多種群協(xié)方差學(xué)習(xí)差分進化算法
doi: 10.11999/JEIT180670
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華僑大學(xué)工學(xué)院? ?泉州? ?362021
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華僑大學(xué)機電及自動化學(xué)院? ?廈門? ?361021
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華僑大學(xué)計算機科學(xué)與技術(shù)學(xué)院? ?廈門? ?361021
Multi-populations Covariance Learning Differential Evolution Algorithm
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College of Engineering, Huaqiao University, Quanzhou 362021, China
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College of Mechanical Engineering and Automation, Huaqiao University, Xiamen 361021, China
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College of Computer Science and Technology, Huaqiao University, Xiamen 361021, China
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摘要: 種群多樣性與交叉算子在差分進化(DE)算法求解全局優(yōu)化問題中具有重要作用,該文提出一種多種群協(xié)方差學(xué)習(xí)差分進化(MCDE)算法。首先,采用多種群機制的種群結(jié)構(gòu),利用每一子種群結(jié)合相應(yīng)的變異策略保證進化過程個體多樣性。然后,通過種群間的協(xié)方差學(xué)習(xí),為交叉操作建立一個適當旋轉(zhuǎn)的坐標系統(tǒng);同時,使用自適應(yīng)控制參數(shù)來平衡種群的勘測與收斂能力。最后,在單峰函數(shù)、多峰函數(shù)、偏移函數(shù)和高維函數(shù)的25個基準測試函數(shù)上進行測試,并同其他先進的進化算法對比,實驗結(jié)果表明該文算法相較于其他算法在求解全局優(yōu)化問題上達到最優(yōu)效果。
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關(guān)鍵詞:
- 差分進化 /
- 多種群 /
- 協(xié)方差學(xué)習(xí) /
- 自適應(yīng)參數(shù)
Abstract: The diversity of the population and the crossover operator algorithm play an important role in solving global optimization problems in Differential Evolution (DE). The Multi-poplutions Covariance learning Differential Evolution (MCDE) algorithm is proposed. Firstly, the population structure is a multi-poplutions mechanism, and each subpopulation combines the corresponding mutation strategy to ensure the individual diversity in the evolutionary process. Then, the covariance learning establishes a proper rotation coordinate system for the crossover operation in the population. At the same time, the adaptive control parameters are used to balance the ability of population survey and convergence. Finally, the proposed algorithm is conducted on 25 benchmark functions including unimodal, multimodal, shifted and high-dimensional test functions and compared with the state-of-the-art evolutionary algorithms. The experimental results show that the proposed algorithm compared with other algorithms has the best effect on solving the global optimization problem. -
表 1 在D=30下3種算法與MCDE的Wilcoxon’s檢測結(jié)果比較
比較算法 R+ R – P值 $\alpha $=0.05 $\alpha $=0.10 JADE 240.5 59.5 0.007012 是 是 CoDE 264.5 60.5 0.005181 是 是 CoBiDE 251.0 74.0 0.016633 是 是 下載: 導(dǎo)出CSV
表 3 30次獨立運行在4種算法的最優(yōu)解平均值及標準差
函數(shù) JADE CoDE CoBiDE MCDE F1 0.00E+00±0.00E+00≈ 0.00E+00±0.00E+00≈ 0.00E+00±0.00E+00≈ 0.00E+00±0.00E+00 F2 1.26E–28±1.22E–28+ 6.77E–15±3.44E–15– 1.60E–12±2.90E–12– 8.49E–28±3.75E–28 F3 8.42E+03±6.58E+03– 5.65E+05±5.66E+04– 7.26E+04±5.64E+04– 2.74E–12±2.82E–11 F4 4.13E–16±3.45E–16– 6.21E–03±4.67E–02– 1.16E–03±2.74E–03– 7.57E–22±4.26E–21 F5 7.59E–08±5.65E–07– 3.16E+02±3.62E+02– 8.03E+02±1.51E+01– 5.38E–10±7.12E–10 F6 1.16E+01±3.16E+01– 3.32E–01±6.57E–01– 4.13E–02±9.21E–02+ 3.19E–01±1.09E–01 F7 8.27E–03±8.22E–03– 7.39E–03±6.45E–03– 1.77E–03±3.73E–03– 1.52E–03±4.11E–03 F8 2.09E+01±1.68E–01≈ 2.01E+01±1.25E–01+ 2.07E+01±3.75E–01+ 2.09E+01±4.21E–02 F9 0.00E+00±0.00E+00+ 0.00E+00±0.00E+00+ 0.00E+00±0.00E+00+ 2.64E–07±5.87E–07 F10 2.42E+01±5.44E+00– 4.21E+01±2.84E+01– 4.41E+01±1.29E+01– 2.28E+01±4.27E+00 F11 2.57E+01±2.21E+00– 1.24E+01±3.55E+00+ 5.62E+00±2.19E+00+ 1.51E+01±6.81e+00 F12 6.45E+03±2.89E+03– 3.21E+03±4.48E+03– 2.94E+03±3.93E+03– 2.12E+03±1.34E+03 F13 1.47E+00±1.15E–01+ 1.66E+00±3.25E–01+ 2.64E+00±1.13E+00– 1.74E+00±2.04E–01 F14 1.23E+01±3.21E–01≈ 1.23E+01±3.56E–01≈ 1.23E+01±4.90E–01≈ 1.23E+01±2.66E–01 F15 3.61E+02±2.24E+02+ 4.00E+02±5.24E+01≈ 4.04E+02±5.03E+01– 4.00E+02±1.09E+02 F16 9.33E+01±1.31E+02– 7.25E+01±6.22E+01+ 7.38E+01±3.66E+01– 5.37E+01±3.01E+01 F17 1.21E+02±1.08E+02– 7.16E+01±2.35E+01– 7.25E+01±2.02e+01– 6.36E+01±6.41E+01 F18 9.04E+02±1.24E–01≈ 9.04E+02±1.34E+00≈ 9.03E+02±1.05E+01≈ 9.03E+02±6.01E–01 F19 9.04E+02±8.32E+00≈ 9.04E+02±3.22E–01≈ 9.03E+02±1.04E+01≈ 9.03E+02±2.31E–01 F20 9.04E+02±7.65E–01≈ 9.04E+02±7.11E–01≈ 9.04E+02±5.95E–01≈ 9.03E+02±2.45E–01 F21 5.00E+02±4.67E–13≈ 5.00E+02±4.68E–13≈ 5.00E+02±4.62E–13≈ 5.00E+02±4.51E–14 F22 8.68E+02±2.24E+01≈ 8.78E+02±3.54E+01≈ 8.69E+02±2.80E+01≈ 8.69E+02±1.89E+01 F23 5.48E+02±8.62E+01– 5.34E+02±4.45E–04≈ 5.34E+02±1.30E–04≈ 5.34E+02±2.49E–13 F24 2.00E+02±2.12E–14≈ 2.00E+02±2.62E–14≈ 2.00E+02±2.90E–14≈ 2.00E+02±2.90E–14 F25 2.11E+02±7.35E–01– 2.11E+02±6.82E–01– 2.10E+02±7.73E–01– 2.09E+02±2.78E–01 +/–/≈ 3/13/9 5/10/10 4/13/8 下載: 導(dǎo)出CSV
表 4 30次獨立運行在CLPSO, CMA-ES, GL-25, MCDE最優(yōu)解平均值及標準差
Function CLPSO CMA-ES GL-25 MCDE F1 0.00E+00±0.00e+00≈ 1.58E–25±3.35E–26– 5.60E–27±1.76E–26– 0.00E+00±0.00E+00 F2 8.40E+02±1.90E+02– 1.12E–24±2.93E–25– 4.04E+01±6.28E+01– 8.49E–28±3.75E–28 F3 1.42E+07±4.19E+06– 5.54E–21±1.69E–21+ 2.19E+06±1.08E+06– 2.74E–12±2.82E–11 F4 6.99E+03±1.73E+03– 9.15E+05±2.16E+06– 9.07E+02±4.25E+02– 7.57E–22±4.26E–21 F5 3.86E+03±4.35E+02– 2.77E–10±5.04E–11+ 2.51E+03±1.96E+02– 5.38E–10±7.12E–10 F6 4.16E+00±3.48E+00– 4.78E–01±1.32E+00– 2.15E+01±1.17E+00– 3.19E–01±1.09E–01 F7 4.51E–01±8.47E–02– 1.82E–03±4.33E–03– 2.78E–02±3.62E–02– 1.52E–03±4.11E–03 F8 2.09E+01±4.41E–02– 2.03E+01±5.72E–01+ 2.09E+01±5.94E–02– 2.09E+01±4.21E–02 F9 0.00e+00±0.00e+00+ 4.45E+02±7.12E+01– 2.45E+01±7.35E+00– 2.64E–07±5.87E–07 F10 1.04E+02±1.53E+01– 4.63E+01±1.16E+01– 1.42E+02±6.45E+01– 2.28E+01±4.27E+00 F11 2.60E+01±1.63E+00– 7.11E+00±2.14E+00+ 3.27E+01±7.79E+00– 1.51E+01±6.81e+00 F12 1.79E+04±5.24E+03– 1.26E+04±1.74E+04– 6.53E+04±4.69E+04– 2.12E+03±1.34E+03 F13 2.06E+00±2.15E–01– 3.43E+00±7.60E–01– 6.23E+00±4.88E+00– 1.74E+00±2.04E–01 F14 1.28E+01±2.48E–01– 1.47E+01±3.31E–01– 1.31E+01±1.84E–01– 1.23E+01±2.66E–01 F15 5.77E+01±2.76E+01– 5.55E+02±3.32E+02– 3.04E+02±1.99E+01+ 4.00E+02±1.09E+02 F16 1.74E+02±2.82E+01– 2.98E+02±2.08E+02– 1.32E+02±7.60E+01– 5.37E+01±3.01E+01 F17 2.46E+02±4.81E+01– 4.43E+02±3.34E+02– 1.61E+02±6.80E+01– 6.36E+01±6.41E+01 F18 9.13E+02±1.42E+00– 9.04E+02±3.01E–01≈ 9.07E+02±1.48E+00– 9.03E+02±6.01E–01 F19 9.14E+02±1.45E+00– 9.16E+02±6.03E+01– 9.06E+02±1.24E+00– 9.03E+02±2.31E–01 F20 9.14E+02±3.62E+00– 9.04E+02±2.71E–01≈ 9.07E+02±1.35E+00– 9.03E+02±2.45E–01 F21 5.00E+02±3.39E–13≈ 5.00E+02±2.68E–12≈ 5.00E+02±4.83E–13≈ 5.00E+02±4.51E–14 F22 9.72E+02±1.20E+01– 8.26E+02±1.46E+01+ 9.28E+02±7.04E+01– 8.69E+02±1.89E+01 F23 5.34E+02±2.19E–04≈ 5.36E+02±5.44E+00≈ 5.34E+02±4.66E–04≈ 5.34E+02±2.49E–13 F24 2.00E+02±1.49E–12≈ 2.12E+02±6.00E+01– 2.00E+02±5.52E–11≈ 2.00E+02±2.90E–14 F25 2.00E+02±1.96E+00+ 2.07E+02±6.07E+00≈ 2.17E+02±1.36E–01– 2.09E+02±2.78E–01 +/–/≈ 2/19/4 5/15/5 1/21/3 下載: 導(dǎo)出CSV
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