Decomposition and Dominance Relation Based Many-objective Evolutionary Algorithm
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摘要:
近年来，超多目标优化问题(MaOPs)成为了进化计算领域的研究热点。然而，在处理各种优化问题中，如何有效地平衡收敛性和多样性仍是一个难题。为了解决上述的问题，该文提出了一种基于分解和支配关系的超多目标进化算法(DdrEA)。首先利用权重向量把整个种群分解为一组子种群，这些子种群将进行协同优化；然后利用角度和角度支配关系计算子种群内每个解的值；最后根据适应度值进行精英选择，即在每个子空间内选取适应度值最小的解作为精英解进入下一代。DdrEA通过与当前较优的NSGA-II/AD, RVEA, MOMBI-II等多个超多目标进化算法进行实验对比，实验结果表明该文算法性能明显优于对比算法，能够有效平衡种群的收敛性和多样性。
Abstract:In recent year, the Many-objective Optimization Problems (MaOPs) have become an increasingly hot research area in evolutionary computation. However, it is still a difficult problem to achieve a good balance between convergence and diversity on solving various kinds of MaOPs. To alleviate this issue mentioned above, a Decomposition and dominance relation based many-objective Evolutionary Algorithm(DdrEA) is proposed in this paper. Firstly, the population is decomposed into numbers of sub-populations by using a set of uniform weight vectors, in which they are optimized in a cooperative manner. Then, the fitness value of solution in each sub-population is calculated by angle dominance relation and angle. Finally, elite selection strategy is performed according to its corresponding fitness value. That is, in each subspace, the solution with the smallest fitness value is selected as the elite solution to enter the next generation. Comparing with several high-dimensional and multi-objective evolutionary algorithms (NSGA-II/AD, RVEA, MOMBI-II), the experimental results show that the performance of the proposed algorithm DdrEA is better than that of the comparison algorithm, and the convergence and diversity of the population can be effectively balanced.
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Key words:
- Many-objective optimization /
- Decomposition /
- Dominance relation /
- Evolutionary algorithm
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表 1 DdrEA算法框架
输入: N (种群规模) 输出: P (最终种群) (1) ${ V} = \operatorname{Uniform\;Reference\;Vector} (N)$; (2) $P = \operatorname{RandomInitialize} (N)$; (3) while 终止条件未满足 do (4) ${\rm{Pool} } = \operatorname{Bianry\;Tournament} (P)$; (5) $O = \operatorname{Variation} ({\rm{Pool}})$; (6) $P = \operatorname{Environmental\;Selection} (P \cup O,V,N)$; (7) end while 
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表 2 环境选择算法框架
输入: P (合并后的种群), V (参考向量); 输出: Q (下一代种群); (1) /*目标值标准化*/
(2) ${f_i}\;(p) = \dfrac{{{f_i}(p) - z_i^{{\rm{id}}}}}{{z_i^{{\rm{na}}} - z_i^{{\rm{id}}}}},\forall p \in P$;(3) /*种群分解*/ (4) for $i = 1\;{\rm{to }}\left| P \right|$ do (5) for $j = 1\;{\rm{to }}N$ do
(6) ${\theta _{i,j} } = \arccos \dfrac{ { {f_i}\;(p) \cdot {v_j} } }{ {\left\| { {f_i}\;(p)} \right\|} }$;(7) end for (8) end for (9) for $i = 1\;{\rm{to }}\left| P \right|$ do (10) $r = \arg \min {\theta _{i,j}},j \in \{ 1,2, ··· ,N\} $; (11) ${\overline P _k} = {\overline P _k} \cup \{ {I_r}\} $; (12) end for (13) /*精英选择*/ (14) 非支配排序获得每个解的AD等级; (15) for $i = 1\;{\rm{to }}N$ do (16) for $j = 1\;{\rm{to } }\left| { { { {\overline P}_k} } } \right|$ do (17) ${\rm{fi}}{{\rm{t}}_{i,j}} = F_i^p + {\theta _{i,j}}$; (18) end for (19) end for (20) for $i = 1\;{\rm{to }}N$ do (21) $r = \arg \min {\rm{fi}}{{\rm{t}}_{i,j}}$ (22) $Q = Q \cup \{ {I_r}\} $ (23) end for
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表 3 各算法在WFG测试集上独立运行30次的HV均值和标准差
测试问题 M DdrEA NSGA-II/AD RVEA MOMBI-II WFG1 5 9.7500e-1 (3.88e-3) 9.7759e-1 (1.72e-3) + 9.9722e-1 (5.54e-4) + 9.4668e-1 (5.46e-2) = 10 9.9821e-1 (4.94e-4) 9.9849e-1 (1.57e-4) + 9.9768e-1 (5.30e-4) – 9.9990e-1 (6.65e-5) + 15 9.9960e-1 (2.23e-4) 9.9977e-1 (5.94e-5) + 9.9546e-1 (1.65e-2) – 9.9900e-1 (9.15e-4) – WFG2 5 9.5980e-1 (4.39e-3) 9.8390e-1 (6.84e-4) + 9.9385e-1 (1.19e-3) + 9.9432e-1 (2.64e-3) + 10 9.9548e-1 (1.94e-3) 9.9819e-1 (3.42e-4) + 9.8944e-1 (2.64e-3) – 9.9459e-1 (4.80e-3) = 15 9.9516e-1 (3.28e-3) 9.9854e-1 (4.21e-4) + 9.7143e-1 (6.37e-3) – 9.4784e-1 (4.42e-2) – WFG3 5 2.1343e-1 (1.95e-2) 2.4953e-1 (4.42e-3) + 1.5392e-1 (2.34e-2) – 9.1612e-2 (1.14e-3) – 10 9.2805e-2 (1.72e-2) 7.7407e-2 (1.47e-2) – 0.0000e+0 (0.00e+0) – 8.8033e-2 (1.38e-2) = 15 8.3502e-4 (3.18e-3) 5.6841e-4 (2.97e-3) = 0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0) = WFG4 5 7.9321e-1 (6.31e-4) 6.4860e-1 (1.20e-3) – 7.9120e-1 (6.98e-4) – 7.9039e-1 (7.50e-3) = 10 9.6871e-1 (4.54e-4) 8.3306e-1 (1.02e-3) – 9.5997e-1 (2.21e-3) – 9.2583e-1 (5.33e-2) – 15 9.8842e-1 (4.27e-3) 9.0197e-1 (2.24e-3) – 9.7572e-1 (3.32e-3) – 6.5742e-1 (6.09e-2) – WFG5 5 7.4375e-1 (4.53e-4) 5.9898e-1 (9.69e-4) – 7.4385e-1 (3.78e-4) = 7.1661e-1 (1.18e-2) – 10 9.0472e-1 (2.50e-4) 7.6876e-1 (7.00e-4) – 9.0333e-1 (3.03e-4) – 8.2511e-1 (1.49e-2) – 15 9.1756e-1 (2.50e-4) 8.2982e-1 (1.15e-3) – 9.1591e-1 (2.52e-4) – 3.0980e-1 (1.03e-1) – WFG6 5 7.2159e-1 (1.32e-2) 5.8627e-1 (1.86e-2) – 7.2535e-1 (1.44e-2) = 7.2502e-1 (2.91e-2) = 10 8.8790e-1 (1.79e-2) 7.4225e-1 (2.01e-2) – 8.7718e-1 (1.67e-2) – 8.6054e-1 (5.34e-2) – 15 8.9955e-1 (2.46e-2) 7.9363e-1 (2.38e-2) – 7.2363e-1 (8.04e-2) – 6.2650e-1 (5.72e-2) – WFG7 5 7.9404e-1 (5.38e-4) 6.4353e-1 (1.20e-3) – 7.8991e-1 (5.70e-4) – 7.8952e-1 (8.36e-3) – 10 9.6965e-1 (2.60e-4) 8.2831e-1 (4.30e-3) – 9.5835e-1 (1.80e-3) – 9.6818e-1 (3.92e-3) = 15 9.9072e-1 (9.04e-4) 8.9903e-1 (1.10e-3) – 9.8234e-1 (5.24e-3) – 7.5033e-1 (8.01e-2) – WFG8 5 6.8107e-1 (1.14e-3) 4.6230e-1 (5.93e-3) – 6.7438e-1 (2.36e-3) – 3.1772e-1 (6.41e-3) – 10 8.7577e-1 (6.90e-3) 6.0416e-1 (6.06e-2) – 8.3098e-1 (8.86e-2) = 6.4422e-1 (1.85e-2) – 15 8.9925e-1 (1.56e-3) 7.6855e-1 (6.04e-2) – 7.7368e-1 (1.23e-1) – 5.1932e-1 (7.33e-2) – WFG9 5 7.4084e-1 (4.55e-2) 6.3546e-1 (3.05e-3) – 7.5200e-1 (5.64e-3) = 5.5025e-1 (5.88e-2) – 10 9.0376e-1 (4.59e-2) 8.1230e-1 (7.64e-3) – 8.8746e-1 (3.28e-2) – 8.0514e-1 (1.40e-2) – 15 9.1408e-1 (3.94e-2) 8.4483e-1 (3.67e-2) – 8.3486e-1 (5.14e-2) – 2.6009e-1 (3.09e-2) – $ + \;/\; - \;/\; = $ 7/19/1 2/20/5 2/18/7
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表 4 各算法在DTLZ测试集上独立运行30次的HV均值和标准差
测试问题 M DdrEA NSGA-II/AD RVEA MOMBI-II DTLZ1 5 8.4928e-1 (4.56e-2) 9.0703e-1 (4.42e-3) + 9.7490e-1 (1.72e-4) + 9.7461e-1 (2.59e-4) + 10 9.8958e-1 (5.26e-3) 9.8084e-1 (1.48e-3) – 9.9968e-1 (1.73e-5) + 9.6198e-1 (3.15e-2) – 15 9.1625e-1 (6.21e-2) 9.9583e-1 (3.87e-4) + 9.9990e-1 (4.62e-5) + 9.0337e-1 (3.90e-2) – DTLZ2 5 7.9489e-1 (2.85e-4) 7.1691e-1 (5.66e-3) – 7.9479e-1 (3.56e-4) = 7.9449e-1 (5.39e-4) – 10 9.7092e-1 (2.45e-4) 9.0347e-1 (4.42e-3) – 9.6974e-1 (1.49e-4) – 9.7085e-1 (2.13e-4) = 15 9.9160e-1 (1.05e-4) 9.5411e-1 (3.00e-3) – 9.9109e-1 (2.99e-4) – 8.8008e-1 (6.31e-2) – DTLZ3 5 6.2295e-1 (4.78e-2) 7.1455e-1 (7.51e-3) = 7.9246e-1 (2.06e-3) + 7.9042e-1 (2.58e-3) + 10 8.8458e-1 (2.77e-2) 8.9986e-1 (5.59e-3) = 9.6952e-1 (3.56e-4) + 8.2910e-1 (9.36e-2) – 15 7.0985e-1 (5.92e-2) 9.4928e-1 (5.01e-3) + 9.9072e-1 (2.96e-4) + 5.5538e-1 (5.08e-2) – DTLZ4 5 7.9479e-1 (4.40e-4) 7.2830e-1 (5.61e-3) – 7.8505e-1 (2.91e-2) = 7.8426e-1 (2.85e-2) – 10 9.7157e-1 (2.23e-4) 9.1475e-1 (4.07e-3) – 9.6980e-1 (1.53e-4) – 9.7226e-1 (2.24e-3) + 15 9.9206e-1 (1.14e-4) 9.6070e-1 (2.35e-3) – 9.9076e-1 (1.18e-3) – 9.9044e-1 (1.51e-3) – DTLZ5 5 1.0799e-1 (3.39e-3) 1.2935e-1 (3.74e-4) + 1.0483e-1 (2.04e-3) – 9.0959e-2 (2.73e-4) – 10 9.1644e-2 (1.16e-3) 1.0062e-1 (2.77e-4) + 9.0893e-2 (1.10e-4) – 9.1359e-2 (7.19e-4) = 15 9.1168e-2 (3.89e-4) 9.4858e-2 (3.18e-4) + 9.0648e-2 (3.24e-3) = 9.1206e-2 (3.78e-4) = DTLZ6 5 1.0371e-1 (5.39e-3) 1.2924e-1 (2.68e-4) + 1.0340e-1 (1.29e-2) = 9.0959e-2 (2.53e-4) – 10 9.1619e-2 (8.82e-4) 1.0053e-1 (2.70e-4) + 9.1873e-2 (8.07e-4) = 9.2808e-2 (1.11e-3) + 15 9.1147e-2 (5.02e-4) 9.4877e-2 (2.90e-4) + 8.7721e-2 (1.72e-2) = 9.1664e-2 (5.93e-4) + DTLZ7 5 2.4710e-1 (5.77e-3) 2.3300e-1 (4.51e-3) – 2.0265e-1 (3.05e-3) – 2.4292e-1 (5.42e-3) – 10 1.9578e-1 (2.23e-3) 1.9180e-1 (3.40e-3) – 1.4878e-1 (2.05e-2) – 1.5877e-1 (1.00e-2) – 15 1.6487e-1 (4.21e-3) 1.5882e-1 (5.21e-3) – 5.8918e-2 (5.77e-2) – 1.2108e-1 (1.74e-3) – $ + \;/\; - \;/\; = $ 9/10/2 6/9/6 5/13/3
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