Particle Swarm Optimization With Square Wave Triggered Exploration and Exploitation
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摘要: 在粒子群优化算法中, 当勘探时间持续过长, 将可能导致种群在解空间过度徘徊; 种群在开发阶段陷入局部最优后, 难以再次进行全局勘探. 针对上述问题, 提出方波触发勘探与开发的粒子群优化算法. 依据方波的周期特性, 在前半个周期内使用标准粒子群优化算法执行全局勘探, 后半个周期使用改进的更新公式执行局部开发. 经过实验验证, 在方波触发机制下, 通过为粒子提供多变步长, 可达到周期性触发勘探与开发的目的. 使用多类型测试函数, 将该算法与新改进粒子群算法、改进人工蜂群算法、改进差分算法在30、50和100维下比较, 实验结果表明, 该算法在收敛速度和精度上更具优势.Abstract: In the particle swarm optimization, when the exploration time continues to be too long, it may cause the particles to be excessively lingered in the solution space. After the population has fallen into local optimum during the exploitation stage, it is difficult to conduct global exploration again. To solve these two problems, an improved particle swarm optimization called particle swarm optimization with square wave triggered exploration and exploitation is proposed. According to the periodic characteristics of the square wave, the exploration is performed using the standard particle swarm optimization equation in the first half of the period, and the exploitation is performed using the improved update equation in the second half of the period. After experimental verification, under the square wave triggered mechanism, by providing variable step sizes to particles, the purpose of triggering exploration and exploitation periodically is achieved. The square wave triggered exploration and exploitation was compared with other state-of-the-art intelligent algorithms, numerical experiment results on several types of benchmark functions of dimensions 30, 50 and 100 showed that the square wave triggered exploration and exploitation had more advantages in terms of accuracy and convergence speed.
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表 1 Penalized 2函数上使用不同的f值寻找全局最优值的迭代次数
Table 1 Numbers of iterations for finding the global optimum using different values of f on the Penalized 2 function
运行次数 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 1 11704 12273 11981 11625 10146 9985 11252 11198 × 2 2 2 2 2 1780 2 2 2 3056 3 13065 11979 12499 11119 2 9847 10795 9500 2 4 1405 2 3 2 10421 801 2 2 8887 5 2 13336 2 11896 1109 2 11033 × 2 6 14060 2 10895 1364 2 10132 3 9472 × 7 2758 14999 1106 1107 12761 2 1333 2 1222 8 865 2 2 2 2476 2 2 9418 426 9 2 12276 13873 11136 2 2 10078 2 2 10 14962 1407 2 2 10834 2 2 10544 × 11 1436 2109 13018 12386 1034 12742 11571 2 9060 12 2 2 2 2 2 2 2 10181 488 13 2 14577 13927 10019 2 11213 10855 1071 2 14 13346 2 2 1457 13428 2 1397 401 8615 15 1633 12661 11396 2 2 10691 2 2 2 16 2 2 2 12466 13939 2 9906 9438 8822 17 14325 13727 15113 2 2 10756 413 424 1319 18 2 1470 2 12205 12848 930 2 2 2 19 13871 2018 12345 2 2 2 11565 9610 × 20 3 505 2 11701 10586 11815 829 2 × 21 3 3 14717 2 2 978 2 11406 5039 22 2 2 2 2 12163 2 10889 601 529 23 × 14773 13978 11976 3 10870 2 2 517 24 14313 270 2 1368 2 2 10413 11753 639 25 927 2 10368 2 11707 11587 857 2 2 26 830 12428 2 12196 2 2 2 14083 9400 27 2 2 13887 2 11969 10566 12339 2 2 28 × 2 953 11848 2 2 2 9706 8173 29 14813 12621 2 2 12454 11169 9447 2 2 30 2 975 10766 12405 960 533 297 13482 14269 平均 4798 5148 6028 5277 5021 4488 4510 4907 3219 
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表 2 Levy函数上使用不同的f值寻找全局最优值的迭代次数
Table 2 Numbers of iterations for finding the global optimum using different values of f on the Levy function
运行次数 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 1 13874 14277 13198 10780 10421 13557 11056 9483 13787 2 2826 1183 2 1449 2 2 2 2 2 3 2 2 13689 2 11314 10167 13358 2 12695 4 × 13732 2042 11042 2 2 2 10550 2 5 × 2 2686 2 11160 11234 13494 1239 × 6 14837 13975 2 9906 2 2 1578 552 3200 7 5154 2 13005 2 11868 9903 2 2 2 8 2 14063 1244 10620 2 2 10440 9624 13760 9 12041 2 2 1362 × 10850 1376 2 2 10 2 × 12501 610 4469 603 2 2 12647 11 × 12550 2 2 2 2 10836 9370 620 12 12924 2 11695 11196 × 9632 2 2 2 13 2 13450 431 1730 4908 1492 9457 13545 13183 14 2 2 3306 2 783 2 2 2 805 15 × 12873 1861 10321 1413 9453 10773 9836 625 16 13729 3 2 2 2 2 701 2 2 17 2 2 × 12235 12042 10359 2 × × 18 13234 14143 12546 2 2 2 8828 12012 10905 19 2780 2 2 2 10954 11268 621 375 2 20 2 14885 12626 12806 2 776 1180 2 8737 21 13953 2 2 2 2 2 2 × 2 22 4594 13092 13093 14701 11422 × 10400 9022 12969 23 2 4313 2 2 2 1381 2 2 446 24 13667 2 11992 11462 × 1442 9952 11454 2 25 2 14440 2 2 2434 2 2 3 × 26 × 754 13095 2 1446 10102 10023 2 13288 27 × 992 956 × 2 959 2 2 2 28 × 1009 2 11072 14639 2 10721 11631 8908 29 × 2 2 2 636 12115 2 2 2 30 14436 12920 12514 11453 2 2 2 10330 12355 平均 5887 5954 5259 4923 4072 4321 4494 4252 5146
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表 3 26个基准测试函数
Table 3 Information of 26 benchmark functions
函数名 函数式 范围 最优解 Sphere ${f_1}(x) = \displaystyle\sum\limits_{i = 1}^D {{x_i}^2} $ [−100, 100] 0 Schwefel2.22 ${f_2}\left( x \right) = \displaystyle\sum\limits_{i = 1}^D {\left| { {x_i} } \right| + \prod_{i = 1}^D {\left| { {x_i} } \right|} }$ [−10, 10] 0 Rosebrock ${f_3}(x) = \displaystyle\sum\limits_{i = 1}^D {\left[ {100{ {\left( { {x_{i + 1} } - x_i^2} \right)}^2} + { {\left( {1 - x_i^2} \right)}^2} } \right]}$ [−5, 10] 0 Quartic ${f_4}(x) = \displaystyle\sum\limits_{i = 1}^D {i{x_i^4} + random\left[ {0,1} \right)}$ [−1.28, 1.28] 0 Schwefel2.26 ${f_5}(x) = 418.98288727243380{\cdot }D{\rm{ - } }\displaystyle\sum\limits_{i \;=\; 1}^D { {x_i}\sin \sqrt {\left| { {x_i} } \right|} }$$\overline { {f} } _5(x) = \displaystyle\sum\limits_{i = 1}^D { - {x_i}\sin \sqrt {\left| { {x_i} } \right|} },\; (D = 100)$ [−500, 500] 0 ~ 418.9826 × D Rastrigin ${f_6}(x) = \displaystyle\sum\limits_{i = 1}^D {\left[ { {x_i^2} - 10\cos 2\pi {x_i} + 10} \right]}$ [−5.12, 5.12] 0 Ackley ${f_7}(x) = - 20{\rm{exp} }\left( { - 0.2\sqrt {\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D { {x_i^2} } } } \right) - {\rm{exp} }\left( {\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D {\cos 2\pi x_i} } \right) + 20 + {\rm{e}}$ [−50, 50] 0 Griewank ${f_8}(x) = \dfrac{1}{{4000}}\displaystyle\sum\limits_{i = 1}^D {{{\left( {{x_i}} \right)}^2}} - \prod _{i = 1}^D\cos \left( {\dfrac{{{x_i}}}{{\sqrt i }}} \right) + 1$ [−600, 600] 0 Penalized 1 ${f_9}(x) = \dfrac{\pi }{D} \bigg\{ {\displaystyle\sum\limits_{i \;=\; 1}^{D - 1} {({y_i} - 1)} ^2}[1 + \sin (\pi {y_{i + 1} })] + {({y_D} - 1)^2} + 10{\sin ^2}(\pi {y_1}) \bigg\} +$$\displaystyle\sum\limits_{i = 1}^D {u({x_i},10,100,4),{y_i} = 1 + \dfrac{ { {x_i} + 1} }{4} }u({x_i},a,k,m) = \left\{ {\begin{aligned} &u({x_i},a,k,m),&{x_i} > a\quad\qquad \\ &0,& - a \le {x_i} \le a\;\\ &k{ {( - {x_i} - a)}^m},&{x_i} < - a \qquad\end{aligned} } \right.$ [−100, 100] 0 Penalized 2 ${f_{10} }\left( x \right) = 0.1\bigg\{ { { {\sin }^2}\left( {\pi {x_1} } \right) + \displaystyle\sum\limits_{i \;=\; 1}^{D - 1} { { {\left( { {x_i} - 1} \right)}^2}[1 + { {\sin }^2}\left( {3\pi {x_{i + 1} } } \right)]+} }$${\left( { {x_D} - 1} \right)^2} {\left[ {1 + { {\sin }^2}\left( {2\pi {x_{i + 1} } } \right)} \right]} \bigg\} + \displaystyle\sum\limits_{i = 1}^D {u\left( { {x_i},5,100,4} \right)}$ [−100, 100] 0 Rotated Schwefel2.26 ${f_{11} }(x) = 4.18.928\cdot D - \displaystyle\sum\limits_{i \;=\; 1}^D { {z_i} } , where \; {z_i} = \left\{ \begin{aligned} & - {y_i}\sin (\sqrt {|{y_i}|} ) if|{y_i}| \le 500 \\ & 0, \quad {\rm{否则} } \end{aligned} \right.$$ {y_i} = y_i^` + 420.96, y_i^` = M \cdot(x - 420.96), M \;is \; an \; orthogonal \; matrix $ [−500, 500] 0 Rotated Rastrigin ${f_{12} }(x) = \displaystyle\sum\limits_{i \;=\; 1}^D {[y_i^2 - 10\cos (2\pi {y_i}) + 10], where\;y = M \cdot x,\;} M \;is\; an\; orthogonal\; matrix$ [−5.12, 5.12] 0 Rotated Ackly ${f_{13} }(x) = - 20\exp \left[ - 0.2\sqrt {\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D {y_i^2} }\right ] - \exp \left [\dfrac{1}{D}\displaystyle\sum\limits_{i \;=\; 1}^D {\cos (2\pi {y_i})} \right ] + (20 + {\rm{e} })$$where\;y = M \cdot x , M \;is\; an\; orthogonal\; matrix$ [−32, 32] 0 Rotated Griewank ${f_{14} }(x) = \dfrac{1}{ {4000} }\displaystyle\sum\limits_{i = 1}^D {y_i^2 - \prod\limits_{i = 1}^D {\cos (\dfrac{ { {y_i} } }{ {\sqrt i } })} + 1, where\; y = M \cdot x,\;} M \;is\; an \; orthogonal \; matrix$ [−600, 600] 0 Elliptic ${f_{15}}\left( x \right) = {\displaystyle\sum\limits_{i = 1}^D {\left( {{{10}^6}} \right)} ^{\dfrac{{i - 1}}{{D - 1}}}}x_{_i}^2$ [−100, 100] 0 SumSquare ${f_{16}}\left( x \right) = \displaystyle\sum\limits_{i = 1}^D {ix_{_i}^2} $ [−10, 10] 0 SumPower ${f_{17} }\left( x \right) = {\displaystyle\sum\limits_{i \;=\; 1}^D {\left| { {x_i} } \right|} ^{\left( {i + 1} \right)} }$ [−1, 1] 0 Schwefel2.21 ${f_{18} }(x) ={\rm max}\left\{ {\left| { {x_i} } \right|,1 \le i \le D} \right\}$ [−100, 100] 0 Step ${f_{19} }(x) = \displaystyle\sum\limits_{i \;=\; 1}^D {\left\lfloor { {x_i} + 0.5} \right\rfloor }$ [−100, 100] 0 Exponential ${f_{20} }\left( x \right) = \exp \left( {0.5\displaystyle\sum\limits_{i\; =\; 1}^D { {x_i} } } \right)$ [−10, 10] 0 NCRastrigin ${f_{21} }\left( x \right) = \displaystyle\sum\limits_{i\; =\; 1}^D {\left[ {y_i^2 - 10\cos \left( {2\pi {y_i} } \right) + 10} \right]}$ [−5.12, 5.12] 0 Alpine ${f_{22} }\left( x \right) = \displaystyle\sum\limits_{i\; = \;1}^{D - 1} {\left| { {x_i}\sin \left( { {x_i} } \right) + 0.1{x_i} } \right|}$ [−10, 10] 0 Levy ${f_{23}}\left( x \right) = \displaystyle\sum\limits_{i = 1}^{D - 1} {{{\left( {{x_i} - 1} \right)}^2}\left[ {1 + {{\sin }^2}\left( {3\pi {x_{i + 1}}} \right)} \right] + {{\sin }^2}\left( {3\pi {x_1}} \right)} + \left| {{x_D} - 1} \right|\left[ {1 + {{\sin }^2}\left( {3\pi {x_D}} \right)} \right]$ [−10, 10] 0 Weierstrass ${f_{24} }\left( x \right) = \displaystyle\sum\limits_{i \;=\; 1}^D {\left( {\displaystyle\sum\limits_{k \;=\; 0}^{k_{max} } {\left[ { {a^k}\cos \left( {2\pi {b^k}\left( { {x_i} + 0.5} \right)} \right)} \right]} } \right)} - D\displaystyle\sum\limits_{k = 0}^{k_{max} } {\left[ { {a^k}\cos \left( {2\pi {b^k}0.5} \right)} \right]}$$a = 0.5;b = 3;k_{max} = 20$ [−1, 1] 0 Himmelballa ${f_{25}}\left( x \right) = \dfrac{1}{D}\displaystyle\sum\limits_1^D {\left( {x_{_i}^4 - 16x_i^2 + 5{x_i}} \right)} $ [−5, 5] −78.332 Michalewice ${f_{26} }\left( x \right) = - \displaystyle\sum\limits_{i \;=\; 1}^D {\sin \left( { {x_i} } \right){ {\sin }^{20} }\left( {\dfrac{ {ix_i^2} }{\pi } } \right)}$ $\left[ {0,\pi } \right]$ −50, $D=50 $
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表 4 30维实验结果对比
Table 4 Comparison of 30-dimensional test results
函数 指标 CLPSO HPSO-TVAC DMS-PSO LFPSO PSOLF RPSOLF H-PSO-SCAC SWTPSO f1 MBF 8.06 × 10−96 2.83 × 10−33 1.53 × 10−113 4.69 × 10−31 0 0 0 0 SD 3.53 × 10−95 3.19 × 10−33 5.14 × 10−113 2.50 × 10−30 0 0 0 0 f2 MBF 5.50 × 10−57 9.03 × 10−20 2.18 × 102 2.64 × 10−17 0 0 4.09 × 10−217 0 SD 1.92 × 10−56 9.58 × 10−20 1.83 × 102 6.92 × 10−17 0 0 0 0 f3 MBF 4.18 × 101 2.39 × 101 3.49 × 101 2.38 × 101 2.68 × 101 1.01 × 101 2.36 × 101 3.99 SD 3.35 × 101 2.65 × 101 2.76 × 101 3.17 × 10−1 1.03 9.69 × 10−1 1.51 × 10−1 2.22 × 101 f4 MBF 1.74 × 10−3 9.82 × 10−2 6.09 × 10−4 2.41 × 10−3 2.60 × 10−5 6.50 × 10−3 3.94 × 10−237 8.89 × 10−4 SD 7.83 × 10−4 3.26 × 10−2 4.18 × 10−4 8.07 × 10−4 2.10 × 10−5 5.50 × 10−3 0 1.58 × 10−3 f5 MBF 3.82 × 10−4 1.59 × 103 3.21 × 103 1.37 × 103 2.00 × 103 2.85 × 103 6.64 × 103 3.51 × 102 SD 1.28 × 10−7 3.26 × 102 6.51 × 102 6.36 × 102 6.08 × 102 3.79 × 102 2.39 × 103 1.14 × 103 f6 MBF 1.27 × 101 9.43 1.49 × 101 4.54 0 0 0 0 SD 4.22 3.48 3.62 1.03 × 101 0 0 0 0 f7 MBF 1.42 × 10−14 7.29 × 10−14 6.06 × 10−12 1.68 × 10−14 8.88 × 10−16 8.88 × 10−16 8.88 × 10−16 5.88 × 10−16 SD 7.46 × 10−15 3.00 × 10−14 3.90 × 10−13 4.84 × 10−15 0 0 0 0 f8 MBF 3.20 × 10−3 9.75 × 10−3 1.85 × 10−3 8.14 × 10−17 0 0 0 0 SD 4.93 × 10−3 8.33 × 10−3 4.07 × 10−3 4.46 × 10−16 0 0 0 0 f9 MBF 1.36 × 10−33 2.71 × 10−29 0 4.67 × 10−31 1.66 × 10−2 1.98 × 10−32 2.10 × 10−1 1.58 × 10−32 SD 2.82 × 10−33 1.88 × 10−29 0 9.01 × 10−31 1.27 × 10−2 9.23 × 10−33 9.67 × 10−1 5.07 × 10−33 f10 MBF 1.65 × 10−33 2.79 × 10−28 6.16 × 10−35 1.51 × 10−28 9.01 × 10−9 1.65 × 10−32 1.09 1.36 × 10−32 SD 4.03 × 10−33 2.18 × 10−28 2.76 × 10−34 8.00 × 10−28 1.11 × 10−8 4.36 × 10−33 1.42 4.84 × 10−33 f11 MBF 4.39 × 103 5.32 × 103 4.04 × 103 5.51 × 103 1.54 × 103 9.98 × 103 4.48 × 103 6.63 × 103 SD 3.51 × 102 7.00 × 102 5.68 × 102 5.64 × 102 4.28 × 102 7.58 × 10−12 1.73 × 102 1.43 × 103 f12 MBF 8.17 × 101 5.29 × 103 4.20 × 101 1.79 0 0 0 0 SD 1.08 × 101 1.25 × 101 9.74 9.81 0 0 0 0 f13 MBF 5.91 × 10−5 9.29 2.42 × 10−14 1.65 × 10−14 0 0 0 0 SD 6.46 × 10−5 2.07 1.52 × 10−14 5.40 × 10−15 0 0 0 0 f14 MBF 7.69 × 10−5 9.26 × 10−3 1.02 × 10−2 1.48 × 10−3 0 0 0 0 SD 7.66 × 10−5 8.80 × 10−3 1.24 × 10−2 6.17 × 10−3 0 0 0 0
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表 6 100维实验结果对比
Table 6 Comparison of 100-dimensional test results
函数 指标 CLPSO HPSO-TVAC DMS-PSO LFPSO PSOLF RPSOLF H-PSO-SCAC SWTPSO f1 MBF 4.16 × 10−75 5.48 × 10−26 4.89 × 10−77 2.65 × 10−10 0 0 0 0 SD 1.80 × 10−74 2.59 × 10−25 1.40 × 10−76 3.89 × 10−8 0 0 0 0 f2 MBF 3.45 × 10−45 3.77 × 10−15 5.83 × 102 1.02 0 1.22 × 10−252 0 0 SD 1.52 × 10−44 1.20 × 10−14 3.54 × 101 3.21 0 0 0 0 f3 MBF 1.46 × 102 1.34 × 102 1.20 × 102 2.56 × 101 9.04 × 101 9.15 × 101 9.27 × 101 1.03 × 101 SD 4.78 × 101 2.28 × 102 3.62 × 101 5.23 × 10−1 2.57 × 101 1.50 4.31 2.55 × 101 f4 MBF 7.00 × 10−3 1.17 × 10−48 6.40 × 10−3 3.21 × 10−3 2.35 × 10−5 8.68 × 10−3 3.25 × 10−125 6.28 × 10−4 SD 1.53 × 10−3 1.27 × 10−47 2.56 × 10−3 3.24 × 10−3 1.92 × 10−5 5.47 × 10−3 0 1.60 × 10−3 f5 MBF 5.36 2.30 × 103 6.58 × 104 3.56 × 104 2.46 × 104 1.29 × 103 2.66 × 104 6.16 × 102 SD 2.35 × 101 2.88 × 103 5.21 × 101 3.56 × 103 5.75 × 103 2.05 × 103 4.11 × 103 2.08 × 103 f6 MBF 7.02 4.90 × 101 1.95 × 101 3.25 × 101 0 0 0 0 SD 1.00 × 102 4.21 × 101 2.59 × 101 5.68 × 101 0 0 0 0 f7 MBF 2.74 × 10−14 3.21 × 10−12 9.91 × 10−1 3.87 × 10−9 5.89 × 10−16 5.89 × 10−16 5.89 × 10−16 5.88 × 10−16 SD 5.17 × 10−15 4.29 × 10−11 4.43 5.89 × 10−9 0 0 0 0 f8 MBF 3.33 × 10−17 2.71 × 10−3 7.40 × 10−4 3.89 × 10−2 0 0 0 0 SD 7.29 × 10−17 2.32 × 10−2 2.28 × 10−3 5.78 × 10−2 0 0 0 0 f9 MBF 9.33 × 10−3 2.39 × 10−24 1.56 × 10−2 9.56 × 10−2 2.15 × 10−2 1.27 × 10−12 5.47 × 10−1 1.90 × 10−31 SD 2.87 × 10−2 7.19 × 10−24 4.23 × 10−2 3.46 × 10−2 3.26 × 10−2 1.24 × 10−11 7.08 × 10−1 5.09 × 10−30 f10 MBF 1.10 × 10−3 1.74 × 10−21 5.49 × 10−3 8.97 × 10−2 6.32 × 10−3 3.25 × 10−3 4.50 2.80 × 10−30 SD 3.38 × 10−3 7.05 × 10−21 1.26 × 10−2 2.65 × 10−2 2.53 × 10−2 9.57 × 10−2 6.71 3.77 × 10−29 f11 MBF 6.32 × 103 2.85 × 104 1.12 × 104 3.56 × 104 4.18 × 104 3.43 × 104 2.07 × 104 2.99 × 104 SD 2.32 × 102 5.14 × 103 3.65 × 103 3.25 × 103 1.67 × 102 3.57 × 103 8.63 × 102 7.92 × 103 f12 MBF 2.56 × 103 5.47 × 102 9.32 × 102 1.25 × 102 0 0 0 0 SD 5.32 × 102 1.03 × 102 1.35 × 102 6.32 × 101 0 0 0 0 f13 MBF 6.87 × 10−3 1.05 × 10−6 3.56 × 10−6 6.89 × 10−6 0 0 0 0 SD 3.98 × 10−3 2.54 × 10−6 3.89 × 10−5 1.58 × 10−5 0 0 0 0 f14 MBF 2.32 × 10−2 3.89 × 10−3 8.25 × 10−1 9.58 × 10−2 0 0 0 0 SD 1.25 × 10−2 5.89 × 10−3 9.23 × 10−1 2.56 × 10−2 0 0 0 0
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表 5 50维实验结果对比
Table 5 Comparison of 50-dimensional test results
函数 指标 CLPSO HPSO-TVAC DMS-PSO LFPSO PSOLF RPSOLF H-PSO-SCAC SWTPSO f1 MBF 2.28 × 10−85 5.50 × 10−11 8.62 × 10−103 9.17 × 10−17 0 6.57 × 10−201 0 0 SD 3.87 × 10−85 1.42 × 10−10 2.82 × 10−102 3.20 × 10−16 0 0 0 0 f2 MBF 4.47 × 10−52 1.12 × 10−7 3.09 × 102 8.00 × 10−1 0 8.01 × 10−124 4.69 × 10−245 0 SD 1.22 × 10−51 1.76 × 10−7 2.87 × 101 4.44 0 2.36 × 10−122 0 0 f3 MBF 6.97 × 101 1.36 × 102 7.03 × 101 4.41 × 101 4.74 × 101 4.41 × 101 4.71 × 101 7.24 SD 4.45 × 101 5.90 × 102 4.49 × 101 2.82 × 10−1 9.84 × 10−1 4.74 × 101 6.75 5.33 × 101 f4 MBF 3.31 × 10−3 3.78 × 10−23 1.15 × 10−3 2.37 × 10−3 1.60 × 10−5 3.26 × 10−3 8.85 × 10−176 9.08 × 10−4 SD 9.05 × 10−4 1.39 × 10−22 3.92 × 10−4 1.01 × 10−3 1.17 × 10−5 5.36 × 10−3 0 2.68 × 10−3 f5 MBF 9.53 9.83 × 102 9.76 × 103 4.36 × 103 5.07 × 103 5.96 × 102 1.20 × 104 5.38 × 102 SD 4.76 × 101 1.76 × 103 5.62 × 102 1.27 × 103 9.74 × 102 1.43 × 103 2.50 × 103 2.01 × 103 f6 MBF 2.42 × 101 2.31 × 101 2.99 × 101 1.26 × 101 0 0 0 0 SD 6.40 3.06 × 101 6.30 1.61 × 101 0 0 0 0 f7 MBF 1.01 × 10−14 1.53 × 10−4 7.75 × 10−8 6.53 × 10−10 8.88 × 10−16 5.89 × 10−16 8.88×10−16 5.89 × 10−16 SD 3.32 × 10−15 1.91 × 10−3 2.79 × 10−7 2.46 × 10−9 0 0 0 0 f8 MBF 4.93 × 10−4 4.50 × 10−4 7.40 × 10−4 4.86 × 10−3 0 0 0 0 SD 2.20 × 10−3 2.80 × 10−2 2.28 × 10−3 1.36 × 10−2 0 0 0 0 f9 MBF 3.11 × 10−3 2.88 × 10−7 3.11 × 10−3 1.94 × 10−17 3.15 × 10−2 2.26 × 10−14 2.96 × 10−1 1.15 × 10−32 SD 1.39 × 10−2 7.95 × 10−6 1.39 × 10−2 7.21 × 10−17 1.66 × 10−2 3.46 × 10−13 5.03 × 10−1 1.79 × 10−32 f10 MBF 5.49 × 10−4 3.49 × 10−8 8.63 × 10−34 2.39 × 10−4 4.23 × 10−8 9.67 × 10−13 2.21 1.56 × 10−32 SD 2.46 × 10−3 4.85 × 10−7 2.30 × 10−33 1.69 × 103 5.84 × 10−8 1.12 × 10−11 3.38 2.39 × 10−32 f11 MBF 5.96 × 103 1.07 × 104 1.78 × 103 5.72 × 103 1.79 × 103 1.32 × 104 8.71 × 103 1.14 × 104 SD 6.22 × 102 2.85 × 103 6.03 × 102 1.64 × 103 4.44 × 102 2.93 × 103 5.30 × 102 3.27 × 103 f12 MBF 1.41 × 103 2.61 × 102 4.91 × 102 2.56 × 102 0 0 0 0 SD 2.35 × 102 6.73 × 101 1.20 × 102 3.44 × 101 0 0 0 0 f13 MBF 2.07 × 10−4 7.31 × 10−9 1.81 × 10−8 7.97 × 10−9 0 0 0 0 SD 5.52 × 10−3 9.51 × 10−9 4.09 × 10−7 2.99 × 10−8 0 0 0 0 f14 MBF 5.22 × 10−3 4.58 × 10−3 4.11 × 10−1 3.74 × 10−3 0 0 0 0 SD 1.19 × 10−2 6.37 × 10−3 3.72 × 10−1 7.97 × 10−3 0 0 0 0
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表 7 各PSO算法的Friedman 检验结果
Table 7 Friedman test results of each POS algorithm
算法 D = 30 (排名) D = 50 (排名) D = 100 (排名) SWTPSO 2.86 (1) 2.46 (1) 2.61 (1) H-PSO-SCAC 3.86 (4) 3.93 (4) 3.71 (4) RPSOLF 3.79 (3) 3.46 (3) 3.46 (2) PSOLF 3.57 (2) 3.39 (2) 3.68 (3) LFPSO 5.43 7) 5.25 (5) 6.50 (8) DMS-PSO 5.32 (6) 5.96 (7) 6.04 (7) HPSO-TVAC 6.50 (8) 5.57 (6) 5.04 (6) CLPSO 4.68 (5) 5.96 (7) 4.96 (5)
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表 8 SWTPSO与9种相关算法实验结果对比
Table 8 Comparison of experimental results of SWTPSO and nine related algorithms
函数 指标 BABC CoDE JADE jDEscop SaDE AABCLS CMAES ABCVSS MPGABC SWTPSO f1 MBF 1.01 × 10−14 1.16 × 10−37 2.50 × 10−87 1.15 × 10−36 1.48 × 10−57 4.60 × 10−35 1.21 × 10−28 6.68 × 10−23 2.47 × 10−51 0 SD 5.07 × 10−14 2.69 × 10−37 8.62 × 10−87 5.75 × 10−36 6.02 × 10−57 1.67 × 10−35 2.09 × 10−29 3.16 × 10−32 9.52 × 10−51 0 f2 MBF 3.14 × 10−29 1.25 × 10−34 4.43 × 10−78 4.77 × 10−36 8.97 × 10−55 4.48 × 10−33 4.88 × 10−23 1.11 × 10−23 4.54 × 10−49 0 SD 4.95 × 10−29 1.50 × 10−34 2.21 × 10−77 1.85 × 10−35 2.38 × 10−54 2.16 × 10−33 9.20 × 10−24 4.17 × 10−23 1.32 × 10−48 0 f3 MBF 1.23 × 10−13 1.94 × 10−38 2.27 × 10−85 1.80 × 10−41 5.22 × 10−58 2.54 × 10−35 4.22 × 10−27 1.77 × 10−33 6.05 × 10−53 0 SD 6.16 × 10−13 4.08 × 10−38 1.14 × 10−82 1.47 × 10−16 1.49 × 10−57 1.24 × 10−35 9.12 × 10−28 7.88 × 10−33 1.39 × 10−52 0 f4 MBF 2.96 × 10−92 6.20 × 10−143 3.01 × 10−100 3.58 × 10−124 3.38 × 10−80 1.52 × 10−50 6.82 × 10−13 7.21 × 10−41 8.51 × 10−103 0 SD 0 3.15 × 10−142 1.00 × 10−99 1.79 × 10−123 1.59 × 10−79 4.88 × 10−50 5.39 × 10−13 3.56 × 10−40 3.61 × 10−102 0 f5 MBF 2.18 × 10−6 1.21 × 10−20 1.89 × 10−41 1.71 × 10−22 1.71 × 10−38 4.98 × 10−18 3.69 × 10−4 2.89 × 10−18 2.95 × 10−28 1.13 × 10−300 SD 1.06 × 10−5 8.94 × 10−21 8.32 × 10−41 8.46 × 10−22 7.47 × 10−38 1.34 × 10−18 1.84 × 10−3 6.52 × 10−18 4.04 × 10−28 0 f6 MBF 7.20 6.00 × 10−5 8.20 × 10−10 2.52 3.42 × 10−1 1.16 × 10−1 6.00 × 10−15 2.06 2.12 × 101 0 SD 2.78 9.51 × 10−5 9.51 × 10−10 5.61 × 10−1 4.67 × 10−1 1.24 × 10−1 7.45 × 10−16 3.77 × 10−1 4.04 0 f7 MBF 0 0 0 0 0 0 1.60 × 10−1 0 0 0 SD 0 0 0 0 0 0 4.73 × 10−1 0 0 0 f8 MBF 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 2.67 × 10−109 5.40 × 10−66 2.67 × 10−109 2.67 × 10−109 2.96 × 10−109 SD 2.62 × 10−116 9.67 × 10−125 9.60 × 10−125 9.65 × 10−125 3.71 × 10−122 2.77 × 10−119 1.99 × 10−65 3.08 × 10−120 3.06 × 10−122 1.14 × 10−111 f9 MBF 5.74 × 10−2 8.17 × 10−3 2.50 × 10−3 3.91 × 10−3 1.33 × 10−2 1.63 × 10−2 2.80 × 10−1 6.14 × 10−2 5.02 × 10−2 6.61 × 10−4 SD 1.11 × 10−2 2.79 × 10−3 1.54 × 10−3 1.47 × 10−3 3.33 × 10−3 4.68 × 10−3 6.70 × 10−2 1.38 × 10−2 7.98 × 10−3 1.37 × 10−2 f10 MBF 6.29 × 10−2 3.32 × 101 3.19 × 10−1 2.58 × 101 9.45 3.09 1.75 × 10−25 1.09 × 10−1 4.32 × 10−1 8.19 SD 1.24 × 10−1 2.26 × 101 1.10 2.68 × 101 2.08 × 101 1.36 × 101 4.31 × 10−26 2.52 × 10−1 7.44 × 10−1 5.68 × 101 f15 MBF 0 7.34 × 10−1 1.78 × 10−11 1.03 × 10−14 6.77 × 10−1 0 3.89 × 102 0 0 0 SD 0 8.82 × 10−1 1.74 × 10−11 1.31 × 10−14 6.87 × 10−1 0 7.11 × 101 0 0 0 f16 MBF 0 2.28 × 101 2.74 × 10−8 8.02 × 10−2 4.40 × 10−1 8.00 × 10−2 3.78 × 102 0 0 0 SD 0 4.68 2.07 × 10−8 2.77 × 10−1 6.51 × 10−1 2.77 × 10−1 4.90 × 101 0 0 0 f17 MBF 0 2.96 × 10−4 7.88 × 10−4 1.47 × 10−16 6.88 × 10−3 4.44 × 10−18 1.38 × 10−3 3.67 × 10−14 0 0 SD 0 1.48 × 10−3 2.82 × 10−3 2.60 × 10−16 1.22 × 10−2 2.22 × 10−17 3.34 × 10−3 1.84 × 10−13 0 0 f18 MBF 3.78 × 10−12 4.74 4.75 6.18 × 101 3.64 × 10−12 1.41 × 10−11 9.01 × 103 4.66 × 10−12 1.05 × 10−11 5.38 × 102 SD 7.28 × 10−13 2.37 × 101 2.32 × 101 1.69 × 102 0 3.53 × 10−12 1.13 × 103 2.47 × 10−12 3.03 × 10−12 2.01 × 103 f19 MBF 7.50 × 10−15 2.81 × 10−15 6.22 × 10−15 1.43 × 101 1.32 2.65 × 10−14 1.99 × 101 1.70 × 10−14 2.51 × 10−14 5.89 × 10−16 SD 2.49 × 10−15 7.11 × 10−16 0 6.50 4.90 × 10−1 3.48 × 10−15 2.63 × 10−2 5.75 × 10−33 4.55 × 10−15 0 f20 MBF 1.22 × 10−13 9.42 × 10−33 2.49 × 10−3 1.91 × 10−31 3.24 × 10−2 9.42 × 10−33 4.98 × 10−3 1.07 × 10−32 9.42 × 10−33 1.15 × 10−32 SD 6.09 × 10−13 1.40 × 10−48 1.24 × 10−2 5.28 × 10−31 9.01 × 10−2 1.40 × 10−48 1.72 × 10−2 5.74 × 10−33 1.40 × 10−48 1.79 × 10−32 f21 MBF 2.50 × 10−15 1.55 × 10−33 1.01 × 10−4 5.02 × 10−32 1.09 × 10−2 1.50 × 10−33 8.01 × 103 1.80 × 10−33 1.50 × 10−33 1.56 × 10−32 SD 1.25 × 10−14 2.47 × 10−34 8.05 × 10−5 5.95 × 10−32 3.00 × 10−2 0 6.25 × 103 1.08 × 10−33 0 2.39 × 10−32 f22 MBF 2.07 × 10−16 4.39 × 10−3 1.01 × 10−4 2.95 × 10−5 1.75 × 10−16 5.32 × 10−11 8.59 × 10−1 2.14 × 10−16 3.21 × 10−7 3.10 × 10−290 SD 7.79 × 10−16 6.44 × 10−3 8.05 × 10−5 3.09 × 10−5 2.63 × 10−16 9.68 × 10−11 8.49 × 10−1 7.43 × 10−16 4.58 × 10−7 0 f23 MBF 1.35 × 10−31 1.35 × 10−31 1.35 × 10−31 1.30 × 10−30 7.39 × 10−2 1.35 × 10−31 3.77 × 10−1 1.35 × 10−31 1.35 × 10−31 1.78 × 10−31 SD 2.23 × 10−47 2.47 × 10−33 2.23 × 10−47 3.69 × 10−30 1.07 × 10−1 2.23 × 10−47 1.22 2.23 × 10−47 2.23 × 10−47 2.83 × 10−31 f24 MBF 0 3.45 3.33 × 10−1 0 4.17 × 10−4 2.56 × 10−6 9.41 0 1.56 × 10−2 0 SD 0 2.94 × 10−1 4.45 × 10−2 0 5.94 × 10−4 7.39 × 10−6 4.19 0 2.04 × 10−2 0 f25 MBF −7.83 × 101 −7.83 × 101 −7.83 × 101 −7.83 × 101 −7.83 × 101 −7.83 × 101 −6.43 × 101 −7.83 × 101 −7.83 × 101 −6.26 × 101 SD 1.16 × 10−14 4.10 × 10−15 2.26 × 10−1 1.12 × 10−13 1.13 × 10−1 4.10 × 10−15 2.63 1.00 × 10−14 1.12 × 10−14 3.31 f26 MBF −5.00 × 101 −4.86 × 101 −4.98 × 101 −5.00 × 101 −4.83 × 101 −5.00 × 101 −4.10 × 101 −5.00 × 101 −5.00 × 101 −3.99 × 101 SD 6.79 × 10−6 4.45 × 10−1 3.86 × 10−2 9.23 × 10−3 2.70 × 10−1 9.13 × 10−4 2.65 3.82 × 10−7 4.26 × 10−4 2.13 × 101
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表 9 8种新相关算法和SWTPSO的Friedman测试结果
Table 9 Friedman test results of 8 new correlation algorithms and SWTPSO
算法 SWTPSO MPGABC AABCLS JADE ABCVSS BABC CoDE jDEscop SaDE CMAES 平均值 (排名) 3.86 (1) 4.41 (2) 5.11 (3) 5.05 (4) 5.07 (5) 5.11 (6) 5.59 (7) 5.91 (8) 6.07 (9) 8.82 (10)
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表 10 6种算法在15个函数上的比较结果
Table 10 Results of the fifteen functions for six algorithms
函数 指标 GDHS ODE IGHS GABC IGPSO SWTPSO f1 MBF 8.07 × 10−3 5.67 × 10−81 2.14 × 10−10 8.58 × 102 4.45 × 10−156 0 SD 8.87 × 10−4 9.55 × 10−81 1.22 × 10−11 1.52 × 103 9.29 × 10−156 0 f2 MBF 7.20 × 10−1 3.60 × 10−6 3.96 × 10−1 8.47 × 101 0 0 SD 4.16 × 10−2 1.13 × 10−5 3.07 × 10−1 3.06 × 101 0 0 f3 MBF 6.17 1.81 × 10−5 2.72 × 10−6 8.49 × 104 0 0 SD 7.70 × 10−1 4.69 × 10−5 9.49 × 10−7 3.51 × 104 0 0 f4 MBF 6.71 × 10−2 9.27 × 10−2 3.53 5.34 × 101 0 0 SD 1.05 × 10−2 2.33 × 10−2 2.67 × 10−1 1.93 × 101 0 0 f5 MBF 1.07 × 102 9.69 × 10−12 1.32 × 102 6.07 × 105 4.18 × 10−2 1.03 × 101 SD 2.39 × 101 3.06 × 10−11 4.49 × 101 8.23 × 105 2.86 × 10−2 2.55 × 101 f6 MBF 0 0 0 2.35 × 103 0 0 SD 0 0 0 2.90 × 103 0 0 f7 MBF 2.87 × 10−3 1.83 × 10−3 1.80 × 10−2 8.57 × 10−1 1.40 × 10−4 6.28 × 10−4 SD 5.74 × 10−4 2.02 × 10−3 2.17 × 10−3 6.27 × 10−1 1.45 × 10−4 1.60 × 10−3 f8 MBF −4.19 × 104 −4.17 × 104 −3.99 × 104 −4.20 × 104 −4.19 × 104 −3.34 × 104 SD 4.57 × 10−2 4.35 × 10−2 9.76 × 10−1 4.57 × 10−2 1.33 × 10−2 1.28 × 104 f9 MBF 1.16 × 10−2 4.52 × 10−8 4.41 × 10−8 4.64 × 101 0 0 SD 1.14 × 10−3 1.41 × 10−7 3.47 × 10−9 1.63 × 101 0 0 f10 MBF 1.44 × 10−2 1.72 × 10−4 5.91 × 10−6 7.03 3.55 × 10−15 5.88 × 10−16 SD 6.71 × 10−4 5.44 × 10−4 1.63 × 10−7 2.47 0 0 f11 MBF 4.50 × 10−3 7.40 × 10−4 2.28 × 10−3 2.14 × 101 0 0 SD 5.18 × 10−4 2.28 × 10−3 4.27 × 10−3 3.00 × 101 0 0 f12 MBF 6.83 × 10−6 3.49 × 10−25 5.26 × 10−13 2.93 × 10−1 1.50 × 10−7 1.90 × 10−31 SD 6.46 × 10−7 8.66 × 10−25 4.25 × 10−14 9.27 × 10−1 3.70 × 10−8 5.09 × 10−30 f13 MBF 3.33 × 10−4 1.07 × 10−15 1.86 × 10−6 2.04 × 105 1.75 × 10−5 2.80 × 10−30 SD 4.66 × 10−5 5.60 × 10−6 5.89 × 10−6 4.44 × 105 5.60 × 10−6 3.77 × 10−29 f14 MBF 1.94 × 101 1.23 × 101 4.06 × 101 3.17 × 101 0 0 SD 1.20 7.70 7.70 5.15 0 0 f15 MBF 1.33 × 10−1 4.29 × 102 3.15 × 101 2.18 × 105 0 0 SD 1.61 × 10−2 8.19 × 102 6.83 × 101 2.42 × 104 0 0
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表 11 6种算法的Friedman检验结果
Table 11 Friedman test results of 6 algorithms
算法 平均值 (排名) SWTPSO 1.83 (1) IGPSO 1.97 (2) ODE 3.13 (3) IGHS 3.93 (4) GDHS 4.20 (5) GABC 5.93 (6)
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表 12 CEC2015函数集
Table 12 CEC2015 test suite
函数序号 函数类型 描述 最优值 1 单峰函数 Bent Cigar 旋转函数 100 2 Discus 旋转函数 200 3 简单多峰函数 Schwefel 偏移旋转函数 300 4 Schwefel偏移旋转函数 400 5 Katsuura 偏移旋转函数 500 6 HappyCat 偏移旋转函数 600 7 HGBat 偏移旋转函数 700 8 Griewank + Rosenbrocl 扩展偏移旋转函数 800 9 Schffer＇s F6 扩展偏移旋转函数 900 10 混合函数 混合函数 1 (N = 3) 1000 11 混合函数 2 (N = 4) 1100 12 混合函数 3 (N = 5) 1200 13 组合函数 组合函数 1 (N = 5) 1300 14 组合函数 2 (N = 3) 1400 15 组合函数 3 (N = 5) 1500
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表 13 CEC2015实验结果对比
Table 13 Test results comparison of CEC2015 test suite
函数符号 PSO FA FFPSO HPSOFF HFPSO SWTPSO 1 MBF 3.9049 × 109 2.8899 × 1010 9.2383 × 1010 4.7539 × 109 1.1795×109 3.0501 × 108 2 MBF 9.9760 × 104 1.3418 × 105 6.9430 × 106 9.7376 × 104 8.5653 × 104 5.6540 × 104 3 MBF 3.3113 × 102 3.3850 × 102 3.4771 × 102 3.3059 × 102 3.2638 × 102 3.2631 × 102 4 MBF 7.7928 × 103 8.0330 × 103 9.6696 × 103 6.8199 × 102 5.1202×103 5.8445×103 5 MBF 5.0418 × 102 5.0425 × 102 5.0586 × 102 5.0422 × 102 5.0410 × 102 5.0403 × 102 6 MBF 6.0096 × 102 6.0410 × 102 6.0755 × 102 6.0090 × 102 6.0076 × 102 6.0053 × 102 7 MBF 7.0405 × 102 7.6997 × 102 8.9401 × 102 7.0750 × 102 7.0074 × 102 7.0046 × 102 8 MBF 4.0013 × 103 2.3491 × 106 1.6032 × 108 1.7191 × 104 2.6354 × 103 1.5070 × 103 9 MBF 9.1358 × 102 9.1381 × 102 9.1427 × 102 9.1365 × 102 9.1337 × 102 9.1328 × 102 10 MBF 7.5602 × 106 2.9938 × 107 3.9391 × 108 1.1337 × 107 5.4690 × 106 1.1662 × 107 11 MBF 1.1614 × 103 1.2889 × 103 2.1094 × 103 1.1551 × 103 1.1336 × 103 1.1267 × 103 12 MBF 2.2417 × 103 2.9655 × 103 4.9876 × 105 2.0617 × 103 1.7752 × 103 1.7819 × 103 13 MBF 1.7719 × 103 1.9596 × 103 3.6329 × 103 1.7390 × 103 1.6866×103 1.6552 × 103 14 MBF 1.6644 × 103 1.7522 × 103 2.1683 × 103 1.6711 × 103 1.6469 × 103 1.6510 × 103 15 MBF 2.5522 × 103 2.8256 × 103 3.9013 × 103 2.6529 × 103 2.4467 × 103 2.4639 × 103
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表 14 Friedman检验结果
Table 14 Results of Friedman test
算法 秩均值 (排名) SWTPSO 1.53 (1) HFPSO 1.73 (2) HPSOFF 3.33 (3) PSO 3.40 (4) FA 5.00 (5) FFPSO 6.00 (6)
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