Page 159 - 《软件学报》2021年第12期

孙哲人 等:面向多目标优化的多样性代理辅助进化算法 3823

Table 1 Results of IGD
表 1 反向迭代距离结果
Problem DSAEA K-RVEA MOEA/D-EGO ParEGO
ZDT1 6.2887e−2(2.73e−2) 8.2035e−1(1.15e−1)− 1.7097e+0(3.74e+0)− 4.7243e−1(2.08e−1)−
ZDT2 1.1052e−1(1.07e−1) 7.4677e−1(1.37e−1)− 1.3118e+0(1.55e+0)− 5.9465e−1(1.43e−1)−
ZDT3 1.9822e−1(1.88e−1) 8.1794e−1(1.59e−1)− 1.4819e+0(1.73e+0)− 5.6781e−1(1.61e−1)−
ZDT4 6.3493e+1(1.78e+1) 7.9403e+1(2.22e+1)− 1.0688e+2(1.28e+1)− 8.5326e+1(1.37e+1)−
ZDT6 3.2883e+0(8.31e−1) 5.4347e+0(2.27e+0)− 6.2323e+0(1.99e+0)− 6.4443e+0(8.96e−1)−
DTLZ1 5.4726e+1(1.71e+1) 7.6495e+1(1.87e+1)− 8.7589e+1(1.64e+1)− 8.9775e+1(2.24e+1)−
DTLZ2 1.4977e−1(3.47e−2) 1.8420e−1(1.88e−2)− 3.3203e−1(2.55e−2)− 2.6802e−1(2.14e−2)−
DTLZ3 1.3896e+2(2.69e+1) 2.1205e+2(7.18e+1)− 2.1185e+2(4.05e+1)− 2.6622e+2(6.05e+1)−
DTLZ4 3.3670e−1(1.01e−1) 3.6212e−1(9.93e−2)≈ 6.4259e−1(7.17e−2)− 4.1291e−1(1.12e−1)−
DTLZ5 2.5984e−2(6.28e−3) 7.5002e−2(1.28e−2)− 2.5429e−1(3.03e−2)− 1.7326e−1(3.06e−2)−
DTLZ6 2.7593e+0(4.34e−1) 3.8069e+0(4.72e−1)− 1.8576e+0(5.81e−1)+ 4.1258e+0(4.68e−1)−
DTLZ7 1.7815e−1(3.03e−2) 1.0011e+0(1.10e−1)− 2.3411e−1(9.23e−2)− 3.3278e−1(6.28e−2)−
+/−/≈ − 0/11/1 1/11/0 0/12/0
Table 2 Results of HV
表 2 超体积结果
Problem DSAEA K-RVEA MOEA/D-EGO ParEGO
ZDT1 6.4567e−1(1.48e−2) 8.9466e−2(7.60e−2)− 3.4343e−1(2.33e−1)− 2.0311e−1(1.29e−1)−
ZDT2 3.2907e−1(8.19e−2) 4.9670e−3(1.05e−2)− 9.4252e−2(1.02e−1)− 4.0968e−2(3.81e−2)−
ZDT3 5.2323e−1(1.39e−1) 1.0257e−1(1.19e−1)− 2.4279e−1(2.35e−1)− 1.9277e−1(1.16e−1)−
DTLZ2 4.3094e−1(4.84e−2) 3.5085e−1(2.65e−2)− 1.3774e−1(4.64e−2)− 1.8637e−1(3.65e−2)−
DTLZ4 2.0978e−1(1.07e−1) 1.2883e−1(1.17e−1)− 8.5741e−3(1.49e−2)− 2.1018e−1(6.82e−2)≈
DTLZ5 1.7894e−1(6.78e−3) 1.4107e−1(1.17e−2)− 2.5012e−2(2.17e−2)− 5.9209e−2(2.35e−2)−
DTLZ7 2.2066e−1(1.15e−2) 1.5166e−1(1.28e−2)− 2.1222e−1(1.71e−2)− 1.5591e−1(2.40e−2)−
+/−/≈ − 0/7/0 0/7/0 0/6/1
Table 3 Results of running time
表 3 运行时间结果
Problem DSAEA K-RVEA MOEA/D-EGO ParEGO
ZDT1 2.7213e+1(2.24e−1) 2.6465e+1(3.81e−1)+ 3.8747e+1(3.09e−1)− 5.3115e+1(4.98e−1)−
ZDT2 2.6538e+1(3.10e−1) 2.5988e+1(4.28e−1)+ 3.8776e+1(3.57e−1)− 5.2682e+1(6.76e−1)−
ZDT3 2.6967e+1(3.53e−1) 2.6303e+1(3.22e−1)+ 3.8663e+1(3.22e−1)− 5.3362e+1(5.72e−1)−
ZDT4 2.7191e+1(2.32e+0) 2.6788e+1(6.96e−1)≈ 3.6354e+1(1.36e+0)− 5.3146e+1(7.36e−1)−
ZDT6 2.7832e+1(5.32e−1) 2.7018e+1(4.51e−1)+ 3.9848e+1(8.02e−1)− 5.3760e+1(8.56e−1)−
DTLZ1 9.3734e+1(2.42e+0) 8.3035e+1(8.01e−1)+ 1.5175e+2(2.18e+0)− 1.1071e+2(1.99e+0)−
DTLZ2 1.0472e+2(2.79e+0) 8.2676e+1(8.58e−1)+ 1.4879e+2(4.22e+0)− 1.1163e+2(1.56e+0)−
DTLZ3 9.3374e+1(2.28e+0) 8.2940e+1(9.74e−1)+ 1.5196e+2(2.16e+0)− 1.1049e+2(1.57e+0)−
DTLZ4 8.4365e+1(2.40e+0) 8.3634e+1(1.37e+0)≈ 1.4511e+2(4.16e+0)− 1.0792e+2(2.04e+0)−
DTLZ5 8.4555e+1(1.11e+0) 8.2550e+1(8.31e−1)+ 1.4961e+2(4.99e+0)− 1.1101e+2(1.15e+0)−
DTLZ6 1.0086e+2(3.40e+0) 8.3866e+1(9.85e−1)+ 1.5024e+2(3.75e+0)− 1.1151e+2(1.08e+0)−
DTLZ7 7.9500e+1(8.09e−1) 8.0235e+1(1.00e+0)− 1.4154e+2(1.46e+0)− 1.1207e+2(1.54e+0)−
+/−/≈ − 9/1/2 0/12/0 0/12/0

图 2、图 3 给出了 DSAEA 与对比算法在 ZDT,DTLZ 问题上的收敛曲线,横坐标为真实评估次数,纵坐标为
IGD 指标.
对于 ZDT 测试问题:
• IGD 和 HV 结果表明:在实验设置一致的情况下,DSAEA 明显表现更好;
• 而收敛曲线表明:DSAEA 的收敛效果在大多数情况下要好于对比算法,只有在 ZDT1-3 问题上,
DSAEA 的收敛效果在 170 次真实评估前不如 ParEGO.
由于 ZDT1-3 问题较为简单,在最初阶段,搜索到比现有种群更优的解很容易,如果算法根据收敛性选解,则
收敛速度会很快.但在靠近真实 PF 的区域内,搜索到比现有种群更优的解相对不易,需要模型能够较好地拟合
真实 PF 附近的区域.不同于 ParEGO,DSAEA 是基于多样性选解的,它更倾向于增加解集的多样性,使代理模型
可以较好拟合目前种群最优解附近的区域.因此,DSAEA 的收敛曲线前 30 次的收敛效果不如 ParEGO,而之后的