西安电子科技大学学报 ›› 2024, Vol. 51 ›› Issue (3): 124-135.doi: 10.19665/j.issn1001-2400.20230902
收稿日期:
2023-03-03
出版日期:
2024-06-20
发布日期:
2023-09-14
通讯作者:
武 燕(1975—),女,副教授,E-mail:wuyan@mail.xidian.edu.cn作者简介:
万梦依(1998—),女,西安电子科技大学硕士研究生,E-mail:18336099322@163.com
基金资助:
Received:
2023-03-03
Online:
2024-06-20
Published:
2023-09-14
摘要:
动态多目标优化问题(DMOPs)是指目标函数随时间变化的一类问题,算法求解的目标是持续跟踪移动的Pareto 最优解集或 Pareto最优前沿。基于预测的方法受到格外的关注,然而这些方法多使用历史环境信息进行预测,考虑到使用历史信息预测会存在预测不准确的问题,加强新环境信息的挖掘和利用,提出了一种新的基于预测的动态多目标进化算法,该算法主要包括两个核心部分,分别记为响应机制和加速机制。响应机制在环境变化后重新初始化群体,一部分的个体由预测策略产生,以生成靠近下一环境Pareto 最优解集的个体来提高算法的寻优能力,剩余部分个体采用局部搜索策略生成以增加种群多样性。加速机制用于静态优化过程以提高算法收敛速度。最后,将动态多目标进化算法与其他3种先进的动态多目标优化算法在具有不同动态特征的一系列测试函数上进行实验对比,结果表明,动态多目标进化算法相比其他3个算法在求解动态多目标优化问题中更具有优势。
中图分类号:
万梦依, 武燕. 一种新的基于预测的动态多目标进化算法[J]. 西安电子科技大学学报, 2024, 51(3): 124-135.
WAN Mengyi, WU Yan. New prediction strategy based evolutionary algorithm for dynamic multi-objective optimization[J]. Journal of Xidian University, 2024, 51(3): 124-135.
表1
4个算法在测试问题上IGD的统计结果"
测试函数 | (nt,τt) | NSGA-Ⅱ-A mean(std) | Tr-MOEA mean(std) | DSS mean(std) | RAM mean(std) |
---|---|---|---|---|---|
FDA1 | (10,10) | 2.75e-2(1.18e-2) | 1.75e-1(9.30e-3) | 1.29e-2(9.00e-4) | 9.90e-3(6.00e-4) |
FDA4 | (10,10) | 6.74e-2(5.00e-4) | 7.34e-2(1.25e-2) | 6.58e-2(1.20e-3) | 6.63e-2(1.60e-3) |
dMOP1 | (10,10) | 3.16e-2(8.60e-3) | 6.21e-2(1.74e-2) | 7.44e-2(1.17e-2) | 3.51e-2(2.33e-2) |
dMOP2 | (10,10) | 4.47e-2(1.37e-2) | 1.08e-1(2.75e-2) | 1.58e-2(1.90e-3) | 1.56e-2(1.10e-3) |
F5 | (10,10) | 1.24e+0(3.13e-1) | 2.67e+0(2.34e-1) | 7.97e-2(9.40e-3) | 6.62e-2(8.20-3) |
F6 | (10,10) | 9.22e-1(1.59e-1) | 2.37e+0(7.86e-2) | 9.47e-2(1.20e-2) | 5.07e-2(1.70e-2) |
F7 | (10,10) | 1.12e+0(2.17e-1) | 3.50e+0(1.12e-1) | 6.94e-2(1.17e-2) | 4.69e-2(1.13e-2) |
F8 | (10,10) | 8.19e-2(1.60e-3) | 9.11e-2(8.90e-3) | 8.05e-2(1.60e-3) | 7.70e-2(1.60e-3) |
F9 | (10,10) | 1.26e+0(2.66e-1) | 2.58e+0(5.18e-2) | 9.99e-2(1.05e-2) | 6.96e-2(8.60e-3) |
F10 | (10,10) | 1.29e+0(2.27e-1) | 4.51e+0(5.50e-1) | 7.80e-2(1.24e-2) | 1.02e-1(2.33e-2) |
F11 | (10,10) | 1.40e+0(2.52e-1) | 3.70e+0(1.76e-1) | 1.27e-1(4.21e-2) | 9.65e-2(2.08e-2) |
F12 | (10,10) | 1.20e+0(3.48e-1) | 2.85e+0(1.17e-1) | 7.94e-2(9.20e-3) | 5.29e-2(7.60e-3) |
dMOP3 | (10,10) | 5.07e-2(2.80e-3) | 2.56e-2(6.00e-3) | 2.26e-2(3.70e-3) | 1.84e-2(2.00e-3) |
ZJZ7 | (10,10) | 1.18e+0(2.07e-1) | 2.56e-1(1.76e-2) | 1.34e-1(3.32e-2) | 1.10e-1(1.75e-2) |
ZJZ8 | (10,10) | 4.99e-1(1.33e-1) | 2.49e-1(8.80e-3) | 1.30e-1(2.69e-2) | 1.12e-1(2.61e-2) |
JY10 | (10,10) | 1.27e+0(3.63e-1) | 7.11e-1(4.07e-2) | 3.10e-1(5.98e-2) | 2.59e-1(8.45e-2) |
表2
4个算法在测试问题上HVD的统计结果"
测试函数 | (nt,τt) | NSGA-Ⅱ-A mean(std) | Tr-MOEA mean(std) | DSS mean(std) | RAM mean(std) |
---|---|---|---|---|---|
FDA1 | (10,10) | 9.91e-2(2.22e-2) | 1.58e-1(7.40e-3) | 2.61e-2(1.90e-3) | 2.02e-2(1.70e-3) |
FDA1 | (10,10) | 9.91e-2(2.22e-2) | 1.58e-1(7.40e-3) | 2.61e-2(1.90e-3) | 2.02e-2(1.70e-3) |
FDA4 | (10,10) | 5.09e-2(4.20e-3) | 5.74e-2(3.80e-3) | 1.83e-1(6.00e-3) | 1.83e-1(7.80e-3) |
dMOP1 | (10,10) | 1.08e-1(2.24e-2) | 6.67e-2(1.37e-2) | 1.51e-1(8.60e-2) | 6.52e-2(3.88e-2) |
dMOP2 | (10,10) | 6.84e-2(4.40e-3) | 9.33e-2(1.75e-2) | 3.53e-2(4.70e-3) | 3.59e-2(3.00e-3) |
F5 | (10,10) | 9.06e+0(4.01e+0) | 6.72e-1(1.49e-2) | 2.86e-1(7.72e-2) | 2.52e-1(7.49e-2) |
F6 | (10,10) | 5.30e+0(2.13e+0) | 7.01e-1(1.15e-2) | 5.58e-1(4.95e-1) | 2.47e-1(2.51e-1) |
F7 | (10,10) | 7.49e+0(1.91e+0) | 7.60e-1(1.02e-2) | 2.47e-1(1.05e-1) | 1.64e-1(8.84e-2) |
F8 | (10,10) | 6.93e-2(3.70e-3) | 6.52e-2(2.60e-3) | 2.44e-1(1.92e-2) | 2.42e-1(1.22e-2) |
F9 | (10,10) | 9.26e+0(3.62e+0) | 5.81e-1(1.12e-2) | 3.40e-1(6.47e-2) | 2.22e-1(3.70e-2) |
F10 | (10,10) | 9.13e+0(3.62e+0) | 5.98e-1(2.59e-2) | 2.62e-1(7.01e-2) | 3.75e-1(1.21e-1) |
F11 | (10,10) | 9.30e+0(2.56e+0) | 8.01e-1(7.00e-3) | 8.40e-1(5.97e-1) | 5.04e-1(2.44e-1) |
F12 | (10,10) | 8.39e+0(4.03e+0) | 7.06e-1(1.69e-2) | 2.89e-1(5.77e-2) | 1.92e-1(5.62e-2) |
dMOP3 | (10,10) | 2.55e-1(1.01e-2) | 2.78e-2(5.80e-3) | 2.64e-2(4.90e-3) | 1.58e-1(5.20e-3) |
ZJZ7 | (10,10) | 8.24e+0(2.31e+0) | 2.77e-1(1.72e-2) | 4.12e-1(1.08e-1) | 3.97e-1(1.16e-1) |
ZJZ8 | (10,10) | 3.19e+0(7.83e-1) | 2.71e-1(1.01e-2) | 4.96e-1(2.14e-1) | 3.89e-1(1.33e-1) |
JY10 | (10,10) | 5.27e+0(7.33e-1) | 4.23e-1(1.20e-1) | 3.49e-1(2.96e-1) | 3.64e-1(6.75e-1) |
表3
4个算法在不同变化频率下IGD的均值和标准差"
测试函数 | (nt,τt) | NSGA-Ⅱ-A mean(std) | Tr-MOEA mean(std) | DSS mean(std) | RAM mean(std) |
---|---|---|---|---|---|
F9 | (10,5) | 2.26e+0(3.16e-1) | 2.74e+0(1.74e-2) | 2.56e-1(3.38e-2) | 1.84e-1(1.82e-2) |
(10,10) | 1.27e+0(2.67e-1) | 2.58e+0(5.18e-2) | 9.99e-2(1.05e-2) | 6.96e-2(8.60e-3) | |
(10,20) | 4.95e-1(1.60e-1) | 2.24e+0(1.31e-1) | 3.34e-2(4.00e-3) | 4.39e-2(3.80e-3) | |
F10 | (10,5) | 2.56e+0(3.89e-1) | 6.72e+0(5.71e-1) | 2.02e-1(2.45e-2) | 3.05e-1(4.79e-2) |
(10,10) | 1.29e+0(2.27e-1) | 4.52e+0(5.51e-1) | 8.14e-2(8.00e-3) | 1.02e-1(2.33e-2) | |
(10,20) | 5.79e-1(1.04e-1) | 2.35e+0(4.32e-1) | 3.52e-2(4.60e-3) | 3.34e-2(6.30e-3) | |
F11 | (10,5) | 2.48e+0(3.69e-1) | 4.47e+0(1.15e-1) | 3.38e-1(1.32e-1) | 3.93e-1(8.38e-2) |
(10,10) | 1.40e+0(2.52e-1) | 3.70e+0(1.76e-1) | 1.27e-1(4.21e-2) | 9.65e-2(2.08e-2) | |
(10,20) | 7.70e-1(1.22e-1) | 3.13e+0(2.64e-1) | 3.99e-2(1.27e-2) | 2.75e-2(3.00e-3) | |
F12 | (10,5) | 2.34e+0(5.71e-1) | 3.18e+0(2.40e-1) | 2.19e-1(3.33e-2) | 1.76e-1(2.94e-2) |
(10,10) | 1.21e+0(3.49e-1) | 2.85e+0(1.17e-1) | 794e-2(9.20e-3) | 5.29e-2(7.60e-3) | |
(10,20) | 5.02e-1(1.61e-1) | 2.37e+0(1.45e-1) | 2.73e-2(3.40e-3) | 2.42e-2(2.30e-3) | |
DMOP3 | (10,5) | 1.04e-1(1.81e-2) | 3.20e-2(8.90e-3) | 6.27e-2(6.20e-3) | 4.97e-2(5.50e-3) |
(10,10) | 5.07e-2(2.80e-3) | 2.56e-2(6.00e-3) | 2.26e-2(3.70e-3) | 1.84e-2(2.00e-3) | |
(10,20) | 3.62e-2(2.20e-3) | 2.14e-2(2.20e-3) | 1.02e-2(9.00e-3) | 1.07e-2(2.00e-3) | |
ZJZ7 | (10,5) | 2.83e+0(3.14e-1) | 3.92e-1(3.00e-2) | 3.94e-1(1.65e-1) | 3.36e-1(5.86e-2) |
(10,10) | 1.18e+0(2.07e-1) | 2.56e-1(1.76e-2) | 1.35e-1(3.32e-2) | 1.10e-1(1.75e-2) | |
(10,20) | 3.00e-1(8.76e-2) | 1.95e-1(4.80e-3) | 4.01e-2(5.70e-3) | 3.26e-2(6.20e-3) | |
ZJZ8 | (10,5) | 1.25e+0(1.97e-1) | 2.81e-1(2.24e-2) | 3.37e-1(5.22e-2) | 2.64e-1(6.15e-2) |
(10,10) | 4.99e-1(1.33e-1) | 2.50e-1(8.80e-3) | 1.30e-1(2.69e-2) | 1.12e-1(2.61e-2) | |
(10,20) | 1.54e-1(5.12e-2) | 1.51e-1(9.80e-3) | 3.97e-2(5.30e-3) | 2.57e-2(6.50e-3) | |
JY10 | (10,5) | 2.38e+0(4.77e-1) | 9.68e-1(6.32e-2) | 4.28e-1(1.95e-1) | 3.18e-1(2.17-1) |
(10,10) | 1.26e+0(3.63e-1) | 7.11e-1(4.07e-2) | 3.10e-1(5.98e-2) | 2.59e-1(8.45e-2) | |
(10,20) | 7.71e-1(2.85e-1) | 5.87e-1(5.67e-2) | 1.33e-1(5.20e-3) | 1.28e-1(8.70e-3) |
表4
4个算法在不同变化强度下IGD的均值和标准差"
测试函数 | (nt,τt) | NSGA-Ⅱ-A mean(std) | Tr-MOEA mean(std) | DSS mean(std) | RAM mean(std) |
---|---|---|---|---|---|
F9 | (5,10) | 1.65e+0(2.12e-1) | 2.63e+0(9.19e-2) | 8.42e-2(1.77e-2) | 7.59e-2(1.66e-2) |
(10,10) | 1.26e+0(2.66e-1) | 2.58e+0(5.18e-2) | 9.99e-2(1.05e-2) | 6.96e-2(8.6e-3) | |
(20,10) | 8.53e-1(2.60e-1) | 2.56e+0(3.16e-2) | 7.47e-2(1.09e-2) | 6.46e-2(1.11e-2) | |
F10 | (5,10) | 1.88e+0(2.80e-1) | 4.03e+0(5.02e-1) | 1.24e-1(2.78e-2) | 1.43e-1(4.40e-2) |
(10,10) | 1.29e+0(2.27e-1) | 4.52e+0(5.51e-1) | 8.14e-2(8.00e-3) | 1.02e-1(2.33e-2) | |
(20,10) | 7.75e-1(1.73e-1) | 4.06e+0(5.34e-1) | 8.83e-2(9.68e-1) | 7.39e-2(1.43e-2) | |
F11 | (5,10) | 2.02e+0(3.52e-1) | 3.99e+0(1.78e-1) | 1.36e-1(2.70e-2) | 1.45e-1(3.57e-2) |
(10,10) | 1.40e+0(2.52e-1) | 3.70e+0(1.76e-1) | 1.27e-1(4.21e-2) | 9.65e-2(2.08e-2) | |
(20,10) | 9.68e-1(2.31e-1) | 3.54e+0(1.75e-1) | 8.86e-2(3.78e-2) | 6.67e-2(2.00e-2) | |
F12 | (5,10) | 1.88e+0(3.14e-1) | 2.85e+0(1.23e-1) | 1.24e-1(2.06e-2) | 6.44e-2(6.60e-3) |
(10,10) | 1.21e+0(3.49e-1) | 2.85e+0(1.17e-1) | 7.94e-2(9.20e-3) | 5.29e-2(7.60e-3) | |
(20,10) | 9.61e-1(4.25e-1) | 2.73e+0(2.78e-2) | 5.97e-2(9.80e-3) | 4.55e-2(7.5e-3) | |
DMOP3 | (5,10) | 8.72e-2(1.40e-2) | 2.53e-2(1.26e-2) | 2.55e-2(3.10e-3) | 1.71e-2(2.50e-3) |
(10,10) | 5.07e-2(2.81e-3) | 2.56e-2(6.00e-3) | 2.26e-2(3.70e-3) | 1.84e-2(2.00e-3) | |
(20,10) | 3.81e-2(2.30e-3) | 2.14e-2(2.20e-3) | 2.13e-2(3.40e-3) | 1.79e-2(2.40e-3) | |
ZJZ7 | (5,10) | 1.19e+0(2.02e+0) | 3.11e-1(9.80e-3) | 1.54e-1(5.63e-2) | 1.02e-1(2.81e-2) |
(10,10) | 1.18e+0(2.07e-1) | 2.56e-1(1.76e-2) | 1.35e-1(3.32e-2) | 1.10e-1(1.75e-2) | |
(20,10) | 1.16e+0(2.09e-1) | 3.10e-1(1.27e-2) | 1.43e-1(5.16e-2) | 1.07e-1(1.83e-2) | |
ZJZ8 | (5,10) | 5.15e-1(1.16e-1) | 3.06e-1(1.28e-2) | 1.61e-1(2.45e-2) | 1.19e-1(2.53e-2) |
(10,10) | 5.00e-1(1.33e-1) | 2.50e-1(8.80e-3) | 1.30e-1(2.69e-2) | 1.12e-1(2.61e-2) | |
(20,10) | 5.42e-1(1.65e-1) | 3.06e-1(1.11e-2) | 1.36e-1(2.69e-2) | 1.17e-1(3.30e-2) | |
JY10 | (5,10) | 7.28e-1(4.67e-1) | 8.75e-1(5.17e-2) | 5.37e-1(2.38e-1) | 4.76e-1(2.28e-1) |
(10,10) | 1.27e+0(3.63e-1) | 7.11e-1(4.07e-2) | 3.10e-1(5.89e-2) | 2.60e-1(8.45e-2) | |
(20,10) | 1.01e+0(4.88e-1) | 6.97e-1(6.24e-2) | 2.94e-1(2.32e-1) | 3.12e-1(2.69e-1) |
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