Multi-objective Evolutionary Optimization With Objective Space Partition Based on Online Perception of Pareto Front
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摘要: 多目标进化优化是求解多目标优化问题的可行方法.但是, 由于没有准确感知并充分利用问题的Pareto前沿, 已有方法难以高效求解复杂的多目标优化问题.本文提出一种基于在线感知Pareto前沿划分目标空间的多目标进化优化方法, 以利用感知的结果, 采用有针对性的进化优化方法求解多目标优化问题.首先, 根据个体之间的拥挤距离与给定阈值的关系感知优化问题的Pareto前沿上的间断点, 并基于此将目标空间划分为若干子空间; 然后, 在每一子空间中采用MOEA/D (Multi-objective evolutionary algorithm based on decomposition)得到一个外部保存集; 最后, 基于所有外部保存集生成问题的Pareto解集.将提出的方法应用于15个基准数值函数优化问题, 并与NSGA-Ⅱ、RPEA、MOEA/D、MOEA/DPBI、MOEA/D-STM和MOEA/D-ACD等比较.结果表明, 提出的方法能够产生收敛和分布性更优的Pareto解集, 是一种非常有竞争力的方法.Abstract: Multi-objective evolutionary optimization is a feasible way to solve multi-objective optimization problems. However, previous methods have di–culties in e–ciently tackling a complex multi-objective optimization problem, since they cannot accurately perceive the shape of the Pareto front and take full advantages of it. A multi-objective evolutionary algorithm with objective space partition based on online perceiving the Pareto front of an optimization problem is proposed in this study. In the proposed method, a series of discontinuous points on a Pareto front which is getting from the relationship between the crowded distance of individuals and a given threshold are flrst detected. Then, the objective space is divided into a number of sub-spaces based on these points. In each sub-space, the multi-objective evolutionary algorithm based on decomposition (MOEA/D) is employed to obtain an external archive set. Finally, the Pareto optimal set of the problem is formed based on all the external sets. The proposed method is compared with NSGA-Ⅱ, RPEA, MOEA/D, MOEA/D-PBI, MOEA/D-STM, and MOEA/D-ACD on 15 benchmark optimization problems. The results empirically demonstrate that the proposed method has capabilities in obtaining a Pareto optimal set with good performance in convergence and diversity, and is a competitive alternative for multi-objective optimization.
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Key words:
- Multi-objective evolutionary optimization /
- Pareto front /
- discontinuous point /
- objective space division /
- MOEA/D
1) 本文责任编委 魏庆来 -
图 8 不同时$\alpha$的IGD、GD和运算时间
Fig. 8 Curves of IGD, GD and time for different values of $\alpha$
图 9 IGD指标随$\alpha$的变化曲线
Fig. 9 Curves of IGD with respect to the number of generations for different values of $\alpha$
图 10 $\alpha$取不同值时, ZDT3的Pareto前沿
Fig. 10 Pareto front of different values of $\alpha$ for ZDT3
表 1 不同方法求解测试函数的IGD值
Table 1 The values of IGD obtained by different algorithms
优化问题 NSGA-Ⅱ RPEA MOEA/D MOEA/D-PBI MOEA/D-STM MOEA/D-ACD MOEA-PPF ZDT1 $6.546 \times 10^{-3}\dagger$ $2.260\times 10^{-3}\dagger$ $8.708 \times 10^{-4}$ $1.608\times 10^{-3}\dagger$ $7.637 \times10^{-4}\dagger$ ${\bf 7.629 \times 10^{-4}}\dagger$ $8.668 \times 10^{-4}$ ZDT2 $1.685\times 10^{-2}\dagger$ $4.700\times 10^{-3}\dagger$ $8.392 \times 10^{-4}$ $1.110\times10^{-3}\dagger$ $7.507 \times 10^{-4}\dagger$ ${\bf 7.502 \times 10^{-4}}\dagger$ $8.368 \times 10^{-4}$ ZDT3 $2.647\times 10^{-3}\dagger$ $3.534\times 10^{-3}\dagger$ $2.038\times 10^{-3}\dagger$ $3.978\times 10^{-3}\dagger$ $2.060\times 10^{-3}\dagger$ $2.050\times10^{-3}\dagger$ ${\bf 1.620 \times10^{-3}}$ ZDT4 $2.463\times 10^{-2}\dagger$ $2.198\times 10^{-3}\dagger$ $7.791 \times 10^{-4}$ $1.277\times 10^{-3}\dagger$ ${\bf 7.637 \times 10^{-4}}\dagger$ $7.819 \times10^{-4}$ $7.719 \times10^{-4}$ ZDT6 $8.319 \times10^{-4}\dagger$ $1.177 \times10^{-1}\dagger$ $3.994 \times 10^{-4}$ $4.006 \times 10^{-4}$ ${\bf 3.811 \times10^{-4}}\dagger$ $5.225 \times 10^{-4}\dagger$ 3.988 $\times 10^{-4}$ KUR $3.962 \times10^{-1}$ $3.284\times 10^{-2}\dagger$ 1.012 $\times 10^{-2}$ $1.029 \times10^{-2}$ $9.779\times 10^{-3}\dagger$ $9.853 \times 10^{-3}$ ${\bf 9.609 \times 10^{-3}}$ WFG1 $6.949 \times10^{-1}\dagger$ $4.685 \times10^{-1}\dagger$ $4.553 \times10^{-3}$ $3.248 \times10^{-2}\dagger$ $8.218 \times10^{-3}$ $4.680 \times10^{-1}\dagger$ ${\bf 3.940 \times 10^{-3}}$ WFG2 $3.968 \times 10^{-3}$ $1.049\times 10^{-3}\dagger$ $5.063 \times 10^{-3}$ $4.850 \times10^{-2}\dagger$ $4.805 \times 10^{-3}$ $5.241 \times 10^{-3}$ ${\bf 3.748 \times10^{-3}}$ WFG3 $3.198 \times10^{-3}$ $5.316 \times10^{-3}\dagger$ $2.953 \times10^{-3}$ $4.498 \times10^{-3}\dagger$ ${\bf 2.597 \times 10^{-3}}\dagger$ $2.747\times10^{-3}\dagger$ $2.951 \times10^{-3}$ WFG4 $2.665 \times10^{-2}\dagger$ $1.816 \times10^{-2}\dagger$ $5.724 \times10^{-3}$ $1.886 \times10^{-2}\dagger$ $8.427 \times10^{-3}\dagger$ $1.215 \times10^{-2}\dagger$ ${\bf 5.122 \times10^{-3}}$ WFG5 $6.206 \times10^{-3}\dagger$ $6.777 \times10^{-2}\dagger$ $6.326 \times10^{-2}$ $6.852 \times10^{-2}$ ${\bf 6.179\times10^{-2}}\dagger$ $6.287 \times10^{-2}\dagger$ $6.510 \times10^{-2}$ WFG6 $5.278 \times10^{-2}\dagger$ $7.205 \times10^{-2}\dagger$ $5.560 \times10^{-2}$ $5.597 \times10^{-2}$ $2.862 \times10^{-3}\dagger$ ${\bf 2.672\times10^{-3}}$ $5.287 \times10^{-2}$ WFG7 $3.186 \times10^{-3}\dagger$ $3.014 \times10^{-2}\dagger$ $2.663 \times10^{-3}$ $3.850 \times10^{-3}\dagger $ ${\bf 2.585\times10^{-3}}\dagger$ $2.666 \times10^{-3}$ $2.689 \times10^{-3}$ WFG8 $1.129 \times10^{-1}$ $1.017\times10^{-1}\dagger$ $1.033 \times10^{-1}\dagger$ $1.074 \times10^{-1}$ $9.793 \times10^{-2}\dagger$ ${\bf 8.302\times10^{-2}}\dagger$ $1.029 \times10^{-1}$ WFG9 $1.915 \times10^{-2}$ $1.272 \times10^{-2}\dagger$ $1.533 \times10^{-2}$ $2.155 \times10^{-2}$ $1.229 \times10^{-2}$ ${\bf 8.062\times10^{-3}}$ $1.522 \times10^{-2}$ †表示对比方法与本文方法的IGD指标具有显著差异(Mann-Whitney U分布检验, 置信水平为0.05) 
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表 2 不同方法获得的CR值
Table 2 Metric CR obtained by different algorithms
优化问题 NSGA-Ⅱ RPEA MOEA/D MOEA/D-PBI MOEA/D-STM MOEA/D-ACD MOEA-PPF ZDT1 0.9400 0.6860 0.9440 0.9360 0.9440 0.9440 0.9440 ZDT2 0.9260 0.6380 0.9960 0.9940 0.9960 0.9960 0.9960 ZDT3 0.6440 0.4340 0.5760 0.6460 0.5760 0.6720 0.7480 ZDT4 0.9340 0.6320 0.9440 0.9340 0.9440 0.9420 0.9440 ZDT6 0.9460 0.6640 0.9960 0.9960 0.9960 0.9960 0.9960 KUR 0.8020 0.6060 0.7800 0.9160 0.7780 0.8940 0.9140 WFG1 0.7320 0.3620 0.8700 0.8240 0.8480 0.4520 0.8720 WFG2 0.6100 0.3680 0.4880 0.6060 0.4860 0.6220 0.7360 WFG3 0.9460 0.8260 0.9920 0.9960 0.9960 0.9960 0.9960 WFG4 0.9280 0.5820 0.9500 0.9520 0.8440 0.8340 0.9560 WFG5 0.9020 0.5760 0.9180 0.9180 0.9540 0.9140 0.9200 WFG6 0.9000 0.5500 0.9460 0.9580 0.9620 0.9620 0.9580 WFG7 0.9240 0.5780 0.9660 0.9700 0.9680 0.9620 0.9680 WFG8 0.5400 0.3500 0.9060 0.9220 0.9140 0.9180 0.9000 WFG9 0.9220 0.6340 0.9260 0.9280 0.8920 0.9220 0.8980
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