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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[1] Kennedy J, Eberhart R C. Particle swarm optimization. In: Proceedings of the 1995 International Conference on Neural Networks. Perth, Australia: 1995. 1942−1948 [2] 贾承丰, 韩华, 吕亚楠, 张路.基于word2vec和粒子群的链路预测算法[J/OL].自动化学报, 已录用 Jia Cheng-Feng, Han Hua, Lv Ya-Nan, Zhang Lu. Link prediction algorithm based on word2vec and particle swarm. Acta Automatica Sinica, to be published [3] Chou J S, Pham A D Nature-inspired metaheuristic optimization in least squares support vector regression for obtaining bridge scour information. Information Sciences, 2017, 399: 64-80 doi: 10.1016/j.ins.2017.02.051 [4] Tran B, Xue B, Zhang M Variable-Length Particle Swarm Optimization for Feature Selection on High-Dimensional Classification. IEEE Transactions on Evolutionary Computation, 2018, 23(3): 473-487 [5] Mousavi-Avval S H, Rafiee S, Sharifi M, Hosseinpour S, Notarnicola B, Tassielli G, et al. Application of multi-objective genetic algorithms for optimization of energy, economics and environmental life cycle assessment in oilseed production. Journal of Cleaner Production, 2017, 140: 804-815 doi: 10.1016/j.jclepro.2016.03.075 [6] Chen S M, Huang Z C. Multiattribute decision making based on interval-valued intuitionistic fuzzy values and particle swarm optimization techniques. Information Sciences, 2017, 397: 206-218 [7] 吴庆, 赵涛, 佃松宜, 郭锐, 李胜川, 方红帏, 等.基于FPSO的电力巡检机器人的广义二型模糊逻辑控制. 自动化学报, 已录用 Wu Qing, Zhao Tao, Dian Song-Yi, Guo Rui, Li Sheng-Chuang, Fang Hong-Wei, et al. General type-2 fuzzy logic control for a power-line inspection robot based on FPSO. Acta Automatica Sinica, to be published [8] Shi Y, Eberhart R. A modified particle swarm optimizer. In: Proceedings of the 1998 IEEE International Conference on Evolutionary Computation. Anchorage, USA: IEEE, 1998. 69−73 [9] Zhan Z H, Zhang J, Li Y, Chung H S H. Adaptive particle swarm optimization. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 2009, 39(6): 1362- 1381 doi: 10.1109/TSMCB.2009.2015956 [10] Zhang L, Tang Y, Hua C, Guan X. A new particle swarm optimization algorithm with adaptive inertia weight based on Bayesian techniques. Applied Soft Computing, 2015, 28: 138-149 doi: 10.1016/j.asoc.2014.11.018 [11] Ratnaweera A, Halgamuge S K, Watson H C. Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Transactions on Evolutionary Computation, 2004, 8(3): 240-255 doi: 10.1109/TEVC.2004.826071 [12] Chen K, Zhou F, Yin L, Wang S, Wang Y, Wan F. A hybrid particle swarm optimizer with sine cosine acceleration coefficients. Information Sciences, 2018, 422: 218-241 doi: 10.1016/j.ins.2017.09.015 [13] Li S F, Cheng C Y. Particle Swarm Optimization with Fitness Adjustment Parameters. Computers & Industrial Engineering, 2017, 113:831-841 [14] Isiet M, Gadala M. Self-adapting control parameters in particle swarm optimization. Applied Soft Computing, 2019, 83: 105653 doi: 10.1016/j.asoc.2019.105653 [15] Kennedy J. Bare bones particle swarms. In: Proceedings of the 2003 IEEE Swarm Intelligence Symposium. Indianapolis, USA: 2003. 80−87 [16] Wang H, Wu Z, Rahnamayan S, Liu Y, Ventresca M. Enhancing particle swarm optimization using generalized opposition-based learning. Information Sciences, 2011, 181(20): 4699-4714 doi: 10.1016/j.ins.2011.03.016 [17] Yan B, Zhao Z, Zhou Y, Yuan W, Li J, Wu J, et al. A particle swarm optimization algorithm with random learning mechanism and Levy flight for optimization of atomic clusters. Computer Physics Communications, 2017, 219:79-86 doi: 10.1016/j.cpc.2017.05.009 [18] Xia X, Xing Y, Wei B, Zhang Y, Li X, Deng X, et al. A fitness-based multi-role particle swarm optimization. Swarm and Evolutionary Computation, 2019, 44: 349-364 doi: 10.1016/j.swevo.2018.04.006 [19] Liu H R, Cui J C, Lu Z D, Liu D Y, Deng Y J. A hierarchical simple particle swarm optimization with mean dimensional information. Applied Soft Computing, 2019, 76: 712-725 doi: 10.1016/j.asoc.2019.01.004 [20] Kennedy J, Mendes R. Neighborhood topologies in fully informed and best-of-neighborhood particle swarms. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 2006, 36(4): 515-519 doi: 10.1109/TSMCC.2006.875410 [21] Mendes R, Kennedy J, Neves J. The fully informed particle swarm: simpler, maybe better. IEEE Transactions on Evolutionary Computation, 2004, 8(3): 204-210 doi: 10.1109/TEVC.2004.826074 [22] Lim W H, Isa N A M, Particle swarm optimization with adaptive time-varying topology connectivity. Applied Soft Computing, 2014, 24: 623-642 doi: 10.1016/j.asoc.2014.08.013 [23] Lin A, Sun W, Yu H, Wu G, Tang H. Global genetic learning particle swarm optimization with diversity enhancement by ring topology. Swarm and Evolutionary Computation, 2019, 44: 571-583 doi: 10.1016/j.swevo.2018.07.002 [24] Zhang K, Huang Q, Zhang Y. Enhancing comprehensive learning particle swarm optimization with local optima topology. Information Sciences, 2019, 471: 1-18 doi: 10.1016/j.ins.2018.08.049 [25] Alatas B, Akin E, Ozer A B. Chaos embedded particle swarm optimization algorithms. Chaos, Solitons & Fractals, 2009, 40(4): 1715-1734 [26] Kao Y T, Zahara E. A hybrid genetic algorithm and particle swarm optimization for multimodal functions. Applied Soft Computing, 2008, 8(2): 849-857 doi: 10.1016/j.asoc.2007.07.002 [27] Aydilek İ B. A hybrid firefly and particle swarm optimization algorithm for computationally expensive numerical problems. Applied Soft Computing, 2018, 66: 232-249 doi: 10.1016/j.asoc.2018.02.025 [28] Lynn N, Suganthan P N. Ensemble particle swarm optimizer. Applied Soft Computing, 2017, 55: 533-548 doi: 10.1016/j.asoc.2017.02.007 [29] Wang F, Zhang H, Li K, et al. A hybrid particle swarm optimization algorithm using adaptive learning strategy. Information Sciences, 2018, 436: 162-177 [30] Cui L, Li G, Lin Q, Du Z, Gao W, Chen J, et al. A novel artificial bee colony algorithm with depth-first search framework and elite-guided search equation. Information Sciences, 2016, 367: 1012-1044 [31] Liang J J, Qin A K, Suganthan P N, Baskar S. Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Transactions on Evolutionary Computation, 2006, 10(3): 281-295 doi: 10.1109/TEVC.2005.857610 [32] Haklı H, Uğuz H. A novel particle swarm optimization algorithm with Levy flight. Applied Soft Computing, 2014, 23: 333-345 doi: 10.1016/j.asoc.2014.06.034 [33] Liang J J, Suganthan P N. Dynamic multi-swarm particle swarm optimizer. In: Proceedings of the 2005 IEEE Swarm Intelligence Symposium. Pasadena, USA: 2005. 124−129 [34] Jensi R, Jiji G W. An enhanced particle swarm optimization with levy flight for global optimization. Applied Soft Computing, 2016, 43: 248-261 doi: 10.1016/j.asoc.2016.02.018 [35] Gao W, Chan F T S, Huang L, Liu S. Bare bones artificial bee colony algorithm with parameter adaptation and fitness-based neighborhood. Information Sciences, 2015, 316: 180-200 doi: 10.1016/j.ins.2015.04.006 [36] Jadon S S, Bansal J C, Tiwari R, Sharma H. Accelerating artificial bee colony algorithm with adaptive local search. Memetic Computing, 2015, 7(3): 215-230 doi: 10.1007/s12293-015-0158-x [37] Kiran M S, Hakli H, Gunduz M, Uguz H. Artificial bee colony algorithm with variable search strategy for continuous optimization. Information Sciences, 2015, 300: 140-157 doi: 10.1016/j.ins.2014.12.043 [38] Cui L, Zhang K, Li G, Fu X, Wen Z, Lu N, et al. Modified Gbest-guided artificial bee colony algorithm with new probability model. Soft Computing, 2018, 22(7): 2217-2243 doi: 10.1007/s00500-017-2485-y [39] Wang Y, Cai Z, Zhang Q. Differential evolution with composite trial vector generation strategies and control parameters. IEEE Transactions on Evolutionary Computation, 2011, 15(1): 55-66 doi: 10.1109/TEVC.2010.2087271 [40] Zhang J, Sanderson A C. JADE: adaptive differential evolution with optional external archive. IEEE Transactions on Evolutionary Computation, 2009, 13(5): 945-958 doi: 10.1109/TEVC.2009.2014613 [41] Brest J, Maucec, M S. Self-adaptive differential evolution algorithm using population size reduction and three strategies. Soft Computing, 2011, 15(11): 2157-2174 doi: 10.1007/s00500-010-0644-5 [42] Qin A K, Huang V L, Suganthan P N. Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Transactions on Evolutionary Computation, 2009, 13(2): 398-417 doi: 10.1109/TEVC.2008.927706 [43] Hansen N, Ostermeier A. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 2001, 9(2): 159-195 doi: 10.1162/106365601750190398 [44] Khalili M, Kharrat R, Salahshoor K, Sefat M H. Global dynamic harmony search algorithm: GDHS. Applied Mathematics and Computation, 2014, 228: 195-219 doi: 10.1016/j.amc.2013.11.058 [45] El-Abd M. An improved global-best harmony search algorithm. Applied Mathematics and Computation, 2013, 222: 94-106 doi: 10.1016/j.amc.2013.07.020 [46] Rahnamayan S, Tizhoosh H R, Salama M M A. Opposition-based differential evolution. IEEE Transactions on Evolutionary computation, 2008, 12(1): 64-79 doi: 10.1109/TEVC.2007.894200 [47] Zhu G, Kwong S. Gbest-guided artificial bee colony algorithm for numerical function optimization. Applied Mathematics and Computation, 2010, 217(7): 3166-3173 doi: 10.1016/j.amc.2010.08.049 [48] Ouyang H, Gao L, Li S, Kong X Y. Improved global-best-guided particle swarm optimization with learning operation for global optimization problems. Applied Soft Computing, 2017, 52: 987-1008 doi: 10.1016/j.asoc.2016.09.030 [49] Liang J J, Qu B Y, Suganthan P N, Chen Q. Problem Definitions and Evaluation criteria for the CEC 2015 Competition on Learning-based Real-parameter Single Objective Optimization. Technical Report 201411A, Computational Intelligence Laboratory, Zhengzhou University, China, Nanyang Technological University, Singapore, 2014. 29: 625−640 [50] Kora P, Krishna K S R. Hybrid firefly and particle swarm optimization algorithm for the detection of bundle branch block. International Journal of the Cardiovascular Academy, 2016, 2(1): 44-48 doi: 10.1016/j.ijcac.2015.12.001 [51] Arunachalam S, AgnesBhomila T, Babu M R. Hybrid particle swarm optimization algorithm and firefly algorithm based combined economic and emission dispatch including valve point effect. In: Proceedings of the 2014 International Conference on Swarm, Evolutionary, and Memetic Computing. Odisha, India: 2014. 647−660 -
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