary Computation, 2006, 10(3): 281-295 [33] Hansen N, Ostermeier A. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 2001, 9(2): 159-195 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=58e04c72ec259f9411f7e06253975afb [34] García-Martínez C, Lozano M, Herrera F, Molina D, Sánchez A M. Global and local real-coded genetic algorithms based on parent-centric crossover operators. European Journal of Operational Research, 2008, 185(3): 1088-1113 doi: 10.1016/j.ejor.2006.06.043 [35] Fan Q Q, Yan X F. Self-adaptive differential evolution algorithm with zoning evolution of control parameters and adaptive mutation strategies. IEEE Transactions on Cybernetics, 2016, 46(1): 219-232 doi: 10.1109/TCYB.2015.2399478 [36] Draa A, Bouzoubia S, Boukhalfa I. A sinusoidal differential evolution algorithm for numerical optimisation. Applied Soft Computing, 2015, 27: 99-126 doi: 10.1016/j.asoc.2014.11.003 [37] Liang J J, Qu B Y, Suganthan P N, Hernández-Díaz A G. Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization, Technical Report 201212, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, 2013. -
图(8) / 表(7)
计量
- 文章访问数: 4587
- HTML全文浏览量: 1405
- PDF下载量: 383
- 被引次数: 0
