基于多区域动态分组的大规模多目标进化算法

doi:10.16182/j.issn1004731x.joss.25-0216

摘要/Abstract

摘要：

大规模多目标优化问题的决策变量维度可达数百上千维。针对现有的基于决策变量分析的大规模多目标进化算法通常需要消耗大量的计算资源用于分组，且未考虑收敛性相关变量和多样性相关变量之间的相互作用，提出一种多区域自适应动态分组的大规模多目标进化算法。利用高斯混合模型将决策空间划分为多个区域；在每个区域内，为每个决策变量构造特征向量，并使用谱聚类算法进行分组。为验证算法的有效性，将其与不同算法在140个大规模基准测试问题上进行了实验对比。结果表明：该算法在收敛性和多样性指标上具有更好的性能。

关键词: 大规模多目标优化, 进化算法, 动态分组, 决策变量分析, 问题转换

Abstract:

The decision variable dimension of large-scale multi-objective optimization problems can reach hundreds or even thousands. For existing large-scale multi-objective evolutionary algorithms based on decision variable analysis, which usually consume a large amount of computational resources for grouping and fail to consider the interactions between convergence-related variables and diversity-related variables, a large-scale multi-objective evolutionary algorithm based on multi-region adaptive dynamic grouping was proposed. The algorithm employed a Gaussian mixture model to partition the decision space into multiple regions; within each region, feature vectors were constructed for each decision variable, and spectral clustering was utilized to perform grouping. To validate the effectiveness of the algorithm, it was compared experimentally with different algorithms on 140 large-scale benchmark test problems. The results demonstrate that the proposed algorithm exhibits better performance in terms of both convergence and diversity metrics.

Key words: large-scale multi-objective optimization, evolutionary algorithm, dynamic grouping, decision variable analysis, problem transformation

中图分类号:

TP391.9

梁斌豪,魏静萱,梁沨琴 . 基于多区域动态分组的大规模多目标进化算法[J]. 系统仿真学报, 2026, 38(4): 974-987.

Liang Binhao,Wei Jingxuan,Liang Fengqin . Large-scale Multi-objective Evolutionary Algorithm Based on Multi-region Dynamic Grouping[J]. Journal of System Simulation, 2026, 38(4): 974-987.

图/表 10

图1

图2

图3

图4

表1

对比算法和MRADG在测试问题LSMOP1~LSMOP9的二目标问题上获得的IGD值统计结果

测试问题	目标维度	CCGDE3	MOEA/DVA	LCSA	LSMOF	FLEA	LERD	MRADG
LSMOP1	100	2.577 8e+0(5.40e-1)-	4.390 6e-2(2.36e-3)-	3.362 3e-1(3.09e-2)-	2.075 1e-2(9.50e-3)-	4.877 9e-1(1.27e-1)-	6.168 8e-3(4.74e-4)+	1.636 9e-2(7.41e-3)
	500	4.445 6e+0(3.46e-1)-	4.825 0e+0(2.10e-1)-	5.719 1e-1(9.33e-2)-	2.832 8e-1(3.25e-2)-	5.846 0e-1(1.11e-1)-	5.724 4e-1(1.28e-1)-	1.900 8e-1(9.19e-3)
	1 000	5.352 1e+0(3.13e-1)-	7.644 6e+0(2.22e-1)-	5.090 5e-1(1.33e-1)-	3.962 6e-1(4.28e-2)-	6.145 7e-1(8.69e-2)-	1.211 8e+0(1.23e-1)-	2.366 8e-1(2.20e-2)
	1 500	5.538 7e+0(2.67e-1)-	8.898 0e+0(1.82e-1)-	5.778 0e-1(1.15e-1)-	4.471 4e-1(6.58e-2)-	6.201 7e-1(8.34e-2)-	1.384 3e+0(1.66e-1)-	2.745 5e-1(2.61e-2)
	2 000	5.743 8e+0(2.43e-1)-	9.637 0e+0(1.26e-1)-	4.729 3e-1(1.26e-1)-	4.518 9e-1(4.01e-2)-	5.916 2e-1(9.14e-2)-	1.490 0e+0(2.05e-1)-	3.173 0e-1(3.58e-2)
LSMOP2	100	2.136 8e-1(8.53e-3)-	2.009 1e-1(2.00e-3)-	8.301 0e-2(4.06e-3)-	2.913 4e-2(2.72e-3)-	7.625 1e-2(9.54e-3)-	3.438 9e-2(1.00e-2)-	2.195 4e-2(5.02e-3)
	500	7.193 7e-2(8.04e-4)-	7.214 0e-2(3.42e-4)-	3.325 5e-2(2.34e-4)-	1.676 8e-2(7.15e-4)-	1.986 2e-2(5.94e-4)-	3.453 9e-2(5.95e-3)-	1.572 5e-2(1.88e-3)
	1 000	3.990 1e-2(4.95e-4)-	4.032 0e-2(4.09e-4)-	1.915 7e-2(3.44e-4)-	1.161 2e-2(2.99e-4)-	1.123 6e-2(4.42e-4)-	2.159 4e-2(1.43e-3)-	1.052 4e-2(1.09e-3)
	1 500	2.802 0e-2(3.98e-4)-	2.852 9e-2(3.17e-4)-	1.455 0e-2(2.72e-4)-	9.837 7e-3(3.92e-4)-	8.967 7e-3(9.26e-4)-	1.621 9e-2(7.95e-4)-	8.299 2e-3(4.73e-4)
	2 000	2.170 1e-2(3.49e-4)-	2.302 7e-2(5.93e-4)-	1.273 4e-2(2.23e-4)-	8.727 4e-3(4.06e-4)-	8.542 2e-3(1.71e-3)-	1.286 2e-2(3.73e-4)-	7.196 5e-3(5.05e-4)
LSMOP3	100	1.272 4e+1(3.08e+0)-	3.653 2e+0(1.81e+0)-	6.488 3e-1(1.20e-3)+	1.425 8e+0(3.51e-2)-	1.511 4e+0(2.57e-3)-	6.708 6e-1(3.48e-2)+	1.101 4e+0(1.29e-1)
	500	2.357 2e+1(2.01e+0)-	7.491 6e+2(6.22e+2)-	1.028 5e+0(3.96e-2)+	1.559 4e+0(2.42e-3)+	1.565 7e+0(1.71e-3)+	4.905 9e+0(5.29e+0)-	1.571 2e+0(2.26e-3)
	1 000	2.814 3e+1(1.07e+0)-	9.563 3e+2(8.64e+2)-	1.483 9e+0(1.61e-2)+	1.569 5e+0(1.52e-3)+	1.573 5e+0(1.08e-3)+	1.161 4e+1(5.77e+0)-	1.580 0e+0(2.31e-3)
	1 500	3.001 3e+1(9.72e-1)-	1.115 4e+3(1.13e+3)-	1.586 0e+0(2.19e-4)-	1.573 6e+0(7.69e-4)+	1.576 1e+0(9.62e-5)+	1.342 5e+1(2.51e+0)-	1.582 0e+0(2.81e-3)
	2 000	3.140 3e+1(1.03e+0)-	8.853 9e+2(9.04e+2)-	1.587 8e+0(1.73e-4)-	1.575 0e+0(9.45e-4)+	1.577 0e+0(7.79e-4)+	1.372 2e+1(2.58e+0)-	1.584 6e+0(1.68e-3)
LSMOP4	100	3.075 3e-1(2.37e-2)-	1.053 8e-1(1.79e-2)-	1.617 6e-1(9.49e-3)-	2.109 5e-2(1.08e-3)≈	1.737 9e-1(1.66e-2)-	2.837 6e-2(1.93e-3)-	2.258 9e-2(2.89e-3)
	500	1.138 9e-1(2.08e-3)-	1.053 2e-1(7.37e-4)-	8.792 4e-2(1.94e-3)-	4.000 4e-2(1.55e-3)-	5.398 4e-2(2.57e-3)-	5.184 6e-2(5.26e-3)-	2.902 2e-2(2.06e-3)
	1 000	6.937 4e-2(1.05e-3)-	6.944 1e-2(4.11e-4)-	5.776 3e-2(9.57e-4)-	2.623 2e-2(7.97e-4)-	3.018 2e-2(1.51e-3)-	3.464 6e-2(3.65e-3)-	2.201 6e-2(3.35e-4)
	1 500	5.128 3e-2(1.12e-3)-	5.238 9e-2(2.40e-4)-	5.002 9e-2(2.44e-4)-	1.993 3e-2(7.25e-4)-	2.174 1e-2(1.05e-3)-	2.513 0e-2(1.97e-3)-	1.661 1e-2(2.98e-4)
	2 000	4.061 5e-2(7.30e-4)-	4.224 8e-2(2.81e-4)-	3.905 0e-2(2.78e-4)-	1.613 1e-2(5.57e-4)-	1.799 8e-2(9.52e-4)-	2.101 9e-2(1.48e-3)-	1.370 7e-2(2.80e-4)
LSMOP5	100	5.597 1e+0(8.04e-1)-	1.032 5e-1(6.73e-3)-	1.925 4e-1(6.03e-2)-	3.642 1e-1(1.71e-1)-	7.420 9e-1(1.11e-16)-	1.117 6e-1(1.33e-1)-	1.017 4e-2(1.01e-3)
	500	1.053 2e+1(6.84e-1)-	1.134 3e+1(5.43e-1)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	6.650 8e-1(2.50e-1)≈	7.420 2e-1(2.92e-4)
	1 000	1.229 8e+1(8.08e-1)-	1.680 4e+1(4.89e-1)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	3.380 3e+0(1.05e+0)-	7.420 9e-1(1.11e-16)
	1 500	1.320 4e+1(9.28e-1)-	1.907 3e+1(3.99e-1)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	4.492 0e+0(6.37e-1)-	7.420 9e-1(1.11e-16)
	2 000	1.324 4e+1(6.85e-1)-	2.057 9e+1(3.12e-1)-	7.420 9e-1(1.11e-16)---	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	5.024 3e+0(7.11e-1)-	7.420 9e-1(1.11e-16)
LSMOP6	100	1.093 6e+0(2.70e-2)-	1.039 5e+0(4.97e-2)-	5.894 2e-1(3.83e-3)-	4.364 9e-1(1.04e-2)≈	4.045 0e-1(2.08e-1)≈	5.393 1e-1(8.20e-2)-	4.119 8e-1(9.58e-2)
	500	8.132 2e-1(2.45e-3)-	1.124 6e+3(8.92e+2)-	6.792 2e-1(7.20e-4)-	3.344 5e-1(1.25e-2)+	2.473 8e-1(1.18e-1)+	7.301 2e-1(6.42e-2)-	6.763 9e-1(4.25e-3)
	1 000	7.752 8e-1(4.21e-4)-	1.540 0e+3(1.11e+3)-	6.764 7e-1(2.41e-4)≈	3.234 5e-1(1.33e-2)+	3.005 5e-1(1.19e-1)+	7.083 1e-1(7.23e-2)-	6.747 4e-1(4.97e-3)
	1 500	7.630 6e-1(1.23e-4)-	2.056 4e+3(1.74e+3)-	6.752 0e-1(1.71e-4)-	3.162 0e-1(6.78e-3)+	3.257 1e-1(9.76e-2)+	7.200 4e-1(4.50e-2)-	6.745 3e-1(2.40e-3)
	2 000	7.573 3e-1(7.71e-5)-	1.856 9e+3(1.96e+3)-	6.744 6e-1(1.09e-4)≈	3.116 6e-1(1.56e-2)+	3.315 1e-1(1.22e-1)+	6.863 9e-1(9.63e-2)-	6.753 3e-1(3.54e-3)
LSMOP7	100	6.491 8e+3(1.93e+3)-	2.085 7e+1(7.35e+0)-	1.163 5e+0(2.03e-1)+	1.411 6e+0(1.78e-2)≈	1.459 5e+0(5.99e-4)-	1.446 3e+0(8.72e-1)-	1.376 6e+0(1.15e-1)
	500	2.124 8e+4(4.82e+3)-	2.804 1e+4(2.08e+3)-	1.505 9e+0(8.65e-4)+	1.501 3e+0(5.04e-4)+	1.602 3e+0(4.19e-1)-	3.037 3e+0(4.60e-1)-	1.508 1e+0(1.81e-3)
	1 000	2.659 0e+4(5.12e+3)-	5.300 4e+4(2.90e+3)-	1.516 2e+0(3.61e-4)-	1.509 9e+0(3.76e-4)+	1.511 6e+0(1.14e-4)+	4.010 4e+0(4.88e-1)-	1.514 8e+0(9.40e-4)
	1 500	2.853 9e+4(4.58e+3)-	6.568 5e+4(3.04e+3)-	1.519 5e+0(2.34e-4)-	1.512 4e+0(2.85e-4)+	1.791 7e+0(6.62e-1)-	4.741 0e+0(5.64e-1)-	1.517 9e+0(2.52e-3)
	2 000	2.974 5e+4(4.77e+3)-	7.195 7e+4(2.37e+3)-	1.521 3e+0(1.38e-4)-	1.513 8e+0(2.00e-4)+	1.514 3e+0(1.40e-4)+	6.914 1e+0(7.98e-1)-	1.518 4e+0(9.06e-4)
LSMOP8	100	5.391 1e+0(1.19e+0)-	8.463 2e-2(3.83e-3)-	2.892 3e-1(1.16e-1)-	1.968 1e-1(1.83e-1)-	7.420 9e-1(1.11e-16)-	4.965 9e-2(1.15e-2)≈	5.119 6e-2(1.39e-2)
	500	7.852 2e+0(4.60e-1)-	9.950 7e+0(3.29e-1)-	7.112 7e-1(6.82e-2)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	1.824 3e-1(3.53e-2)+	3.005 2e-1(3.55e-2)
	1 000	9.153 6e+0(6.59e-1)-	1.434 3e+1(3.98e-1)-	7.414 4e-1(1.94e-3)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	9.295 6e-1(2.02e-1)-	7.008 3e-1(4.48e-2)
	1 500	9.944 2e+0(5.79e-1)-	1.639 5e+1(3.79e-1)-	7.420 9e-1(1.11e-16)+	7.420 9e-1(1.11e-16)+	7.420 9e-1(1.11e-16)+	1.735 5e+0(6.65e-1)-	7.436 0e-1(2.46e-3)
	2 000	1.012 2e+1(5.59e-1)-	1.750 2e+1(2.61e-1)-	7.420 9e-1(1.11e-16)-	7.420 9e-1(1.11e-16)-	7.421 3e-1(1.79e-4)-	2.419 5e+0(4.61e-1)-	7.423 8e-1(6.49e-4)
LSMOP9	100	1.006 2e+1(4.35e+0)-	3.191 0e-1(2.88e-2)+	8.100 4e-1(1.11e-16)-	8.100 4e-1(1.11e-16)-	8.836 3e-1(2.22e-1)-	1.124 4e+0(1.37e+0)-	8.099 9e-1(0.00e+0)
	500	2.338 7e+1(4.52e+0)-	2.246 9e+1(1.39e+0)-	8.100 4e-1(1.11e-16)-	8.019 8e-1(1.98e-2)-	7.721 7e-1(1.62e-1)-	8.997 9e-1(2.50e-1)-	8.013 2e-1(4.60e-3)
	1 000	2.788 2e+1(3.71e+0)-	3.914 5e+1(2.21e+0)-	8.095 6e-1(2.41e-4)-	8.003 1e-1(7.13e-3)-	7.915 9e-1(5.15e-2)-	8.628 8e-1(2.35e-2)-	7.781 1e-1(3.79e-2)
	1 500	3.095 6e+1(3.81e+0)-	4.723 8e+1(1.32e+0)-	8.092 0e-1(4.21e-4)-	7.983 4e-1(1.50e-2)-	7.665 3e-1(8.64e-2)-	9.388 3e-1(4.78e-2)-	7.635 4e-1(4.21e-2)
	2 000	3.269 0e+1(3.42e+0)-	5.112 6e+1(1.31e+0)-	8.089 1e-1(4.74e-4)-	8.007 8e-1(5.67e-3)-	7.743 1e-1(6.01e-2)-	1.615 6e+0(1.48e+0)-	7.781 9e-1(3.35e-2)
+/-/≈		0/45/0	1/44/0	6/37/2	13/29/3	11/33/1	3/40/2

表1

图5

表2

对比算法和MRADG在测试问题LSMOP1~LSMOP9的三目标问题上获得的IGD值统计结果

测试问题	目标维度	CCGDE3	MOEA/DVA	LCSA	LSMOF	FLEA	LERD	MRADG
LSMOP1	100	5.125 2e+0(1.57e+0)-	8.016 3e-2(4.27e-3)+	2.262 5e-1(7.15e-3)+	3.884 5e-1(5.39e-2)-	6.306 3e-1(1.82e-1)-	2.588 9e-1(5.82e-2)+	2.914 8e-1(1.86e-2)
	500	8.736 7e+0(1.14e+0)-	4.846 0e+0(3.67e-1)-	4.078 8e-1(7.13e-2)+	6.730 0e-1(5.12e-2)-	6.898 1e-1(1.44e-1)-	9.639 5e-1(2.88e-1)-	5.465 3e-1(4.70e-2)
	1 000	9.145 0e+0(9.21e-1)-	7.703 9e+0(2.28e-1)-	5.944 3e-1(3.97e-2)-	7.140 8e-1(1.25e-2)-	7.511 3e-1(1.38e-1)-	1.418 0e+0(2.17e-1)-	5.798 3e-1(5.01e-2)
	1 500	9.386 0e+0(1.25e+0)-	9.027 8e+0(2.83e-1)-	6.490 9e-1(4.56e-2)-	7.246 7e-1(1.19e-2)-	7.272 4e-1(1.65e-1)-	1.420 4e+0(2.24e-1)-	6.175 0e-1(4.98e-2)
	2 000	9.584 8e+0(1.08e+0)-	9.807 0e+0(2.59e-1)-	7.079 4e-1(3.27e-2)-	7.297 2e-1(1.27e-2)-	8.399 9e-1(1.71e-1)-	1.452 4e+0(2.18e-1)-	6.440 9e-1(4.72e-2)
LSMOP2	100	2.251 1e-1(5.89e-3)-	1.483 1e-1(3.94e-3)+	2.229 6e-1(1.37e-2)-	2.001 3e-1(3.98e-3)-	1.697 6e-1(1.72e-2)≈	1.720 6e-1(5.63e-3)≈	1.683 1e-1(1.14e-2)
	500	8.216 2e-2(5.08e-3)-	7.887 6e-2(2.65e-3)-	8.834 0e-2(2.96e-3)-	7.508 6e-2(1.94e-3)-	8.288 5e-2(4.80e-3)-	6.929 1e-2(1.51e-3)≈	6.948 5e-2(5.06e-3)
	1 000	6.415 8e-2(4.74e-3)-	6.410 7e-2(2.45e-3)-	7.248 0e-2(4.19e-3)-	6.082 1e-2(1.72e-3)≈	6.967 5e-2(3.78e-3)-	5.388 6e-2(3.07e-4)+	6.095 3e-2(2.61e-3)
	1 500	5.961 2e-2(4.13e-3)≈	6.129 6e-2(3.03e-3)≈	6.744 9e-2(3.35e-3)-	5.675 8e-2(1.72e-3)+	6.745 1e-2(3.41e-3)-	4.901 8e-2(2.17e-4)+	6.184 4e-2(4.85e-3)
	2 000	5.612 2e-2(3.10e-3)≈	5.955 7e-2(4.34e-3)≈	6.483 8e-2(4.30e-3)-	5.506 1e-2(1.44e-3)+	6.477 5e-2(3.23e-3)-	4.700 2e-2(2.11e-4)+	5.842 0e-2(3.64e-3)
LSMOP3	100	1.401 3e+1(2.23e+0)-	1.596 4e+0(2.51e-1)-	4.889 0e-1(1.31e-1)+	8.369 5e-1(3.11e-2)≈	8.593 6e-1(5.90e-3)-	1.267 6e+0(9.59e-1)≈	8.299 9e-1(4.10e-2)
	500	2.101 9e+1(2.96e+0)-	1.690 7e+2(1.19e+2)-	7.587 4e-1(2.95e-2)+	8.607 1e-1(2.21e-5)≈	1.113 1e+0(7.71e-1)≈	9.599 1e+0(3.94e-1)-	8.521 7e-1(1.89e-2)
	1 000	2.036 6e+1(2.93e+0)-	2.510 3e+2(1.84e+2)-	8.416 8e-1(4.73e-3)+	8.607 1e-1(2.88e-5)+	1.597 2e+0(1.30e+0)≈	9.772 7e+0(3.98e-1)-	8.882 4e-1(1.25e-1)
	1 500	2.161 3e+1(3.46e+0)-	2.396 1e+2(2.01e+2)-	8.607 2e-1(9.29e-6)+	8.607 2e-1(0.00e+0)+	2.304 7e+0(1.77e+0)≈	9.975 4e+0(3.72e-1)-	8.986 4e-1(1.03e-1)
	2 000	2.210 7e+1(4.70e+0)-	3.213 2e+2(2.59e+2)-	8.607 2e-1(0.00e+0)+	8.607 1e-1(3.27e-5)+	1.785 2e+0(1.62e+0)≈	9.979 6e+0(2.64e-1)-	8.966 9e-1(1.20e-1)
LSMOP4	100	5.576 6e-1(1.14e-2)-	1.967 4e-1(1.65e-2)+	3.135 9e-1(1.51e-2)+	3.805 7e-1(9.59e-3)-	4.165 1e-1(2.45e-2)-	2.276 0e-1(7.81e-3)+	3.430 0e-1(1.83e-2)
	500	2.121 1e-1(7.37e-3)-	1.777 7e-1(5.48e-3)-	2.258 9e-1(5.81e-3)-	1.800 4e-1(5.69e-3)-	1.516 7e-1(6.20e-3)-	1.543 2e-1(7.29e-3)-	1.322 8e-1(3.45e-3)
	1 000	1.302 9e-1(5.01e-3)-	1.233 3e-1(2.26e-3)-	1.454 1e-1(3.58e-3)-	1.156 2e-1(2.89e-3)-	1.092 1e-1(5.39e-3)-	1.059 3e-1(3.45e-3)-	8.780 3e-2(2.40e-3)
	1 500	1.021 5e-1(5.32e-3)-	9.898 2e-2(1.78e-3)-	1.143 0e-1(3.83e-3)-	8.995 2e-2(3.14e-3)-	9.197 3e-2(4.99e-3)-	8.344 1e-2(2.30e-3)-	7.352 8e-2(2.91e-3)
	2 000	8.610 8e-2(5.98e-3)-	8.707 9e-2(4.03e-3)-	9.777 3e-2(4.08e-3)-	7.834 3e-2(2.00e-3)-	8.145 9e-2(5.08e-3)-	7.226 7e-2(1.27e-3)-	6.749 0e-2(3.85e-3)
LSMOP5	100	7.788 8e+0(2.29e+0)-	1.699 9e-1(1.01e-2)-	4.223 5e-1(5.98e-2)-	3.459 1e-1(2.45e-2)-	9.515 9e-1(2.74e-1)-	3.583 5e-1(9.82e-4)-	1.593 5e-1(5.00e-2)
	500	1.456 2e+1(2.18e+0)-	9.368 4e+0(4.67e-1)-	5.409 6e-1(3.76e-5)-	5.499 5e-1(1.22e-1)-	1.134 4e+0(3.66e-1)-	1.004 8e+0(8.06e-1)-	4.696 7e-1(2.29e-2)
	1 000	1.672 0e+1(2.15e+0)-	1.373 7e+1(6.32e-1)-	5.410 8e-1(3.88e-5)-	5.731 0e-1(1.19e-1)-	1.115 1e+0(3.17e-1)-	2.023 5e+0(1.07e+0)-	5.244 2e-1(1.53e-2)
	1 500	1.679 3e+1(2.21e+0)-	1.595 4e+1(6.16e-1)-	5.410 7e-1(3.32e-5)-	5.791 3e-1(1.22e-1)-	1.075 9e+0(3.55e-1)-	2.355 1e+0(9.02e-1)-	5.386 8e-1(4.08e-3)
	2 000	1.784 9e+1(1.98e+0)-	1.713 1e+1(4.46e-1)-	5.410 6e-1(3.77e-5)-	5.797 3e-1(1.22e-1)-	9.024 9e-1(3.26e-1)-	3.157 7e+0(3.92e-1)-	5.392 6e-1(4.68e-3)
LSMOP6	100	1.640 8e+3(1.33e+3)-	7.994 4e+0(2.41e+0)-	9.865 0e-1(5.99e-3)≈	7.914 9e-1(1.26e-1)+	4.965 5e+1(1.90e+2)≈	1.046 5e+0(4.98e-1)+	1.030 1e+0(1.46e-1)
	500	2.173 5e+4(5.90e+3)-	1.311 5e+4(2.29e+3)-	1.280 8e+0(1.17e-3)+	1.241 7e+0(1.03e-1)≈	1.243 7e+0(2.65e-1)+	4.172 4e+0(4.16e+0)-	1.292 9e+0(2.04e-3)
	1 000	2.696 8e+4(8.19e+3)-	2.393 4e+4(4.21e+3)-	1.310 9e+0(7.78e-4)+	1.244 5e+0(1.39e-1)≈	1.278 2e+0(2.44e-1)+	5.631 5e+1(1.13e+2)-	1.313 8e+0(2.15e-3)
	1 500	2.604 6e+4(5.51e+3)-	2.839 5e+4(3.84e+3)-	1.319 9e+0(6.45e-4)≈	1.221 5e+0(1.70e-1)≈	2.624 4e+1(7.88e+1)≈	4.695 6e+1(3.41e+1)-	1.320 7e+0(3.86e-3)
	2 000	2.813 2e+4(5.85e+3)-	3.414 9e+4(5.29e+3)-	1.324 2e+0(2.40e-4)-	1.122 6e+0(2.36e-1)+	9.742 5e+0(3.68e+1)+	1.351 4e+2(1.24e+2)-	1.323 4e+0(1.55e-3)
LSMOP7	100	2.903 2e+0(3.05e-1)-	3.398 6e+0(3.29e+0)-	1.007 2e+0(3.80e-2)-	9.117 9e-1(6.14e-2)-	1.731 1e+0(4.33e-1)-	8.568 2e-1(8.43e-2)-	8.340 2e-1(3.64e-2)
	500	1.306 0e+0(1.48e-2)-	1.041 0e+3(8.05e+2)-	9.014 2e-1(5.87e-3)-	8.225 0e-1(6.74e-2)≈	9.443 6e-1(1.39e-1)≈	9.670 7e-1(1.01e-1)-	8.552 4e-1(2.60e-2)
	1 000	1.104 9e+0(4.74e-3)-	1.403 7e+3(9.57e+2)-	8.680 8e-1(1.11e-3)-	8.162 0e-1(1.05e-1)-	9.418 7e-1(9.05e-2)-	9.744 0e-1(7.76e-2)-	8.096 4e-1(5.76e-2)
	1 500	1.044 7e+0(2.41e-3)-	1.478 8e+3(9.75e+2)-	8.565 6e-1(5.68e-4)-	8.040 3e-1(9.58e-2)≈	9.659 6e-1(1.59e-2)-	9.848 4e-1(5.18e-2)-	8.259 3e-1(3.47e-2)
	2 000	1.017 5e+0(1.56e-3)-	2.068 6e+3(1.60e+3)-	8.511 0e-1(5.08e-4)-	7.863 6e-1(9.85e-2)+	9.731 2e-1(2.09e-2)-	9.799 8e-1(2.77e-2)-	8.216 8e-1(3.70e-2)
LSMOP8	100	9.793 7e-1(1.02e-1)-	1.378 6e-1(6.51e-3)+	3.546 7e-1(1.87e-2)-	3.060 1e-1(4.04e-2)-	4.030 5e-1(1.43e-1)-	3.498 1e-1(3.13e-2)-	2.260 8e-1(4.34e-2)
	500	9.612 5e-1(2.34e-2)-	6.585 0e-1(5.57e-2)-	3.581 1e-1(8.29e-3)-	2.890 7e-1(5.67e-2)≈	5.484 7e-1(3.27e-2)-	3.009 9e-1(5.46e-2)≈	2.793 6e-1(4.18e-2)
	1 000	8.973 9e-1(9.10e-2)-	6.796 7e-1(6.06e-2)-	3.331 3e-1(2.22e-2)-	2.513 6e-1(3.05e-2)-	5.583 3e-1(1.00e-1)-	2.463 1e-1(3.59e-2)-	2.355 5e-1(1.51e-2)
	1 500	9.276 4e-1(6.40e-2)-	6.950 0e-1(6.94e-2)-	3.331 1e-1(2.71e-2)-	2.466 2e-1(3.45e-2)-	7.144 6e-1(1.66e-1)-	2.741 8e-1(1.03e-1)-	2.314 4e-1(5.86e-3)
	2 000	9.077 5e-1(7.43e-2)-	6.868 8e-1(5.56e-2)-	3.108 4e-1(2.98e-2)-	2.338 1e-1(4.09e-3)-	8.223 4e-1(1.54e-1)-	4.139 5e-1(1.51e-1)-	2.302 3e-1(7.54e-3)
LSMOP9	100	3.045 3e+1(9.57e+0)-	6.146 8e-1(5.08e-2)≈	1.443 0e+0(2.85e-1)-	1.479 1e+0(1.40e-1)-	1.557 5e+0(8.53e-2)-	1.481 5e+0(1.46e+0)-	8.610 5e-1(2.75e-1)
	500	6.125 5e+1(7.57e+0)-	5.487 8e+1(2.92e+0)-	1.537 9e+0(2.22e-16)-	1.498 6e+0(1.18e-1)-	1.303 6e+0(3.58e-1)-	1.378 2e+0(1.72e-1)-	1.081 4e+0(1.64e-1)
	1 000	7.260 4e+1(8.20e+0)-	9.518 9e+1(4.50e+0)-	1.537 9e+0(2.22e-16)-	1.184 6e+0(1.18e-1)-	1.474 6e+0(2.09e-1)-	1.636 5e+0(1.42e-1)-	1.080 9e+0(1.65e-1)
	1 500	7.787 9e+1(9.90e+0)-	1.154 1e+2(2.24e+0)-	1.490 4e+0(2.07e-1)-	1.282 5e+0(1.87e-1)-	1.205 0e+0(4.19e-1)-	2.418 1e+0(1.38e+0)-	1.082 1e+0(1.64e-1)
	2 000	8.081 4e+1(8.52e+0)-	1.245 9e+2(3.25e+0)-	1.537 9e+0(2.22e-16)-	1.203 9e+0(1.40e-1)-	1.338 6e+0(3.68e-1)-	3.472 8e+0(1.75e+0)-	1.084 2e+0(1.66e-1)
+/-/≈		0/43/2	4/38/3	10/33/2	8/28/9	3/34/8	6/35/4

表2

图6

表3

对比算法和MRADG在测试问题IMF1~IMF10的三目标问题上获得的IGD值统计结果

测试问题	目标维度	CCGDE3	MOEADVA	LCSA	LSMOF	FLEA	LERD	MRADG
IMF1	100	3.491 1e+0(8.21e-1)-	6.291 2e-2(5.99e-3)-	2.564 9e-1(7.69e-2)-	4.621 5e-1(1.46e-2)-	3.205 1e-1(1.07e-1)-	5.622 9e-3(2.08e-4)+	8.020 2e-3(8.16e-4)
	500	6.126 8e+0(5.54e-1)-	6.384 5e+0(4.71e-1)-	5.296 6e-1(4.98e-3)-	5.368 5e-1(5.40e-3)-	3.829 8e-1(1.01e-1)-	3.912 8e-1(1.75e-2)-	2.165 8e-1(2.09e-2)
	1 000	6.863 5e+0(5.60e-1)-	1.021 5e+1(4.37e-1)-	5.706 3e-1(1.91e-3)-	5.505 0e-1(8.44e-3)-	4.995 3e-1(1.57e-1)-	4.722 1e-1(1.05e-2)-	3.282 4e-1(3.09e-2)
	1 500	7.088 9e+0(5.90e-1)-	1.217 3e+1(2.83e-1)-	5.815 1e-1(1.69e-3)-	5.560 9e-1(5.96e-3)-	3.871 6e-1(1.23e-1)≈	5.399 5e-1(7.97e-2)-	3.878 3e-1(2.18e-2)
	2 000	7.558 9e+0(5.72e-1)-	1.324 6e+1(4.10e-1)-	5.861 4e-1(1.06e-3)-	5.579 1e-1(4.37e-3)-	4.171 0e-1(1.49e-1)-	6.525 5e-1(1.31e-1)-	4.124 7e-1(2.36e-2)
IMF2	100	5.419 4e+0(1.02e+0)-	9.244 3e-2(1.03e-2)+	6.094 9e-1(0.00e+0)+	6.094 9e-1(0.00e+0)+	6.094 9e-1(0.00e+0)+	6.753 1e-1(2.87e-1)+	6.094 9e-1(0.00e+0)
	500	8.554 4e+0(6.90e-1)-	8.761 2e+0(4.30e-1)-	6.094 9e-1(0.00e+0)≈	6.094 9e-1(0.00e+0)+	6.094 9e-1(0.00e+0)+	6.056 4e-1(3.71e-3)+	6.094 9e-1(2.17e-8)
	1 000	9.782 5e+0(7.25e-1)-	1.342 5e+1(5.09e-1)-	6.094 9e-1(0.00e+0)+	6.094 9e-1(0.00e+0)+	6.094 9e-1(2.30e-5)+	5.807 5e-1(7.69e-3)+	6.094 9e-1(1.54e-8)
	1 500	1.003 3e+1(6.47e-1)-	1.578 1e+1(4.54e-1)-	6.094 9e-1(0.00e+0)+	6.094 9e-1(0.00e+0)+	6.102 5e-1(1.73e-3)+	5.949 1e-1(1.73e-2)+	6.099 1e-1(1.29e-3)
	2 000	1.067 0e+1(7.11e-1)-	1.707 2e+1(4.83e-1)-	6.094 9e-1(0.00e+0)+	6.094 9e-1(0.00e+0)+	6.152 7e-1(1.76e-2)≈	6.938 9e-1(1.31e-1)-	6.100 4e-1(1.42e-3)
IMF3	100	5.558 0e+0(1.01e+0)-	1.723 6e-1(2.38e-2)-	4.911 3e-3(1.58e-4)+	6.745 5e-3(6.72e-4)-	2.731 6e-2(4.44e-2)-	4.039 5e-3(2.31e-4)+	5.965 5e-3(9.25e-4)
	500	8.439 4e+0(7.72e-1)-	8.915 7e+0(5.24e-1)-	1.071 3e-2(3.35e-4)-	1.000 5e-2(7.34e-4)-	1.773 0e-2(1.88e-2)-	1.739 1e-2(5.15e-3)-	8.469 8e-3(1.02e-3)
	1 000	9.500 6e+0(6.22e-1)-	1.373 1e+1(5.56e-1)-	1.746 2e-2(3.04e-4)-	1.199 3e-2(7.30e-4)-	3.469 3e-2(4.12e-2)-	5.378 5e-2(1.87e-2)-	9.533 9e-3(1.01e-3)
	1 500	1.028 5e+1(7.58e-1)-	1.582 4e+1(4.83e-1)-	2.054 2e-2(2.50e-4)-	1.293 5e-2(1.24e-3)-	2.114 6e-2(2.84e-2)-	1.144 5e-1(3.20e-2)-	9.792 0e-3(6.45e-4)
	2 000	1.087 6e+1(7.77e-1)-	1.713 3e+1(3.68e-1)-	2.204 6e-2(2.29e-4)-	1.370 0e-2(1.88e-3)-	2.958 8e-2(2.29e-2)-	3.231 3e-1(2.22e-1)-	1.022 3e-2(7.62e-4)
IMF4	100	5.895 2e+1(9.98e+0)-	5.644 1e-1(9.07e-2)-	5.375 1e-1(1.52e-2)-	5.417 9e-1(1.38e-3)-	8.712 7e-1(1.50e-1)-	8.788 2e-2(2.22e-2)+	1.949 5e-1(4.41e-2)
	500	5.201 5e+2(4.80e+1)-	4.847 3e+2(2.50e+1)-	5.409 1e-1(2.22e-5)+	5.617 2e-1(8.81e-2)-	9.273 7e-1(8.09e-2)-	5.594 5e-1(7.39e-2)-	5.412 4e-1(3.65e-4)
	1 000	1.192 9e+3(9.52e+1)-	1.464 9e+3(4.03e+1)-	5.409 2e-1(3.06e-5)+	5.816 8e-1(1.21e-1)≈	9.257 6e-1(8.79e-2)-	4.140 0e+1(2.80e+1)-	5.414 6e-1(5.78e-4)
	1 500	1.875 1e+3(1.59e+2)-	2.607 7e+3(6.92e+1)-	5.409 2e-1(1.16e-5)+	6.299 0e-1(1.61e-1)≈	2.424 7e+0(3.50e+0)-	1.251 7e+2(6.61e+1)-	5.415 1e-1(4.20e-4)
	2 000	2.712 1e+3(2.10e+2)-	3.726 1e+3(7.76e+1)-	5.409 1e-1(1.60e-5)+	6.171 1e-1(1.51e-1)≈	3.249 5e+1(5.62e+1)-	2.646 8e+2(1.58e+2)-	5.414 6e-1(6.96e-4)
IMF5	100	1.821 8e-1(1.20e-2)-	2.094 0e-2(1.14e-3)-	3.255 3e-2(2.05e-3)-	2.592 3e-2(2.50e-3)-	1.282 1e-1(1.45e-2)-	5.728 8e-3(2.24e-4)+	1.046 1e-2(4.51e-4)
	500	2.520 4e-1(1.08e-2)-	2.011 3e-1(2.15e-3)-	7.938 5e-2(2.70e-3)-	4.488 0e-2(7.58e-4)-	1.421 6e-1(4.17e-3)-	4.236 0e-2(7.87e-3)-	3.119 6e-2(1.19e-3)
	1 000	2.697 0e-1(1.15e-2)-	2.632 7e-1(3.21e-3)-	1.037 5e-1(4.12e-3)-	4.621 4e-2(5.16e-4)-	1.408 6e-1(5.20e-3)-	7.712 0e-2(1.34e-2)-	4.026 4e-2(9.37e-4)
	1 500	2.796 1e-1(1.45e-2)-	2.914 5e-1(2.61e-3)-	1.144 3e-1(2.28e-3)-	4.668 5e-2(5.15e-4)-	1.293 1e-1(2.01e-2)-	9.177 7e-2(1.42e-2)-	4.369 9e-2(5.77e-4)
	2 000	2.745 0e-1(1.26e-2)-	3.079 9e-1(3.20e-3)-	1.251 4e-1(9.78e-3)-	4.672 0e-2(3.17e-4)-	1.330 6e-1(1.64e-2)-	8.641 1e-2(9.92e-3)-	4.486 3e-2(4.64e-4)
IMF6	100	2.992 8e-1(3.08e-2)-	3.654 6e-2(1.87e-3)-	5.585 9e-2(3.54e-3)-	6.347 2e-2(6.35e-3)-	1.188 3e-1(4.64e-3)-	4.221 0e-2(7.36e-3)-	2.585 5e-2(3.91e-3)
	500	4.285 1e-1(2.49e-2)-	3.321 5e-1(4.57e-3)-	1.479 2e-1(1.06e-2)-	9.357 6e-2(6.10e-3)-	1.263 4e-1(3.98e-3)-	1.362 1e-1(3.60e-2)-	6.719 5e-2(3.25e-3)
	1 000	4.509 7e-1(1.77e-2)-	4.492 3e-1(6.46e-3)-	1.521 0e-1(2.37e-2)-	9.700 1e-2(6.32e-3)-	1.268 7e-1(3.83e-3)-	1.786 0e-1(4.46e-2)-	8.217 3e-2(1.42e-3)
	1 500	4.612 0e-1(2.06e-2)-	5.093 7e-1(7.11e-3)-	1.429 9e-1(2.43e-2)-	9.882 6e-2(7.26e-3)-	1.259 7e-1(4.38e-3)-	1.786 2e-1(4.47e-2)-	8.732 0e-2(1.18e-3)
	2 000	4.735 6e-1(2.96e-2)-	5.424 5e-1(6.34e-3)-	1.590 0e-1(2.17e-2)-	1.005 5e-1(6.71e-3)-	1.241 9e-1(4.05e-3)-	2.002 8e-1(3.44e-2)-	9.046 4e-2(8.83e-4)
IMF7	100	2.890 1e-1(7.84e-3)-	9.676 9e-2(6.75e-3)-	5.095 3e-3(2.01e-4)+	8.315 7e-3(7.73e-4)-	2.372 4e-2(1.71e-2)-	4.870 0e-3(3.02e-4)+	7.284 2e-3(9.84e-4)
	500	3.463 7e-1(8.72e-3)-	3.420 2e-1(5.04e-3)-	9.601 7e-3(1.36e-3)+	1.648 9e-2(1.13e-3)-	4.028 8e-2(3.17e-2)-	1.655 4e-2(1.25e-2)+	1.499 7e-2(1.23e-3)
	1 000	3.691 2e-1(1.93e-2)-	4.050 2e-1(3.90e-3)-	1.434 6e-2(1.19e-3)+	2.139 8e-2(1.61e-3)-	4.653 8e-2(4.18e-2)-	2.150 1e-1(5.10e-2)-	1.758 0e-2(1.00e-3)
	1 500	3.766 1e-1(1.82e-2)-	4.432 8e-1(4.99e-3)-	1.730 1e-2(1.33e-3)+	2.244 9e-2(2.01e-3)-	3.762 0e-2(2.70e-2)-	2.290 0e-1(3.25e-2)-	1.921 0e-2(1.35e-3)
	2 000	3.709 4e-1(1.19e-2)-	4.677 6e-1(4.19e-3)-	1.875 6e-2(1.11e-3)≈	2.385 2e-2(1.92e-3)-	5.650 5e-2(3.02e-2)-	2.483 8e-1(3.42e-2)-	1.966 6e-2(1.79e-3)
IMF8	100	2.551 6e+0(5.39e-1)-	2.896 7e-1(1.44e-2)+	3.571 7e-1(1.00e-4)-	3.561 7e-1(4.96e-4)+	9.325 0e-1(5.85e-2)-	2.503 3e-1(9.74e-2)≈	3.566 7e-1(2.60e-4)
	500	1.751 5e+1(3.31e+0)-	1.085 1e+1(5.75e-1)-	3.572 1e-1(1.87e-5)-	3.571 8e-1(4.93e-5)-	8.519 8e-1(1.28e-1)-	5.204 5e-1(1.72e-1)-	3.571 7e-1(3.30e-5)
	1 000	3.852 3e+1(6.87e+0)-	3.513 5e+1(1.10e+0)-	3.572 1e-1(1.77e-5)-	3.571 9e-1(3.65e-5)-	8.519 8e-1(1.28e-1)-	8.238 1e-1(2.93e-1)-	3.571 6e-1(1.52e-4)
	1 500	6.018 9e+1(1.01e+1)-	6.195 1e+1(1.37e+0)-	3.572 0e-1(2.14e-5)≈	3.572 0e-1(2.19e-5)≈	8.251 3e-1(1.34e-1)-	1.790 3e+0(7.61e-1)-	3.572 0e-1(2.51e-5)
	2 000	8.084 3e+1(1.56e+1)-	8.966 2e+1(1.76e+0)-	3.572 0e-1(2.02e-5)≈	3.572 2e-1(1.68e-5)-	7.848 7e-1(1.31e-1)-	3.827 7e+0(1.42e+0)-	3.572 0e-1(1.97e-5)
IMF9	100	4.437 0e-1(3.51e-2)-	1.516 7e-1(4.72e-2)-	1.588 1e-2(3.53e-4)-	1.386 2e-2(1.46e-3)-	6.213 3e-2(9.70e-2)-	6.662 1e-3(2.63e-4)-	5.533 6e-3(3.47e-4)
	500	6.136 7e-1(9.19e-3)-	5.930 3e-1(7.22e-3)-	7.852 7e-2(1.37e-2)-	4.841 2e-2(4.89e-3)-	7.870 1e-2(1.15e-1)-	4.937 2e-2(4.74e-3)-	6.280 4e-3(3.33e-4)
	1 000	7.227 8e-1(1.19e-2)-	7.845 6e-1(6.63e-3)-	2.120 8e-1(7.44e-2)-	8.159 5e-2(1.28e-2)-	1.035 4e-1(1.03e-1)-	1.270 8e-1(2.34e-2)-	6.482 0e-3(2.27e-4)
	1 500	8.271 7e-1(1.18e-2)-	9.900 6e-1(1.30e-2)-	4.060 8e-1(4.90e-3)-	1.492 0e-1(4.27e-2)-	1.372 7e-1(1.19e-1)-	1.985 4e-1(4.03e-2)-	6.883 8e-3(2.60e-4)
	2 000	9.477 4e-1(1.53e-2)-	1.188 9e+0(2.44e-2)-	4.296 8e-1(8.48e-3)-	2.324 8e-1(7.47e-2)-	2.244 2e-1(2.17e-1)-	2.555 6e-1(4.42e-2)-	7.607 9e-3(2.76e-4)
IMF10	100	5.275 6e+2(7.81e+1)-	9.967 0e+1(4.67e+0)-	8.254 5e-1(4.21e-5)+	8.254 9e-1(2.21e-5)+	1.412 1e+2(1.82e+2)≈	2.436 4e+2(3.75e+2)≈	4.589 8e+1(1.65e+1)
	500	4.124 4e+3(2.06e+2)-	3.390 7e+3(3.50e+1)-	8.347 4e-1(4.08e-3)+	8.373 1e-1(4.08e-3)+	2.476 8e+2(6.19e+2)+	1.426 9e+3(6.85e+2)-	2.335 5e+2(5.61e+1)
	1 000	9.224 3e+3(4.69e+2)-	9.280 3e+3(1.12e+2)-	8.379 3e-1(3.23e-3)+	8.389 5e-1(2.71e-3)+	9.241 6e+2(1.28e+3)≈	1.966 5e+3(1.15e+3)-	4.291 9e+2(1.45e+2)
	1 500	1.427 9e+4(6.13e+2)-	1.547 0e+4(1.40e+2)-	8.391 2e-1(2.36e-3)+	8.403 0e-1(1.11e-16)+	6.784 1e+2(1.16e+3)+	2.495 1e+3(1.11e+3)-	6.970 1e+2(4.18e+1)
	2 000	1.986 8e+4(5.96e+2)-	2.187 1e+4(1.91e+2)-	8.395 0e-1(1.92e-3)+	8.403 0e-1(1.11e-16)+	1.123 0e+3(3.46e+3)+	3.026 2e+3(9.84e+2)-	9.186 9e+2(4.61e+1)
+/-/≈		0/50/0	2/48/0	18/28/4	11/35/4	7/39/4	10/38/2

表3

图7

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