通信学报 ›› 2023, Vol. 44 ›› Issue (6): 34-46.doi: 10.11959/j.issn.1000-436x.2023113
麦伟杰1, 刘伟莉2, 钟竞辉1
修回日期:2023-05-16
出版日期:2023-06-25
发布日期:2023-06-01
作者简介:麦伟杰（1985- ），男，广东广州人，华南理工大学博士生，主要研究方向为演化计算、机器学习基金资助:Weijie MAI1, Weili LIU2, Jinghui ZHONG1
Revised:2023-05-16
Online:2023-06-25
Published:2023-06-01
Supported by:摘要：
种群的局部最优与停滞状态会严重影响差分进化（DE）算法的性能。为了消除这2种状态引起的不利因素，提出一种带有种群状态处理措施的改进 DE 算法。当种群处于局部最优状态时，运用限制记忆的拟牛顿（LBFGS）方法对种群中的个体进行随机学习提高解的全局质量，通过高斯变异生成新个体，促使种群跳出局部最优；当算法处于停滞状态时，运用种群的协方差矩阵，通过空间坐标旋转对目标个体进行重组，从而抑制种群停滞状态，加强算法全局搜索能力。此外，算法设计一种新型的选择策略，该选择策略设置一个存放经贪心选择后被遗弃个体的外部存档。当实验个体劣于目标个体时，算法则不再以贪心选择策略生成下一代，而是围绕外部存档进行合理的智能选择，使算法向全局最优收敛。实验表明，通过与先进的8个DE算法在29个标准的测试函数比较，所提算法在解的精确度和收敛速度均具有更好的性能。
中图分类号:
麦伟杰, 刘伟莉, 钟竞辉. 基于种群状态信息的自适应差分进化算法[J]. 通信学报, 2023, 44(6): 34-46.
Weijie MAI, Weili LIU, Jinghui ZHONG. Self-adaptive differential evolution algorithm based on population state information[J]. Journal on Communications, 2023, 44(6): 34-46.
图1
LBFGS随机优化示意"
图2
二维个体通过旋转坐标系随机间接重组"
表1
NSPSDE与8个变种DE算法在29个测试函数比较结果"
| 函数 | 量度 | CODE | SADE | SinDE | RankDE | AEPDE-JADE | MPEDE | CSDE | ADEDE | NSPSDE | ||||||||
| f1 | mean | 2.84×10-30 | - | 4.06×10-39 | - | 4.36×10-58 | - | 1.18×10-31 | - | 4.16×10-163 | - | 1.93×10-60 | - | 2.01×10-58 | - | 2.9×10-34 | - | 5.21×10-278 |
| std | 2.39×10-30 | 5.79×10-39 | 2.40×10-58 | 7.78×10-32 | 0 | 4.21×10-60 | 2.66×10-58 | 1.65×10-34 | 0 | |||||||||
| f2 | mean | 8.77×10-27 | - | 1.41×10-36 | - | 1.30×10-55 | - | 1.72×10-28 | - | 2.03×10-159 | - | 4.10×10-56 | - | 2.42×10-57 | - | 1.47×10-30 | - | 2.33×10-238 |
| std | 6.62×10-27 | 1.15×10-36 | 1.05×10-55 | 1.45×10-28 | 4.54×10-159 | 5.45×10-56 | 3.33×10-57 | 1.09×10-30 | 0 | |||||||||
| f3 | mean | 4.99×10-24 | - | 5.53×10-34 | - | 1.23×10-52 | - | 4.13×10-25 | - | 5.17×10-150 | - | 1.63×10-54 | - | 6.96×10-54 | - | 2.65×10-28 | - | 2.45×10-279 |
| std | 5.19×10-24 | 4.66×10-34 | 6.81×10-53 | 7.41×10-25 | 1.16×10-149 | 2.04×10-54 | 1.25×10-53 | 1.45×10-28 | 0 | |||||||||
| f4 | mean | 3.19×10-7 | - | 1.11×10-4 | - | 6.65×100 | - | 5.85×10-7 | - | 1.86×10-21 | - | 3.19×10-39 | - | 3.30×10-5 | - | 8.74×10-2 | - | 1.21×10-200 |
| std | 4.61×10-7 | 1.16×10-4 | 2.31×100 | 1.62×10-7 | 4.16×10-21 | 5.55×10-39 | 3.17×10-5 | 5.56×10-2 | 0 | |||||||||
| f5 | mean | 4.54×10-15 | - | 2.76×10-21 | - | 2.51×10-32 | - | 9.07×10-15 | - | 1.82×10-78 | - | 4.79×10-27 | - | 5.34×10-28 | - | 8.37×10-20 | - | 7.00×10-149 |
| std | 2.68×10-15 | 1.66×10-21 | 1.18×10-32 | 5.10×10-15 | 3.12×10-78 | 4.71×10-27 | 1.17×10-27 | 2.35×10-20 | 3.79×10-148 | |||||||||
| f6 | mean | 2.50×10-7 | - | 2.69×10-4 | - | 5.12×10-7 | - | 2.39×10-4 | - | 4.80×10-3 | - | 7.27×10-23 | - | 4.51×10-10 | - | 4.50×10-2 | - | 8.97×10-137 |
| std | 2.13×10-7 | 5.43×10-4 | 4.50×10-7 | 1.59×10-4 | 5.81×10-3 | 8.64×10-23 | 3.39×10-10 | 9.65×10-2 | 7.91×10-137 | |||||||||
| f7 | mean | 1.11×10-27 | - | 1.82×10-38 | - | 5.89×10-57 | - | 3.39×10-30 | - | 1.18×10-159 | - | 1.41×10-57 | - | 8.75×10-56 | - | 4.64×10-33 | - | 1.80×10-277 |
| std | 2.21×10-27 | 3.31×10-38 | 6.99×10-57 | 2.68×10-30 | 2.41×10-159 | 2.69×10-57 | 1.42×10-55 | 2.96×10-33 | 0 | |||||||||
| f8 | mean | 8.37×10-30 | - | 5.33×10-39 | - | 7.63×10-59 | - | 1.94×10-31 | - | 3.33×10-164 | - | 3.15×10-60 | - | 1.90×10-56 | - | 8.66×10-34 | - | 2.49×10-277 |
| std | 7.58×10-30 | 7.40×10-39 | 5.24×10-59 | 1.78×10-31 | 0 | 4.33×10-60 | 3.32×10-56 | 5.95×10-34 | 0 | |||||||||
| f9 | mean | 5.03×10-26 | - | 5.57×10-33 | - | 1.12×10-52 | - | 2.27×10-27 | - | 1.72×10-130 | + | 3.82×10-41 | - | 2.41×10-46 | - | 3.12×10-31 | - | 7.40×10-97 |
| std | 6.28×10-26 | 2.77×10-33 | 3.94×10-53 | 2.80×10-27 | 3.28×10-130 | 6.84×10-41 | 4.53×10-46 | 9.70×10-32 | 2.07×10-98 | |||||||||
| f10 | mean | -1.00×100 | ≈ | -1.00×100 | ≈ | -1.00×100 | ≈ | -1.00×100 | - | -1.00×100 | - | -1.00×100 | ≈ | -1.00×100 | ≈ | -1.00×100 | ≈ | -1.00×100 |
| std | 0 | 0 | 0 | 1.57×10-16 | 1.57×10-16 | 0 | 0 | 0 | 0 | |||||||||
| f11 | mean | 7.22×10-28 | - | 9.36×10-34 | - | 1.95×10-45 | - | 4.86×10-29 | - | 3.21×10-144 | - | 7.08×10-63 | - | 3.07×10-48 | - | 2.49×10-26 | - | 1.72×10-253 |
| std | 5.98×10-28 | 1.07×10-33 | 1.28×10-45 | 1.84×10-29 | 7.19×10-144 | 1.42×10-62 | 4.18×10-48 | 2.10×10-26 | 0 | |||||||||
| f12 | mean | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 |
| std | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||
| f13 | mean | 6.16×10-3 | + | 8.38×10-3 | + | 5.12×10-3 | + | 5.70×10-3 | + | 4.81×10-3 | + | 1.25×10-3 | + | 3.34×10-2 | + | 9.43×10-3 | + | 6.21×10-2 |
| std | 2.18×10-3 | 2.52×10-3 | 8.26×10-4 | 1.57×10-3 | 1.72×10-3 | 3.24×10-4 | 6.23×10-3 | 1.43×10-3 | 2.32×10-2 | |||||||||
| f14 | mean | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 |
| std | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||
| f15 | mean | 2.04×10-6 | - | 4.62×10-15 | - | 9.94×101 | - | 1.20×102 | - | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 |
| std | 2.01×10-6 | 6.36×10-15 | 3.44×100 | 3.88×101 | 0 | 0 | 0 | 0 | 0 | |||||||||
| f16 | mean | 4.23×10-3 | - | 8.94×10-6 | - | 9.81×10-3 | - | 2.65×10-9 | - | 4.44×10-17 | - | 2.75×10-7 | - | 1.44×10-7 | - | 5.51×10-5 | - | 2.70×10-147 |
| std | 2.03×10-3 | 7.01×10-6 | 4.33×10-3 | 5.53×10-9 | 9.93×10-17 | 4.07×10-7 | 2.72×10-7 | 5.82×10-5 | 4.05×10-147 | |||||||||
| f17 | mean | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 0 | ≈ | 2.22×10-17 | - | 0 | ≈ | 0 |
| std | 0 | 0 | 0 | 0 | 0 | 0 | 4.97×10-17 | 0 | 0 | |||||||||
| f18 | mean | 1.99×10-1 | - | 1.60×10-1 | - | 1.99×10-1 | - | 1.99×10-1 | - | 1.99×10-1 | - | 1.19×10-1 | - | 1.99×10-1 | - | 1.99×10-1 | - | 1.06×10-133 |
| std | 1.13×10-9 | 5.47×10-2 | 7.71×10-10 | 3.68×10-9 | 1.69×10-9 | 4.47×10-2 | 3.10×10-13 | 1.36×10-10 | 4.94×10-134 | |||||||||
| f19 | mean | 2.77×100 | - | 3.63×100 | - | 1.52×101 | - | 1.99×101 | - | 2.45×10-1 | - | 3.59×10-1 | - | 3.78×10-1 | - | 4.29×100 | - | 2.45×10-80 |
| std | 1.57×100 | 6.48×10-1 | 1.02×100 | 2.28×100 | 4.48×10-2 | 1.71×10-1 | 3.40×10-1 | 9.99×10-1 | 0 | |||||||||
| f20 | mean | 5.51×10-15 | - | 2.66×10-15 | - | 3.38×10-15 | - | 4.09×10-15 | - | 6.22×10-15 | - | 2.66×10-15 | - | 2.04×10-14 | - | 2.66×10-15 | - | 8.88×10-16 |
| std | 1.59×10-15 | 0 | 1.59×10-15 | 1.95×10-15 | 0 | 0 | 2.15×10-14 | 0 | 0 | |||||||||
| f21 | mean | 1.78×100 | - | 1.52×101 | - | 2.68×101 | - | 3.75×101 | - | 1.09×10-1 | - | 2.02×10-1 | - | 8.79×100 | - | 1.33×101 | - | 0 |
| Std | 2.47×10-1 | 4.05×10-1 | 1.32×100 | 2.07×100 | 8.16×10-2 | 4.91×10-2 | 1.94×100 | 9.85×10-1 | 0 | |||||||||
| f22 | mean | 1.61×10-3 | - | 1.11×10-1 | - | 3.33×10-1 | - | 1.55×100 | - | 2.65×10-5 | - | 1.75×10-4 | - | 3.09×10-2 | - | 1.17×10-1 | - | 0 |
| std | 2.51×10-4 | 1.24×10-2 | 5.18×10-2 | 2.14×10-1 | 5.40×10-5 | 7.55×10-5 | 4.90×10-3 | 2.58×10-2 | 0 | |||||||||
| f23 | mean | 3.15×10-1 | + | 3.59×10-1 | + | 3.54×10-1 | + | 3.76×10-1 | + | 2.37×10-1 | + | 2.30×10-1 | + | 2.18×10-1 | + | 3.39×10-1 | + | 6.82×10-1 |
| std | 6.92×10-2 | 5.81×10-2 | 3.43×10-2 | 5.74×10-2 | 1.56×10-2 | 1.71×10-2 | 2.90×10-2 | 2.24×10-2 | 5.74×10-1 | |||||||||
| f24 | mean | 2.77×10-1 | ≈ | 3.64×10-1 | ≈ | 3.13×10-1 | ≈ | 3.80×10-1 | ≈ | 2.13×10-1 | ≈ | 3.62×10-1 | ≈ | 4.39×10-1 | ≈ | 3.50×10-1 | ≈ | 5.00×10-1 |
| std | 5.31×10-2 | 3.50×10-2 | 5.07×10-2 | 1.63×10-1 | 4.73×10-2 | 9.12×10-2 | 1.67×10-1 | 6.77×10-3 | 0 | |||||||||
| f25 | mean | 1.55×100 | - | 1.71×100 | - | 4.02×100 | - | 6.57×100 | - | 5.16×10-1 | - | 7.57×10-1 | - | 5.76×10-1 | - | 2.09×100 | - | 0 |
| std | 3.55×10-1 | 2.24×10-1 | 6.89×10-1 | 2.84×10-1 | 5.26×10-2 | 6.37×10-2 | 1.10×10-1 | 2.81×10-1 | 0 | |||||||||
| f26 | mean | 1.25×100 | - | 1.71×100 | - | 4.21×100 | - | 6.57×100 | - | 5.16×10-1 | - | 6.96×10-1 | - | 4.47×10-1 | - | 2.01×100 | - | 0 |
| std | 3.16×10-1 | 1.29×10-1 | 4.03×10-1 | 4.01×10-1 | 5.26×10-2 | 6.01×10-2 | 1.09×10-1 | 2.60×10-1 | 0 | |||||||||
| f27 | mean | 9.85×100 | - | 3.32×10-9 | - | 5.84×101 | - | 9.98×101 | - | 0 | ≈ | 0 | ≈ | 4.60×100 | - | 0 | ≈ | 0 |
| std | 4.07×100 | 3.99×10-9 | 6.11×100 | 2.20×101 | 0 | 0 | 1.52×100 | 0 | 0 | |||||||||
| f28 | mean | 1.32×10-31 | - | 1.57×10-32 | ≈ | 1.57×10-32 | ≈ | 2.01×10-32 | - | 1.57×10-32 | ≈ | 1.57×10-32 | ≈ | 4.38×10-31 | - | 1.57×10-32 | ≈ | 1.57×10-32 |
| std | 1.09×10-31 | 0 | 0 | 8.44×10-33 | 0 | 0 | 3.05×10-31 | 0 | 0 | |||||||||
| f29 | mean | 9.57×10-30 | - | 1.35×10-31 | ≈ | 1.35×10-31 | ≈ | 4.03×10-31 | - | 1.35×10-31 | ≈ | 1.35×10-31 | ≈ | 1.35×10-31 | ≈ | 1.35×10-31 | ≈ | 1.35×10-31 |
| std | 9.19×10-30 | 0 | 0 | 2.20×10-31 | 0 | 0 | 0 | 0 | 0 | |||||||||
| +、-、=的数目 | 2、23、4 | 2、21、6 | 2、21、6 | 2、23、4 | 3、19、7 | 2、19、8 | 2、23、4 | 2、20、7 | — | |||||||||
| 最优解数目 | 4 | 6 | 6 | 3 | 8 | 9 | 5 | 8 | 25 |
表2
NSPSDE与8个DE变种算法在D=30结果的Friedman检验分析"
| 算法 | 等级 | 排名 |
| CODE | 6.379 3 | 8 |
| SADE | 5.620 7 | 5 |
| SinDE | 5.672 4 | 6 |
| RankDE | 7.172 4 | 9 |
| AEPDE-JADE | 3.206 9 | 2 |
| MPEDE | 3.465 5 | 3 |
| CSDE | 4.982 8 | 4 |
| ADEDE | 6.034 5 | 7 |
| NSPSDE | 2.465 5 | 1 |
表3
NSPSDE与8个DE变种算法在D=30结果的Wilcoxon’s rank-sum检验分析"
| 对比算法 | R+ | R- | p | α=0.05 |
| CODE | 383.0 | 52.0 | 1.39×10-4 | Yes |
| SADE | 376.5 | 58.5 | 2.84×10-4 | Yes |
| SinDE | 383.5 | 51.5 | 1.36×10-4 | Yes |
| RankDE | 388.0 | 47.0 | 7.73×10-5 | Yes |
| AEPDE-JADE | 347.0 | 88.0 | 4.10×10-3 | Yes |
| MPEDE | 342.0 | 93.0 | 5.99×10-3 | Yes |
| CSDE | 354.0 | 52.0 | 2.74×10-4 | Yes |
| ADEDE | 369.0 | 66.0 | 6.07×10-4 | Yes |
图3
NSPSDE与8个DE变种算法在D=30处的收敛曲线"
图3
NSPSDE与8个DE变种算法在D=30处的收敛曲线（续）"
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