Interval Multimodal Particle Swarm Optimization Algorithm Assisted by Heterogeneous Ensemble Surrogate
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摘要: 现实生活中的很多黑盒优化问题可归为高计算代价的多模态优化问题(Multimodal optimization problem, MMOP), 即昂贵多模态优化问题(Expensive MMOP, EMMOP). 在处理该类问题时, 决策者希望以尽量少的计算代价(即尽量少的真实函数评价次数)找到多个高质量的最优解. 然而, 已有代理辅助的进化优化算法(Surrogate-assisted evolutionary algorithm, SAEA)很少考虑问题的多模态属性, 运行一次仅可获得问题的一个最优解. 鉴于此, 研究一种异构集成代理辅助的区间多模态粒子群优化(Interval multimodal particle swarm optimization algorithm assisted by heterogeneous ensemble surrogate, IMPSO-HES)算法. 首先, 借助异构集成的思想构建一个由多个基础代理模型组成的模型池; 随后, 依据待评价粒子与已发现模态之间的匹配关系, 从模型池中自主选择部分基础代理模型进行集成, 并使用集成后的代理模型预测该粒子的适应值. 进一步, 为节约代理模型管理的代价, 设计一种增量式的代理模型管理策略; 为减少代理模型预测误差对算法性能的影响, 首次将区间排序关系引入到进化过程中. 将所提算法与当前流行的5种代理辅助进化优化算法和7 种最先进的多模态优化算法进行对比, 在20个测试函数和1个建筑节能实际问题上的实验结果表明, 所提算法可以在较少计算代价下获得问题的多个高竞争最优解.Abstract: Many real-world black-box optimization problems can be classified as multimodal optimization problems (MMOPs) with high computational cost, that is, expensive multimodal optimization problems (EMMOPs). When dealing with such problems, decision-makers hope to find multiple high-quality solutions with less computational cost (i.e., the least number of real function evaluations). However, existing surrogate-assisted evolutionary algorithms (SAEAs) seldom consider the multimodal properties of problem, and they can only obtain one optimal solution of the problem at a time. In view of this, this paper studies an interval multimodal particle swarm optimization (PSO) algorithm assisted by heterogeneous ensemble surrogate (IMPSO-HES). Firstly, a model pool composed of multiple basic surrogate models is constructed with the idea of heterogeneous ensemble. Then, according to the matching relationship between the particle to be evaluated and the discovered modalities, some basic surrogate models will be selected from the model pool for integration, and the integrated surrogate model is utilized to predict the fitness value of the particle. Furthermore, in order to save the cost of model management, an incremental surrogate model management strategy is designed. In order to reduce the influence of prediction error of surrogate model on the algorithm＇s performance, the interval ordering relation is introduced into the evolutionary process for the first time. The proposed algorithm is compared with five SAEAs and seven state-of-the-art multimodal algorithms, experimental results on 20 benchmark functions and the building energy conservation problem show that the proposed algorithm can obtain multiple highly-competitive optimal solutions at a low computational cost.
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图 2 精确评价和区间评价策略下IMPSO-HES所得GS值
Fig. 2 GS values obtained by IMPSO-HES under precision and interval evaluation
图 3 精确评价和区间评价策略下IMPSO-HES所得VR值
Fig. 3 VR values obtained by IMPSO-HES under precision and interval evaluation
图 4 IMPSO-HES/D和IMPSO-HES得到的GS值
Fig. 4 GS values obtained by IMPSO-HES/D and IMPSO-HES
图 5 IMPSO-HES/D和IMPSO-HES得到的VR值
Fig. 5 VR values obtained by IMPSO-HES/D and IMPSO-HES
表 1 基准问题
Table 1 Benchmark problems
问题 测试函数 维数 变量空间 全局/局部解个数 全局最优解的目标值 F1 Ellipsoid 10/20 $\boldsymbol{X} \in [-1,1]^{D}$ 1/0 0 F2 Ackley 10/20 $\boldsymbol{X} \in [-30,30]^{D}$ 1/many 0 F3 Rastrigin 10/20 $\boldsymbol{X }\in [-5.12,5.12]^{D}$ 1/many 0 F4 Rosenbrock 10/20 $\boldsymbol{X} \in [-2.048,2.048]^{D}$ 1/many 0 F5 Griewank 10/20 $\boldsymbol{X} \in [-600,600]^{D}$ 1/many 0 F6 Reverse five-uneven-peak trap 1 $\boldsymbol{X} \in [0,30] $ 2/3 −200 F7 Reverse equal maxima 1 $\boldsymbol{X} \in [0,1] $ 5/0 −1 F8 Reverse uneven decreasing maxima 1 $\boldsymbol{X} \in [0,1] $ 1/4 −1 F9 Reverse himmelblau 2 $\boldsymbol{X} \in [-6,6]^{D}$ 4/0 −200 F10 Six-hump camel 2 $x_1\in[-1.9,1.9], x_2\in[-1.1,1.1] $ 2/2 −1.031 6 F11 Reverse shubert 2 $\boldsymbol{X} \in [-10,10]^{D}$ 18/many −186.73 F12 Reverse vincent 2 $\boldsymbol{X} \in [0.25,10]^{D}$ 36/0 −1 F13 Reverse modified rastrigin 2 $\boldsymbol{X} \in [0,1]^{D}$ 12/0 2 F14 Reverse CF1 2 $\boldsymbol{X}\in [-5,5]^D$ 6/0 0 F15 Reverse CF2 2 $\boldsymbol{X}\in [-5,5]^D$ 8/0 0 F16 Reverse CF3 2 $\boldsymbol{X} \in[-5,5]^D $ 6/0 0 F17 Reverse CF4 3 $\boldsymbol{X}\in [-5,5]^D$ 8/0 0 F18 UrsemF4 back 2 $\boldsymbol{X }\in [-2,2]^{D}$ 2/0 −0.267 9 F19 Branin RCOS 2 $x_1\in[-5,10], x_2\in[0,15] $ 3/0 0.397 8 F20 Waves 2 $x_1\in[-0.9,1.2], x_2\in[-1.2,1.2]$ 1/9 −7.776 
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表 2 F6 ~ F20的幅值精度和距离精度
Table 2 Amplitude accuracy and distance accuracy for F6 ~ F20
F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 $R_{v}$ 1 0.05 0.1 0.5 0.05 10 0.1 0.5 1 1 1 1 0.1 0.1 0.5 $R_{d}$ 1 0.05 0.5 0.5 0.2 2 0.5 0.5 1 1 1 1 0.5 1 0.2
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表 3 不同$g_{{\rm{max}}}$取值下IMPSO-HES所得的性能指标值
Table 3 Performance values obtained by IMPSO-HES under different $g_{{\rm{max}}}$ values
问题 $g_{{\rm{max}}}$ $GS $均值(标准差) $VR $均值 耗时(s) F5 (D = 10) 3 3.800 7 (3.5E+00)+ — 64 6 1.174 5 (3.7E−02) — 85 9 1.108 3 (2.5E−02) = — 116 F5 (D = 20) 3 8.198 0 (9.8E+00) + — 776 6 1.075 7 (1.6E−02) — 1 400 9 0.807 9 (2.8E−01) − — 2 045 F9 3 −199.93 (3.1E−03) = 0.68 11 6 −199.99 (1.0E−04) 0.70 19 9 −200.00 (1.4E−03) = 0.63 36 F10 3 −1.031 6 (1.7E−06) = 1.00 19 6 −1.031 6 (9.8E−07) 1.00 28 9 −1.031 6 (9.8E−07) = 1.00 38 F12 3 −0.999 0 (7.1E−06) = 0.13 10 6 −0.999 9 (1.0E−06) 0.13 14 9 −0.999 9 (2.2E−06) = 0.11 25 注: 加粗字体表示各组的最优结果值.
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表 4 不同Q取值下IMPSO-HES所得的性能指标值
Table 4 Performance values obtained by IMPSO-HES under different Q values
问题 Q GS 均值(标准差) VR 均值 耗时(s) F5 (D = 10) K/5 1.658 1 (2.2E−01) + — 64 K/4 1.174 5 (3.7E−02) — 85 K/3 1.382 1 (1.5E−01) + — 108 K/2 1.269 6 (5.1E−02) + — 160 F5 (D = 20) K/5 1.980 0 (1.0E+00) + — 1137 K/4 1.075 7 (1.6E−02) — 1400 K/3 1.832 1 (1.1E+00) + — 1920 K/2 1.835 2 (1.7E+00) + — 2700 F9 K/5 −199.98 (7.2E−04) = 0.53 17 K/4 −199.99 (1.0E−04) 0.70 19 K/3 −199.98 (4.6E−04) = 0.55 24 K/2 −199.14 (6.8E+00) + 0.33 34 F10 K/5 −1.031 6 (1.1E−09) = 1.00 28 K/4 −1.031 6 (9.8E−07) 1.00 28 K/3 −1.031 6 (9.8E−07) = 1.00 30 K/2 −1.030 0 (1.4E−03) + 0.85 48 F12 K/5 −0.999 1 (2.3E−06) + 0.12 12 K/4 −0.999 9 (1.0E−06) 0.13 14 K/3 −0.999 6 (8.5E−07) + 0.10 18 K/2 −0.994 9 (9.2E−05) + 0.10 24 注: 加粗字体表示各组的最优结果值.
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表 5 异构集成与同质集成下IMPSO-HES所得结果
Table 5 Performance values obtained by IMPSO-HES under heterogeneous and homogeneous ensemble
问题 算法 GS均值(标准差) VR均值 耗时(s) F5 (D = 10) IMPSO-PR 1.631 0 (7.1E−01) + — 86 IMPSO-RBFN 45.27 2 (8.9E+02) + — 39 IMPSO-HES 1.174 5 (3.7E−02) — 85 F5 (D = 20) IMPSO-PR 2.003 7 (2.9E+00) + — 1 478 IMPSO-RBFN 116.7 8 (9.5E+02) + — 180 IMPSO-HES 1.075 7 (1.6E−02) — 1 400 F9 IMPSO-PR −196.81 (9.5E+00) + 0.05 16 IMPSO-RBFN −199.99 (4.7E−07) = 0.65 22 IMPSO-HES −199.99 (1.0E−04) 0.70 19 F10 IMPSO-PR −0.962 0 (2.5E−03) + 0.2 17 IMPSO-RBFN −1.031 6 (9.8E−09) = 1.00 20 IMPSO-HES −1.031 6 (9.8E−07) 1.00 28 F12 IMPSO-PR −0.988 6 (1.5E−04) + 0.06 11 IMPSO-RBFN −0.999 5 (9.4E−07) + 0.09 19 IMPSO-HES −0.999 9 (1.0E−06) 0.13 14 注: 加粗字体表示各组的最优结果值.
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表 6 不同更新概率$p_{m}$下IMPSO-HES所得结果
Table 6 Performance values obtained by IMPSO-HES under different $p_{m}$ values
问题 $p_{m}$ GS 均值 (标准差) VR 均值 耗时(s) F5 (D = 10) 固定 1.439 3 (3.8E−01) + — 84 自适应 1.174 5 (3.7E−02) — 85 F5 (D = 20) 固定 1.750 3 (1.7E+00) + — 1313 自适应 1.075 7 (1.6E−02) — 1400 F9 固定 −199.91 (2.6E−02) + 0.40 19 自适应 −199.99 (1.0E−04) 0.70 19 F10 固定 −1.031 6 (4.7E−08) = 1.00 26 自适应 −1.031 6 (9.8E−07) 1.00 28 F12 固定 −0.996 9 (4.8E−05) + 0.12 14 自适应 −0.999 9 (1.0E−06) 0.13 14 注: 加粗字体表示各组的最优结果值.
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表 7 使用All-S和Mod-S时IMPSO-HES所得结果
Table 7 Performance values obtained by IMPSO-HES with All-S and Mod-S
问题 集成策略 GS 均值 (标准差) VR 均值 耗时(s) F5 (D = 10) All-S 3.878 5 (3.8E+00) + — 243 Mod-S 1.174 5 (3.7E−02) — 85 F5 (D = 20) All-S 8.838 7 (8.1E+00) + — 3 362 Mod-S 1.075 7 (1.6E−02) — 1 400 F9 All-S −187.33 (2.0E+2) + 0.05 80 Mod-S −199.99 (1.0E−04) 0.70 19 F10 All-S −0.9751 (1.4E−02) + 0.70 57 Mod-S −1.031 6 (9.8E−07) 1.00 28 F12 All-S −0.973 7 (1.9E−02) + 0.08 42 Mod-S −0.999 9 (1.0E−06) 0.13 14 注: 加粗字体表示各组的最优结果值.
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表 8 不同模型更新策略下IMPSO-HES所得结果
Table 8 Performance values obtained by IMPSO-HES under different model update strategies
问题 更新策略 GS 均值 (标准差) VR 均值 耗时(s) F5 (D = 10) All-up 1.500 9 (3.9E−02) + — 97 Inc-up 1.174 5 (3.7E−02) — 85 F5 (D = 20) All-up 32.184 (2.4E+04) + — 1 509 Inc-up 1.075 7 (1.6E−02) — 1 400 F9 All-up −200.00 (3.6E-10) = 0.63 30 Inc-up −199.99 (1.0E−04) 0.70 19 F10 All-up −1.031 6 (1.2E−04) = 0.95 30 Inc-up −1.031 6 (9.8E−07) 1.00 28 F12 All-up −0.999 8 (2.7E−07) = 0.11 16 Inc-up −0.999 9 (1.0E−06) 0.13 14 注: 加粗字体表示各组的最优结果值.
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表 9 IMPSO-HES与5种SAEA所得GS值(均值(方差))
Table 9 GS values obtained by IMPSO-HES and 5 SAEAs (mean (variance))
问题 D IMPSO-HES SA-COSO CAL-SAPSO Gr-based SAPSO PESPSO ESPSO F1 10 3.660 0 3.160 0− 0.115 3− 0.147 6− 0.296 2− 0.664 5− (4.2E+00) (6.5E−02) (4.9E−02) (1.1E−03) (1.3E−03) (5.0E−02) 20 21.398 11.017− 0.229 2− 0.027 9− 1.377 0− 1.866 4− (6.1E+01) (1.2E+01) (1.9E−02) (8.2E−06) (1.2E−01) (2.4E−01) F2 10 17.990 17.248= 18.606+ 15.910− 11.820− 13.786− (1.1E+00) (4.1E−02) (4.8E−01) (6.4E−01) (4.3E+00) (2.0E+00) 20 18.866 18.025− 18.421= 14.717− 12.584− 15.958− (9.0E−01) (4.4E−01) (2.4E+00) (1.1E+00) (2.3E+01) (1.6E+01) F3 10 78.266 97.683+ 79.727= 94.349+ 82.325= 89.952= (1.3E+02) (5.8E+02) (1.6E+03) (7.3E+01) (1.2E+02) (2.0E+02) 20 173.97 177.43= 128.71− 168.14= 173.99= 175.65= (2.4E+02) (6.6E+02) (4.0E+03) (1.6E+02) (1.7E+02) (1.1E+02) F4 10 37.310 537.31+ 39.003= 173.66+ 90.531+ 66.581+ (1.1E+02) (2.4E+04) (2.0E+02) (3.3E+02) (6.7E+02) (1.0E+02) 20 41.469 891.97+ 42.758= 330.37+ 97.508+ 195.90+ (5.7E+02) (1.7E+04) (2.0E+02) (3.9E+03) (6.8E+02) (1.9E+03) F5 10 1.174 5 66.556+ 1.736 4+ 1.310 6+ 2.798 7+ 2.317 2+ (3.7E−02) (1.8E+02) (1.4E−01) (1.7E−02) (2.4E+00) (3.9E−01) 20 1.075 7 43.897+ 2.255 3+ 1.057 2= 6.701 8+ 10.373+ (1.6E−02) (1.9E+02) (3.2E−01) (2.0E−05) (7.4E+00) (6.2E+00) F6 1 −199.15 −200.00− −200.00− −190.91+ −200.00− −200.00− (4.6E+00) (2.1E-10) (1.6E−09) (3.2E+01) (1.2E-13) (1.0E-11) F7 1 −0.999 9 −1.00= −0.505 2+ −0.999 1+ −0.999 9= −0.999 8= (3.1E−06) (0.0E+00) (1.2E−01) (1.1E−07) (2.7E−05) (3.8E−06) F8 1 −0.985 4 −0.980 8= −0.511 4+ −0.944 7+ −0.948 6+ −0.948 6+ (1.3E−05) (1.0E-10) (8.0E−02) (7.4E−04) (5.1E−04) (5.1E−04) F9 2 −199.99 −196.14+ −157.69+ −199.93+ −199.98= −199.74+ (1.0E−04) (3.8E+01) (8.6E+02) (5.1E−04) (2.7E−04) (6.4E−03) F10 2 −1.031 6 −0.995 6+ −0.464 6+ −1.030 6+ −1.030 3+ −1.029 2+ (9.8E−07) (1.6E−03) (1.3E−01) (1.9E−06) (1.7E−07) (5.3E−07) F11 2 −158.32 −89.368+ −52.464+ −113.85+ −130.53+ −94.463+ (1.9E+03) (2.4E+03) (2.6E+03) (3.5E+04) (2.5E+03) (1.5E+03) F12 2 −0.999 9 −0.979 8+ −0.719 4+ −0.984 5+ −0.995 4+ −0.980 0+ (1.0E−06) (5.6E−04) (9.0E−02) (1.9E−04) (2.0E−06) (5.5E−05) F13 2 2.232 9 2.890 3+ 7.846 7+ 2.298 5= 2.022 8− 2.060 9− (2.3E−01) (6.4E−02) (3.0E+01) (1.0E−01) (4.6E−03) (3.1E−03) F14 2 0.087 9 40.011+ 197.39+ 23.774+ 7.588 4+ 9.961 7+ (5.0E−01) (2.6E+02) (9.2E+03) (6.3E+03) (1.1E+02) (3.0E+02) F15 2 36.423 89.091+ 183.14+ 80.557+ 26.116= 57.889+ (3.7E+03) (2.7E+02) (3.6E+03) (1.1E+03) (7.6E+02) (2.8E+03) F16 2 0.242 3 90.430+ 350.88+ 60.296+ 1.162 1+ 18.280+ (1.3E−01) (1.2E+04) (4.8E+04) (3.2E+03) (2.5E+00) (1.2E+03) F17 3 32.566 88.270+ 173.56+ 57.380+ 26.079= 37.233= (2.0E+04) (5.3E+02) (2.6E+04) (2.1E+03) (6.2E+02) (6.0E+02) F18 2 −0.267 9 −0.245 7+ −0.130 4+ −0.267 1+ −0.267 8= −0.267 8= (1.6E−06) (3.6E−04) (5.6E−03) (6.8E−08) (1.6E−06) (5.4E−09) F19 2 0.399 9 1.148 8+ 2.260 3+ 0.425 9+ 0.424 9+ 0.513 6+ (2.4E−05) (8.6E−01) (6.2E+00) (1.3E−03) (1.2E−03) (5.3E−02) F20 2 −7.429 9 −7.776 0− −7.775 3− −6.340 8+ −7.294 3+ −7.451 1= (1.7E−02) (0.0E+00) (4.2E−06) (8.4E−01) (2.2E−01) (2.7E−01) 注: 加粗字体表示各行GS值的最优结果值.
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问题 IMPSO-HES SA-COSO CAL-SAPSO Gr-based SAPSO PESPSO ESPSO F2 ~ F5 好/平/差 — 5/2/1 3/4/1 4/2/2 4/2/2 4/2/2 Rank 2.500 0 5.500 0 3.000 0 3.125 0 3.125 0 3.750 0 Adjusted p-value — 0.006 6 0.689 2 0.689 2 0.689 2 0.393 8 F6 ~ F20 好/平/差 — 11/2/2 13/0/2 14/1/0 8/5/2 9/4/2 Rank 1.833 3 4.166 6 5.433 3 4.000 0 2.266 6 3.300 0 Adjusted p-value — 0.001 6 0.000 0 0.002 5 0.525 8 0.039 5 注: 加粗字体表示各组的最优结果值.
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表 11 处理F1 ~ F5时IMPSO-HES与7种多模态进化算法所得GS值(均值(方差))
Table 11 GS values obtained by IMPSO-HES and the 7 multimodal EAs on F1 ~ F5 (mean (variance))
问题 D IMPSO-HES LIPS EMO-MMO R3PSO FERPSO NCDE NSDE ANDE F1 10 3.6600 3.3110 −5.0580+ 5.9282 +4.3713 +5.7227 +5.8277 +5.2888 +(4.2E+00) (7.8E-01) (1.3E+00) (2.3E+00) (1.2E+00) (6.4E+00) (1.6E+00) (2.6E+00) 20 21.398 19.528= 26.709+ 31.059+ 18.792- 28.868+ 29.060+ 32.311+ (6.1E+01) (9.8E+00) (2.2E+01) (2.2E+01) (1.2E+01) (5.8E+01) (1.5E+01) (5.5E+01) F2 10 17.990 18.046= 18.022= 19.159+ 18.073= 19.411+ 19.432+ 19.523+ (1.1E+00) (8.1E−01) (7.0E−01) (3.9E−01) (1.06E+00) (1.3E+00) (3.0E−01) (1.5E−01) 20 18.866 18.924= 18.922= 19.663+ 19.313+ 19.895+ 20.108+ 19.950+ (9.0E-01) (3.6E+01) (1.7E−01) (6.5E−02) (2.5E−01) (9.9E−02) (4.9E−02) (8.2E−06) F3 10 78.266 95.069+ 89.325= 108.58+ 100.83+ 110.95+ 101.33+ 106.90+ (1.3E+02) (6.3E+01) (1.2E+02) (2.2E+02) (8.2E+01) (5.5E+02) (1.3E+02) (1.3E+02) 20 173.97 212.48+ 207.09+ 258.90+ 225.25+ 251.77+ 262.26+ 268.57+ (2.4E+02) (2.6E+02) (2.8E+02) (3.3E+02) (5.1E+02) (3.2E+02) (6.5E+02) (1.1E+02) F4 10 37.310 343.96+ 257.96+ 670.32+ 451.41+ 812.90+ 982.18+ 523.1+ (1.1E+02) (4.2E+05) (3.6E+05) (1.3E+05) (2.8E+04) (1.0E+05) (1.1E+05) (2.7E+05) 20 41.469 1431.9 +1399.6 +2853.3 +1722.6 +3031.2 +2737.0 +2416.1 +(5.7E+02) (1.1E+05) (1.5E+05) (3.6E+05) (5.1E+04) (6.9E+05) (7.1E+05) (1.6E+05) F5 10 1.1745 66.246+ 65.750+ 94.936+ 71.342+ 129.69+ 115.66+ 109.05+ (3.7E-02) (3.1E+02) (6.7E+02) (4.7E+02) (4.8E+02) (3.5E+02) (8.8E+02) (6.6E+01) 20 1.0757 160.00+ 156.27+ 305.74+ 194.22+ 298.18+ 300.28+ 300.13+ (1.6E-02) (4.8E+02) (1.1E+03) (7.0E+02) (1.5E+03) (3.7E+03) (2.1E+03) (2.2E+03) 注: 加粗字体表示各组的最优结果值.
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表 12 处理F6 ~ F20时IMPSO-HES与7种多模态进化算法所得结果
Table 12 Results of IMPSO-HES and the 7 multimodal EAs on F6 ~ F20
问题 D IMPSO-HES LIPS EMO-MMO R3PSO FERPSO NCDE NSDE ANDE F6 GS 均值 −199.15 −185.64+ −196.52+ −190.93+ −186.31+ −191.25+ −197.86+ −195.52+ (标准差) (4.6E+00) (8.8E+01) (1.0E+02) (6.1E+01) (1.0E+02) (3.4E+02) (4.5E+01) (5.0E+02) VR 均值 0.80 0.20+ 0.40+ 0.10+ 0.00+ 0.65+ 0.75= 0.40+ F7 GS 均值 −0.999 9 −0.999 4+ −0.999 5+ −0.999 1+ −0.998 6+ −0.998 7+ −0.998 4+ −0.998 0+ (标准差) (3.1E−06) (7.3E−07) (2.5E−07) (7.2E−07) (1.0E−06) (8.5E−07) (5.6E−06) (4.6E−06) VR均值 0.78 0.78= 0.76= 0.70= 0.66+ 0.74= 0.78= 0.67+ F8 GS 均值 −0.985 4 −0.969 3+ −0.993 7− −0.993 1− −0.975 8+ −0.966 0+ −0.948 3+ −0.968 3+ (标准差) (1.3E−04) (6.8E−04) (2.5E−04) (6.7E−05) (4.1E−04) (8.9E−04) (5.1E−03) (3.1E−03) VR均值 1.00 0.80+ 0.90+ 1.00= 1.00= 0.90+ 0.60+ 0.80+ F9 GS 均值 −199.99 −197.58+ −197.79+ −196.99+ −196.92+ −197.04+ −196.10+ −197.22+ (标准差) (1.0E−04) (1.7E+00) (9.9E+00) (1.3E+01) (8.6E+00) (5.2E+00) (1.6E+01) (1.3E+01) VR均值 0.70 0.02+ 0.05+ 0.07+ 0.07+ 0.10+ 0.05+ 0.05+ F10 GS 均值 −1.031 6 −1.004 7+ −1.001 6+ −1.003 2+ −0.994 9+ −0.987 8+ −0.973 0+ −1.002 0+ (标准差) (9.8E−07) (3.6E−04) (2.8E−03) (2.8E−03) (8.8E−04) (8.7E−03) (5.0E−03) (3.4E−02) VR均值 1.00 0.55+ 0.10+ 0.45+ 0.30+ 0.40+ 0.35+ 0.5+ F11 GS 均值 −158.32 −105.20+ −134.50= −90.154+ −114.099+ −123.777+ −111.92+ −132.37= (标准差) (1.9E+03) (1.3E+03) (1.7E+03) (5.4E+02) (1.3E+03) (1.0E+03) (2.3E+03) (1.6E+03) VR均值 0.02 0.01= 0.01= 0.00+ 0.00+ 0.00+ 0.00+ 0.01= F12 GS 均值 −0.999 9 −0.973 3+ −0.975 3+ −0.972 7+ −0.976 4+ −0.976 4+ −0.989 0+ −0.988 7+ (标准差) (1.0E−06) (3.2E−04) (4.9E−04) (4.6E−04) (5.8E−04) (5.2E−04) (3.0E−04) (4.6E−03) VR均值 0.13 0.08+ 0.05+ 0.07+ 0.07+ 0.08+ 0.10+ 0.09+ F13 GS 均值 2.232 9 2.714 6+ 2.560 4+ 2.438 4+ 2.590 3+ 2.481 7+ 2.344 6= 2.579 2+ (标准差) (2.3E−01) (3.2E−01) (2.3E+00) (2.1E−01) (2.4E−01) (7.3E−01) (8.7E−01) (2.2E+00) VR均值 0.09 0.08= 0.08= 0.07= 0.08= 0.13+ 0.09= 0.08= F14 GS 均值 0.087 9 44.360+ 45.829+ 43.836+ 38.669+ 40.250+ 38.149+ 41.010+ (标准差) (5.0E−01) (4.0E+03) (4.8E+03) (4.5E+03) (4.5E+03) (4.3E+03) (1.6E+03) (1.2E+02) VR均值 0.24 0.01+ 0.01+ 0.00+ 0.01+ 0.01+ 0.00+ 0.00+ F15 GS 均值 36.423 103.12+ 85.620+ 108.46+ 82.451+ 67.647+ 75.308+ 89.100+ (标准差) (3.7E+03) (1.4E+03) (6.8E+03) (3.2E+03) (2.7E+03) (1.7E+03) (6.6E+03) (1.8E+03) VR均值 0.03 0.00+ 0.01= 0.00+ 0.01= 0.00+ 0.00+ 0.00+ F16 GS 均值 0.242 3 74.272+ 52.296+ 132.800+ 52.555+ 81.104+ 114.04+ 67.231+ (标准差) (1.3E−01) (8.2E+03) (8.1E+03) (6.6E+03) (3.2E+03) (9.0E+03) (1.6E+03) (1.6E+03) VR均值 0.15 0.00+ 0.02+ 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ F17 GS 均值 32.566 127.50+ 141.05+ 165.93+ 148.05+ 192.72+ 162.20+ 100.12+ (标准差) (2.0E+04) (2.3E+03) (2.5E+04) (5.7E+03) (2.0E+03) (8.5E+03) (5.2E+03) (3.2E+03) VR均值 0.13 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ F18 GS 均值 −0.267 9 −0.264 2+ −0.260 3+ −0.257 9+ −0.262 5+ −0.254 8+ −0.260 4+ −0.263 0+ (标准差) (1.6E−06) (6.0E−05) (6.8E−05) (9.1E−05) (4.9E−05) (1.6E−04) (1.8E−05) (1.5E−04) VR均值 1.00 1.00= 0.95= 0.95= 1.00= 0.80+ 0.80+ 0.85+ F19 GS 均值 0.399 9 0.529 2+ 0.882 5+ 0.797 9+ 0.776 3+ 0.789 5+ 1.337 5+ 0.885 8+ (标准差) (2.4E−05) (1.9E−01) (1.8E−01) (1.8E−01) (1.8E−01) (1.7E+00) (8.6E−02) (2.0E−01) VR均值 0.60 0.03+ 0.03+ 0.10+ 0.06+ 0.16+ 0.06+ 0.2+ F20 GS 均值 −7.429 9 −6.619 2+ −6.649 6+ −6.664 4+ −6.728 0+ −6.679 1+ −6.420 4+ −6.981 8+ (标准差) (1.7E−02) (2.9E−01) (8.6E−01) (4.0E−01) (3.3E−01) (4.5E−01) (3.8E−01) (4.6E−01) VR均值 0.40 0.26+ 0.26+ 0.26+ 0.20+ 0.29+ 0.27+ 0.28+ 注: 加粗字体表示各组的最优结果值.
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表 13 IMPSO-HES与7种多模态进化算法的统计对比结果
Table 13 Statistical comparison results of IMPSO-HES and the 7 multimodal EAs
问题 IMPSO-HES LIPS EMO-MMO R3PSO FERPSO NCDE NSDE ANDE F1 ~ F5 好/平/差 GS — 6/3/1 7/3/0 10/0/0 8/1/1 10/0/0 10/0/0 10/0/0 Rank 1.300 0 2.800 0 3.000 0 6.100 0 3.300 0 6.300 0 6.800 0 6.700 0 Adjusted p-value — 0.315 3 0.116 0 0.000 2 0.0937 0.000 1 0.000 0 0.000 0 好/平/差 GS — 15/0/0 13/1/1 14/0/1 15/0/0 15/0/0 14/1/0 14/1/0 F6 ~ F20 VR — 11/4/0 10/5/0 11/4/0 11/4/0 14/1/0 12/3/0 13/2/0 Rank 1.258 6 4.827 5 4.268 9 5.551 7 4.603 4 6.103 4 4.862 0 4.224 1 Adjusted p-value — 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0
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表 14 问题的决策变量信息
Table 14 Decision variable information of the problem
决策变量 单位 范围 房屋方向 $( ^{ {\circ} } )$ [0, 360) 窗户的长 m (0, 3.6) 窗户的高 m (0, 3.9) 窗户的传热系数 ${\rm{W} }/({\rm{m} }^{2}\cdot{\rm{K} })$ [2, 6] 窗户的日射热取得率 — (0, 0.7) 墙体外保温层厚度 m (0, 0.1] 墙体日射吸收率 — [0.1, 1] 人员密度 ${{\text{人}}/\rm{m} }^{2}$ [0.1, 1) 照明功率密度 ${\rm{W} }/{\rm{m} }^{2}$ [6, 12] 设备功率密度 $\rm{W}/{\rm{m} }^{2}$ [10, 18] 空调供热设置温度 ℃ [18, 23] 空调制冷设置问题 ℃ [24, 28]
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表 15 处理建筑节能设计问题时两种算法所得的实验结果
Table 15 Results of the two algorithms on building energy conservation
GS Optimal solutions 时间(s) IMPSO-HES 5.02 X = 71.8, 1.06, 1.85, 3.64, 0.0382 ,0.0905 ,0.2212 ,0.1033 , 6.5, 14.0, 22.3, 26.4, f = 5.1450 X = 297.3, 2.53, 1.63, 4.0065 ,0.0556 ,0.0402 ,0.5983 ,0.1027 , 6.0, 17.2, 19.6, 24.0, f = 5.1X = 351.7, 3.50, 0.38, 2.266, 0.1604 ,0.0567 ,0.8882 ,0.1062 , 6.1, 17.3, 22.6, 24.6, f = 5.11EMO-MMO 4.96 X = 183.2, 1.19, 2.36, 2.32, 0.3439 ,0.0489 ,0.9743 ,0.1085 , 6.18, 12.3, 21.1, 26.3, f = 5.0142 357 X = 215.1, 2.41, 2.09, 5.38, 0.2847 ,0.0532 ,0.4720 ,0.1015 , 6.44, 11.8, 19.3, 27.1, f = 5.02X = 134.7, 1.07, 2.87, 3.73, 0.3129 ,0.0418 ,0.9553 ,0.1015 , 6.02, 12.8, 20.4, 25.3, f = 5.02
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