《计算机应用》唯一官方网站 ›› 2022, Vol. 42 ›› Issue (8): 2511-2518.DOI: 10.11772/j.issn.1001-9081.2021061079
收稿日期:
2021-06-25
修回日期:
2022-02-25
接受日期:
2022-03-04
发布日期:
2022-03-17
出版日期:
2022-08-10
通讯作者:
李和成
作者简介:
沈瑜(1995—),女,重庆人,硕士研究生,主要研究方向:最优化;基金资助:
Yu SHEN, Hecheng LI(), Lijuan CHEN
Received:
2021-06-25
Revised:
2022-02-25
Accepted:
2022-03-04
Online:
2022-03-17
Published:
2022-08-10
Contact:
Hecheng LI
About author:
SHEN Yu, born in 1995, M. S. candidate. Her research interests include optimization.Supported by:
摘要:
双层规划涉及上层和下层两个最优化问题,上层规划问题的约束域由下层规划问题隐式确定,双层优化以上层目标为主,而下层目标在下层变量方面必须达到最优。双层规划问题的递阶结构使其具有很高的计算复杂度,特别是频繁计算下层问题会累计很大的计算量。为了有效求解这类问题,提出一种基于近似技术的进化算法。首先,采取多种群协同进化,分别利用交叉和变异算子平衡算法的开采和勘探能力;其次,基于灵敏度分析理论,设计了新个体的近似评价方式以减少算法的下层求解次数。一个算例的近似效果演示结果表明,由近似技术得到的近似后代个体与精确后代个体的位置大部分是重合的。除此之外,在10个常用算例上的结果显示,所提算法比多值映射算法获得了更好的最优解;并且根据CPU时间比较,说明近似技术有效地提高了找到最优解的速度,减少了运行时间,验证了所提算法采取的近似技术的有效性。
中图分类号:
沈瑜, 李和成, 陈黎娟. 基于近似技术的双层规划进化算法[J]. 计算机应用, 2022, 42(8): 2511-2518.
Yu SHEN, Hecheng LI, Lijuan CHEN. Evolutionary algorithm based on approximation technique for solving bilevel programming problems[J]. Journal of Computer Applications, 2022, 42(8): 2511-2518.
父代个体 | 近似后代个体 | 精确后代个体 |
---|---|---|
(0.00,2.00),(1.00,4.00) | (0.70,3.40) | (0.70,3.40) |
(0.10,2.20),(0.80,3.60) | (0.24,2.48) | (0.24,2.48) |
(0.40,2.80),(0.90,3.80) | (0.55,3.10) | (0.55,3.10) |
(1.20,3.90),(1.60,3.70) | (1.40,3.80) | (1.40,3.80) |
(1.80,3.60),(2.40,3.30) | (2.20,3.40) | (2.20,3.40) |
(2.00,3.50),(2.60,3.20) | (2.40,3.30) | (2.40,3.30) |
(3.00,3.00),(3.50,2.50) | (3.20,2.80) | (3.20,2.80) |
(3.60,2.40),(4.10,1.90) | (3.90,2.10) | (3.90,2.10) |
(4.50,1.50),(4.90,1.10) | (4.70,1.30) | (4.70,1.30) |
(1.75,3.63),(3.25,2.75) | (2.50,3.19) | (2.50,3.25) |
表1 近似下层解和精确下层解
Tab. 1 Approximate and precise follower solutions
父代个体 | 近似后代个体 | 精确后代个体 |
---|---|---|
(0.00,2.00),(1.00,4.00) | (0.70,3.40) | (0.70,3.40) |
(0.10,2.20),(0.80,3.60) | (0.24,2.48) | (0.24,2.48) |
(0.40,2.80),(0.90,3.80) | (0.55,3.10) | (0.55,3.10) |
(1.20,3.90),(1.60,3.70) | (1.40,3.80) | (1.40,3.80) |
(1.80,3.60),(2.40,3.30) | (2.20,3.40) | (2.20,3.40) |
(2.00,3.50),(2.60,3.20) | (2.40,3.30) | (2.40,3.30) |
(3.00,3.00),(3.50,2.50) | (3.20,2.80) | (3.20,2.80) |
(3.60,2.40),(4.10,1.90) | (3.90,2.10) | (3.90,2.10) |
(4.50,1.50),(4.90,1.10) | (4.70,1.30) | (4.70,1.30) |
(1.75,3.63),(3.25,2.75) | (2.50,3.19) | (2.50,3.25) |
4.80 | 4.80 | 0.00 | 0.048 |
2.96 | 2.96 | 0.00 | 0.030 |
4.20 | 4.20 | 0.00 | 0.042 |
6.60 | 6.60 | 0.00 | 0.066 |
7.80 | 7.80 | 0.00 | 0.078 |
8.10 | 8.10 | 0.00 | 0.081 |
9.20 | 9.20 | 0.00 | 0.092 |
9.90 | 9.90 | 0.00 | 0.099 |
9.40 | 9.40 | 0.00 | 0.094 |
8.19 | 8.25 | 0.06 | 0.082 |
表2 目标函数值的误差
Tab. 2 Errors of objective function values
4.80 | 4.80 | 0.00 | 0.048 |
2.96 | 2.96 | 0.00 | 0.030 |
4.20 | 4.20 | 0.00 | 0.042 |
6.60 | 6.60 | 0.00 | 0.066 |
7.80 | 7.80 | 0.00 | 0.078 |
8.10 | 8.10 | 0.00 | 0.081 |
9.20 | 9.20 | 0.00 | 0.092 |
9.90 | 9.90 | 0.00 | 0.099 |
9.40 | 9.40 | 0.00 | 0.094 |
8.19 | 8.25 | 0.06 | 0.082 |
参数 | 含义 | 取值 |
---|---|---|
种群规模 种群聚类组数 | 50 5 | |
好个体数 | 5 | |
交叉概率 | 0.8 | |
变异概率 | 0.2 | |
拟合精度系数 | 1/100 | |
Max_gen | 运行的最大代数 | 20 |
表3 相关参数及取值
Tab. 3 Related parameter and their values
参数 | 含义 | 取值 |
---|---|---|
种群规模 种群聚类组数 | 50 5 | |
好个体数 | 5 | |
交叉概率 | 0.8 | |
变异概率 | 0.2 | |
拟合精度系数 | 1/100 | |
Max_gen | 运行的最大代数 | 20 |
算法 | 上层目标函数 | 算例1 | 算例2 | 算例3 | 算例4 | 算例5 | 算例6 | 算例7 | 算例8 | 算例9 | 算例10 | 算例11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
本文算法 | 49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.679 9 | -29.2 | 0 | 748.89 | |
49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.679 9 | -29.2 | 0 | 748.89 | ||
49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.679 9 | -29.2 | 0 | 748.89 | ||
文献[31-32,29,33]算法 | 49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.678 7 | -29.2 | 0 | 741.56 |
表4 上层目标函数值的最好值、最差值和均值
Tab. 4 Best values, worst values and mean values of leader objective functions
算法 | 上层目标函数 | 算例1 | 算例2 | 算例3 | 算例4 | 算例5 | 算例6 | 算例7 | 算例8 | 算例9 | 算例10 | 算例11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
本文算法 | 49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.679 9 | -29.2 | 0 | 748.89 | |
49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.679 9 | -29.2 | 0 | 748.89 | ||
49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.679 9 | -29.2 | 0 | 748.89 | ||
文献[31-32,29,33]算法 | 49 | -1 011.666 7 | -22 | -20 | -243.5 | 225 | 0 | -18.678 7 | -29.2 | 0 | 741.56 |
算法 | 算例1 | 算例2 | 算例3 | 算例4 | 算例5 | 算例6 | 算例7 | 算例8 | 算例9 | 算例10 |
---|---|---|---|---|---|---|---|---|---|---|
本文算法 | (3,4,2,0,3,0) | (0,1,0,1,0.00,75.00,21.67) | (2,2) | (8,6) | (3,8, 0.5,0) | (20,5, 10,5) | (0,30, -10,10) | (0,2,1.877 0,0.910 0) | (0.000 1,0.899 9,0,0.599 9,0.400 1) | (0,30, -10,10) |
文献[31-32,29] 算法 | (3,4,2,0,3,0) | (0,1,0,1,0.00,75.00,21.67) | (2,2) | (8,6) | (3,8, 0.5,0) | (20,5, 10,5) | (0,30, -10,10) | (0,2,1.876 8,0.907 6) | (0.000 1,0.899 9,0,0.599 9,0.400 1) | (0,30, -10,10) |
表5 本文算法和比较算法的最优解对比
Tab. 5 Best solutions comparison of the proposed algorithm and compared algorithms
算法 | 算例1 | 算例2 | 算例3 | 算例4 | 算例5 | 算例6 | 算例7 | 算例8 | 算例9 | 算例10 |
---|---|---|---|---|---|---|---|---|---|---|
本文算法 | (3,4,2,0,3,0) | (0,1,0,1,0.00,75.00,21.67) | (2,2) | (8,6) | (3,8, 0.5,0) | (20,5, 10,5) | (0,30, -10,10) | (0,2,1.877 0,0.910 0) | (0.000 1,0.899 9,0,0.599 9,0.400 1) | (0,30, -10,10) |
文献[31-32,29] 算法 | (3,4,2,0,3,0) | (0,1,0,1,0.00,75.00,21.67) | (2,2) | (8,6) | (3,8, 0.5,0) | (20,5, 10,5) | (0,30, -10,10) | (0,2,1.876 8,0.907 6) | (0.000 1,0.899 9,0,0.599 9,0.400 1) | (0,30, -10,10) |
算法 | 算例1 | 算例2 | 算例3 | 算例4 | 算例5 | 算例6 | 算例7 | 算例8 | 算例9 | 算例10 |
---|---|---|---|---|---|---|---|---|---|---|
本文算法 | -27 | -4 673.36 | 2 | -6 | -19.5 | 100 | 100 | -1.026 8 | 3.2 | 100 |
文献[31-32,29]算法 | -27 | -4 673.36 | 2 | -6 | -19.5 | 100 | 100 | -1.015 6 | 3.2 | 100 |
表6 最优解对应的下层目标函数值
Tab. 6 Follower objective function values corresponding to best solutions
算法 | 算例1 | 算例2 | 算例3 | 算例4 | 算例5 | 算例6 | 算例7 | 算例8 | 算例9 | 算例10 |
---|---|---|---|---|---|---|---|---|---|---|
本文算法 | -27 | -4 673.36 | 2 | -6 | -19.5 | 100 | 100 | -1.026 8 | 3.2 | 100 |
文献[31-32,29]算法 | -27 | -4 673.36 | 2 | -6 | -19.5 | 100 | 100 | -1.015 6 | 3.2 | 100 |
算例 | CPU/s | UACPU/s | 算例 | CPU/s | UACPU/s |
---|---|---|---|---|---|
1 | 0.4 | 0.7 | 7 | 0.5 | 0.9 |
2 | 0.3 | 0.8 | 8 | 0.5 | 0.9 |
3 | 0.5 | 0.9 | 9 | 0.4 | 0.9 |
4 | 0.3 | 0.7 | 10 | 0.6 | 1.0 |
5 | 0.5 | 0.9 | 11 | 200.0 | 310.0 |
6 | 0.4 | 0.8 |
表7 本文算法的CPU时间和UACPU时间比较
Tab. 7 Comparison of CPU time and UACPU time of the proposed algorithm
算例 | CPU/s | UACPU/s | 算例 | CPU/s | UACPU/s |
---|---|---|---|---|---|
1 | 0.4 | 0.7 | 7 | 0.5 | 0.9 |
2 | 0.3 | 0.8 | 8 | 0.5 | 0.9 |
3 | 0.5 | 0.9 | 9 | 0.4 | 0.9 |
4 | 0.3 | 0.7 | 10 | 0.6 | 1.0 |
5 | 0.5 | 0.9 | 11 | 200.0 | 310.0 |
6 | 0.4 | 0.8 |
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