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摘要: 针对花朵授粉算法（Flower pollination algorithm，FPA）易陷入局部极值、后期收敛速度慢的不足，提出一种基于引力搜索机制的花朵授粉算法.该算法在基本花朵授粉算法的全局寻优部分，采用花朵个体间的万有引力和算法本身的莱维飞行共同实现个体位置的更新，使花朵受莱维飞行和个体间引力的双重影响，个体在通过优化信息的共享向质量大（最优位置）的个体靠近，且个体间的万有引力牵制莱维飞行的随机游走.同时又利用莱维飞行的跳跃及不均匀性步长避免个体陷入局部极值，从而提高算法的寻优能力.通过对高维单峰函数、高维多峰函数、低维函数及多峰复杂函数的优化实验结果表明，改进算法的寻优性能显著优于基本的花朵授粉算法，其收敛速度、收敛精度、鲁棒性均较对比算法有较大提升.最后，利用改进算法对弹簧张力设计问题、压力管设计问题2个工程实例进行测试，获得了较好的结果.仿真实验结果佐证了改进算法的有效性和可行性.Abstract: A flower pollination algorithm (FPA) based on gravity search mechanism is presented to overcome the problems of being easily trapped into local extremum and low speed of convergence. In the global optimization of FPA, the algorithm uses the gravity between individuals and the Lévy flight of algorithm itself to update individual locations. The flower is influenced by both gravity and Lévy flight, and individual is close to the individual which is in a higher quality (optimal position) through optimization of the information, and the random walk of Lévy flight is contained by the gravity between individuals. At the same time, the improved algorithm uses the jump and irregular step length of Lévy to limit the individual into local extremum so as to improve the algorithm's optimization ability. High dimensional unimodal function, high dimensional multimodal function, low dimensional function and multi peak complex function optimization results show that the improved algorithm has better global searching ability than the basic flower pollution algorithm, and faster convergence and more precise convergence than those of comparison algorithm. Finally, the improved algorithm is applied to two engineering examples, the tension spring design problem and the pressure vessel design problem, and obtains desired outcomes. Simulation results have proven the effectiveness and feasibility of the improved algorithm.
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Key words:
- Flower pollination algorithm (FPA) /
- optimization ability /
- gravity /
- fitness
1) 本文责任编委 刘艳军 -
图 2 5种算法在部分函数上的适应度值收敛曲线
Fig. 2 Convergence curves of the fitness value of 5 kinds of algorithms on partial functions
图 3 5种算法在部分函数上的全局最优值方差分析
Fig. 3 Anova test of minimum for 5 kinds of algorithms on partial functions
图 4 5种算法在部分函数上的最优适应度值比较图
Fig. 4 Comparison of optimal fitness for 5 kinds of algorithms on partial functions
图 5 ${{f}_{6}}\sim{{f}_{8}}$ 函数优化均值随函数维数变化曲线
Fig. 5 ${{f}_{6}}\sim{{f}_{8}}$ function optimization with function dimension change curves
表 1 基准测试函数
Table 1 Benchmark test function
类型 测试函数 搜索空间 理论最优值 ${{f}_{1}}(x)=\sum\limits_{i=1}^{n}{x_{i}^{2}}$ [-100, 100] 0 高维 ${{f}_{2}}(x)=\sum\limits_{i=1}^{n}{\left| {{x}_{i}} \right|}+\prod\limits_{i=1}^{n}{\left| {{x}_{i}} \right|}$ [-10, 10] 0 单峰 ${{f}_{3}}(x)=\sum\limits_{i=1}^{n}{{(\sum\limits_{j-1}^{i}{{{x}_{j}}})^{2}}}$ [-100, 100] 0 函数 ${{f}_{4}}(x)={{\max }_{i}}\{\left| {{x}_{i}} \right|,1\le i\le n\}$ [-100, 100] 0 Ⅰ ${{f}_{5}}(x)=\sum\limits_{i=1}^{n}{ix_{i}^{4}}+{\rm random}[0,1)$ [-1.28, 1.28] 0 ${{f}_{6}}(x)=\sum\limits_{i=1}^{n}{[x_{i}^{2}-10\cos (2\pi {{x}_{i}})+10]}$ [-5.12, 5.12] 0 高维 ${{f}_{7}}(x)=-20\exp \left(-0.2\sqrt{\frac{1}{n}\sum\limits_{i}^{n}{x_{i}^{2}}}\right)-\exp (\frac{1}{n}\sum\limits_{i}^{n}{\cos (2\pi {{x}_{i}}}))+20+\,{\rm e}$ [-32, 32] 0 多峰 ${{f}_{8}}(x)=1+\frac{1}{4\,000}\sum\limits_{i=1}^{n}{x_{i}^{2}}-\prod\limits_{i=1}^{n}{\cos (\frac{{{x}_{i}}}{\sqrt{i}})}$ [-600, 600] 0 函数 ${{f}_{9}}(x)=\sum\limits_{i=1}^{n}{\left| {{x}_{i}}\sin ({{x}_{i}})+0.1{{x}_{i}} \right|}$ [-10, 10] 0 Ⅱ ${{f}_{10}}(x)=\left\{ \left[ \sum\limits_{i=1}^{n}{{{\sin }^{2}}({{x}_{i}})} \right]-\exp (-\sum\limits_{i=1}^{n}{x_{i}^{2}}) \right\} \exp \left[ -\sum\limits_{i=1}^{n}{{{\sin }^{2}}\sqrt{\left| {{x}_{i}} \right|}} \right]$ [-10, 10] -1 ${{f}_{11}}(x)=\frac{\pi }{D}{{\{10\sin (\pi {{y}_{1}})+ \sum\limits_{i=1}^{D-1}{({{y}_{i}}}-1)}^{2}}[1+10{{\sin }^{2}}(\pi {{y}_{i+1}})]+{{({{y}_{n}}-1)}^{2}}\}+\\ \;\;\;\;\;\;\; \sum\limits_{i=1}^{n}{u({{x}_{i}}},10,100,4)$ ${y_i} = 1 + \frac{{{x_i} + 1}}{4},u({x_i},a,k,m) = \left\{ \begin{array}{l} k{({x_i} - a)^m}{x_i} > a\\ 0 - a < {x_i} < a\\ k{( - {x_i} - a)^m}{x_i} < a \end{array} \right.$ [-50, 50] 0 ${{f}_{12}}(x)=0.1\{{\sin}^{2}(3\pi {{x}_{1}})+\sum\limits_{j=1}^{n-1}({x}_{i}-1)^{2} [1+10{{\sin}^{2}}(\pi {{x}_{i+1}})]+\\ \;\;\;\;\;\; {{({{x}_{n}}-1)}^{2}}[1+{{\sin}^{2}}(2\pi {{x}_{n}})]\}+ \sum\limits_{i=1}^{n}{u({{x}_{i}},5,100,4)}$ [-50, 50] 0 低维 ${{f}_{13}}(x)=\dfrac{{{\sin }^{2}}\sqrt{{{x}_{1}}^{2}+{{x}_{2}}^{2}}-0.5}{{{[1+0.001({{x}_{1}}^{2}+{{x}_{2}}^{2})]}^{2}}}-0.5$ [-100, 100] -1 函数 ${{f}_{14}}(x)=-\dfrac{1+\cos (12\sqrt{x_{1}^{2}+x_{2}^{2}})}{0.5(x_{1}^{2}+x_{2}^{2})+2}$ [-5.12, 5.12] -1 Ⅲ ${{f}_{15}}(x)=100{{(x_{1}^{2}-{{x}_{2}})}^{2}}+{{(1-{{x}_{1}})}^{2}}$ [-2.048, 2.048] 0 ${{f}_{16}}(x)=4{{x}_{1}}^{2}-2.1{{x}_{1}}^{4}+({{x}_{1}}^{6})/3+{{x}_{1}}{{x}_{2}}-4{{x}_{2}}^{2}+4{{x}_{2}}^{4}$ [-5, 5] -1.0316 
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表 2 高维单峰测试函数的寻优性能比较
Table 2 Comparison of optimization performance of high dimensional single peak test function
函数 维数 算法 最优值 优化均值 最差值 标准方差 寻优成功率 (%) ABC 9.2427E-05 4.8174 66.7505 14.1214 24 PSO 0.1180 0.2232 0.3785 0.0605 0 ${{f}_{1}}$ 30 DEBA 5.2325E-04 0.0014 0.0028 5.4156E-04 18 FPA 1.1778E+03 2.2957E+03 3.4901E+03 646.8865 0 GSFPA 0 0 0 0 100 ABC 0.0152 0.6900 2.5079 0.8117 0 PSO 1.4780 3.3433 9.5861 1.6890 0 ${{f}_{2}}$ 30 DEBA 0.0025 0.0052 0.0106 0.0015 0 FPA 13.5573 23.4681 36.6924 5.3017 0 GSFPA 0 0 0 0 100 ABC 1.6156E+04 3.0746E+04 4.1250E+04 5.3292E+03 0 PSO 88.8952 683.7598 4.8081E+03 936.5027 0 ${{f}_{3}}$ 30 DEBA 2.2154E+04 3.4653E+04 4.8471E+04 7.6307E+03 0 FPA 584.5559 1.3876E+03 2.6430E+03 538.4057 0 GSFA 0 0 0 0 100 ABC 47.6764 65.2530 78.1880 7.0160 0 PSO 1.0993 4.1731 9.0819 1.4612 0 ${{f}_{4}}$ 30 DEBA 8.1257 15.6942 81.0078 10.8027 0 FPA 13.9639 20.7465 29.1326 3.4006 0 GSFPA 0 0 0 0 100 ABC 0.1712 0.4903 0.8525 0.1273 0 PSO 0.0637 0.1832 0.3938 0.0746 0 ${{f}_{5}}$ 30 DEBA 0.0433 0.0828 0.1603 0.0213 0 FPA 0.1585 0.3661 1.0155 0.1922 0 GSFPA 2.2586E-06 1.8984E-04 7.0389E-04 1.5126E-04 100
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表 3 高维多峰测试函数的寻优性能比较
Table 3 Comparison of optimization performance of high dimensional multi peak test function
函数 维数 算法 最优值 优化均值 最差值 标准方差 寻优成功率 (%) ABC 6.6684 39.1469 60.9988 8.2963 0 PSO 27.3446 47.8415 70.8488 11.7886 0 ${{f}_{6}}$ 30 DEBA 34.7258 42.5556 55.3031 5.0813 0 FPA 143.1259 175.2331 216.0925 16.1655 0 GSFPA 0 0 0 0 100 ABC 3.2920 5.6133 7.9734 1.1488 0 PSO 4.3528 7.3247 10.4073 1.4364 0 ${{f}_{7}}$ 30 DEBA 0.1303 3.5707 16.8599 4.3272 0 FPA 3.8093 7.2448 11.4055 1.6964 0 GSFPA 8.8818E-16 8.8818E-16 8.8818E-16 0 100 ABC 0.0356 0.4391 2.2319 0.5126 0 PSO 187.3601 254.1564 330.4583 24.6100 0 ${{f}_{8}}$ 30 DEBA 0.0011 0.0113 0.0338 0.0075 0 FPA 9.0459 19.8711 34.9888 5.8755 0 GSFA 0 0 0 0 100 ABC 0.0561 1.3706 2.7080 0.6986 0 PSO 0.1329 1.7424 5.6551 1.1923 0 ${{f}_{9}}$ 30 DEBA 0.0139 0.0263 0.0761 0.0105 0 FPA 11.6981 18.6693 24.3616 2.6186 0 GSFPA 0 0 0 0 100 ABC 2.1281E-13 3.3409E-13 4.4923E-13 6.2213E-14 0 PSO 1.1895E-14 2.6204E-14 9.4151E-14 1.2864E-14 0 ${{f}_{10}}$ 30 DEBA 9.0212E-12 1.1764E-09 1.3729E-08 2.2561E-09 0 FPA 4.9276E-11 3.9801E-10 1.4113E-09 2.7759E-10 0 GSFPA -1 -1 -1 0 100 ABC 1.6714E-05 0.3305 1.7371 0.5093 46 PSO 1.4931 4.0850 6.7058 1.1217 0 ${{f}_{11}}$ 30 DEBA 0.0131 0.1532 0.7424 0.1544 0 FPA 17.0920 1.2323E+04 1.0090E+05 2.4877E+04 0 GSFPA 0.5621 1.0204 1.6052 0.2623 0 ABC 3.1508E-05 0.7497 4.8903 1.4447 32 PSO 0.1163 16.4369 44.1686 12.3563 0 ${{f}_{12}}$ 30 DEBA 0.1822 0.5158 2.5593 0.3937 0 FPA 2.2491E+04 6.8604E+05 3.5217E+06 6.9315E+05 0 GSFPA 2.9911 2.9964 2.9998 0.0024 0
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表 4 低维多峰测试函数的寻优性能比较
Table 4 Comparison of optimization performance of low dimensional multi peak test function
函数 维数 算法 最优值 优化均值 最差值 标准方差 寻优成功率 (%) ABC -0.999999999999991 -0.9947280936 -0.9902752472 0.0046 26 PSO -0.999999985912173 -0.9909729558 -0.8217776974 0.0270 54 ${{f}_{13}}$ 2 DEBA -1 -0.9902840887 -0.9902839963 0.0041 86 FPA -0.9902840887 -0.9902840887 -0.9902840887 0.0038 0 GSFPA -1 -1 -1 0 100 ABC -1 -0.9958895987 -0.9362452095 0.0133 70 PSO -0.999999996387299 -0.9987217811 -0.9362453268 0.0090 98 ${{f}_{14}}$ 2 DEBA -1 -0.9999933301 -0.9996665051 4.7163E-05 100 FPA -0.9999993778 -0.9992595211 -0.9884197110 1.8405E-03 84 GSFPA -1 -1 -1 0 100 ABC 3.2521E-05 0.0213 0.2329 0.0401 22 PSO 5.2032E-08 7.7110E-06 7.0335E-05 1.3433E-05 100 ${{f}_{15}}$ 2 DEBA 6.0310E-122 0.0049894963 0.2494748168 0.0352810669 98 FPA 4.4709E-14 1.2759E-08 5.0346E-07 7.1014E-08 100 GSFA 0 0 0 0 100 ABC -1.0316284535 -1.0316284534 -1.0316284535 4.6186E-16 100 PSO -1.0316284501 -1.0316279534 -1.0316207215 1.3119E-06 100 ${{f}_{16}}$ 2 DEBA -1.0316284535 -1.0316284535 -1.0316284535 2.3738E-16 100 FPA -1.0316284528 -1.0316283539 -1.0316276702 1.4887E-07 100 GSFPA -1.031628172 -1.0316011959 -1.0315001067 3.0222E-05 92
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表 5 多峰函数在不同高维上的优化均值比较
Table 5 Comparison of optimal mean for multi peak functions on different high dimensions
维数 函数 算法 50 100 150 200 250 300 平均变化率 (%) ${{f}_{6}}$ FPA 3.48071E+02 8.1029E+02 1.2901E+03 1.7671E+03 2.2710E+03 2.7899E+03 0.5607 GSFPA 0 0 0 0 0 0 0 ${{f}_{7}}$ FPA 6.8115 7.8408 8.1632 8.4016 8.1426 8.0703 0.0363 GSFPA 8.8818E-16 8.8818E-16 8.8818E-16 8.8818E-16 8.8818E-16 8.8818E-16 0 ${{f}_{8}}$ FPA 4.7387E+01 1.3265E+02 1.905E+02 2.6661E+02 3.4341E+02 4.1927E+02 0.6231 GSFPA 0 0 0 0 0 0 0
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表 6 7种群智能算法对基准函数的优化性能比较
Table 6 Comparison of optimization performance of the 7 swarm intelligence algorithms on the benchmark functions
函数 PSO-RM FETLBO PSO_CG DMPSO MEABC DDIFPA GSFPA ${{f}_{1}}$ Mean 0 3.43E-231 0 2.602520E-133 4.85E-40 4.62E-289 0 Std.Dve 0 3.21E-231 - 1.389667E-132 2.31E-40 0 0 ${{f}_{2}}$ Mean 0 4.16E-116 - 6.434905E-11 1.25E-21 0 2.8396E-223 Std.Dve 0 1.07E-115 - 1.771972E-10 3.56E-21 0 0 ${{f}_{3}}$ Mean 0 0 2.6132E-07 3.240222E-59 9.81E+03 4.44E-15 0 Std.Dve 0 0 - 2.268155E-58 2.49E+03 0 0 ${{f}_{5}}$ Mean - - - - 2.29E-02 3.7294E-03 9.1393E-04 Std.Dve - - - - 1.38E-02 9.9611E-04 7.4609E-04 ${{f}_{6}}$ Mean 3.22E-14 - 19.6502 3.979836 0 0 0 Std.Dve 8.66E-14 - - 13.49628 0 0 0 ${{f}_{7}}$ Mean 1.13E-14 4.44E-15 0.1737 1.140421E-14 2.90E-14 4.44E-15 8.8818E-16 Std.Dve 9.12E-15 0 - 8.672115E-15 1.32E-14 0 0 ${{f}_{8}}$ Mean 0.0094 0 4.7268E-13 3.774758E-17 0 0 0 Std.Dve 0.0096 0 - 9.049510E-17 0 0 0 ${{f}_{11}}$ Mean - - - - 3.02E-17 1.57E-32 1.2855 Std.Dve - - - - 0 2.89E-48 0.0574
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表 7 5种群智能算法对复合基准函数的寻优性能对比
Table 7 Comparison of optimization performance of the 5 swarm intelligence algorithms on the composite benchmark function
函数 算法 平均值 标准差 最优值 最差值 Wilcoxon ABC 1.29E+02 8.86E+01 5.22E-01 2.25E+02 + ASFA 4.80E+02 1.15E+02 2.99E+02 6.69E+02 + CF1 PSO 2.00E+02 1.05E+02 1.53E-01 3.00E+02 + FPA 1.98E+02 7.44E+01 1.06E+02 3.12E+02 + GSFPA 1.28E+02 3.50E+01 3.80E+01 1.86E+02 ABC 1.49E+02 4.94E+01 7.78E+01 2.26E+02 + ASFA 5.49E+02 1.08E+02 4.03E+02 7.19E+02 + CF2 PSO 1.76E+02 8.59E+01 1.02E+02 3.89E+02 + FPA 2.27E+02 1.82E+02 5.78E+01 6.40E+02 + GSFPA 1.19E+02 5.26E+01 7.42E+01 2.36E+02 ABC 2.90E+02 7.53E+01 1.51E+02 4.03E+02 - ASFA 1.03E+03 1.39E+02 8.18E+02 1.24E+03 + CF3 PSO 6.27E+02 2.32E+02 2.69E+02 9.02E+02 $\approx$ FPA 4.81E+02 9.02E+01 3.00E+02 5.80E+02 + GSFPA 4.35E+02 6.10E+01 3.29E+02 5.28E+02 ABC 4.98E+02 9.10E+01 3.93E+02 6.26E+02 + ASFA 9.99E+02 9.35E+01 8.41E+02 1.14E+03 + CF4 PSO 5.67E+02 7.01E+01 4.18E+02 6.50E+02 $\approx$ FPA 5.27E+02 1.63E+02 3.72E+02 7.76E+02 + GSFPA 4.95E+02 3.58E+01 4.41E+02 5.67E+02 ABC 1.20E+02 1.19E+02 4.38E+01 4.13E+02 - ASFA 5.56E+02 2.12E+02 1.95E+02 9.74E+02 + CF5 PSO 1.75E+02 1.78E+02 4.18E+00 5.04E+02 + FPA 1.90E+02 1.03E+02 4.89E+01 3.81E+02 + GSFPA 1.29E+02 6.19E+01 6.80E+01 2.61E+02 ABC 7.80E+02 1.82E+02 5.11E+02 9.07E+02 + ASFA 9.25E+02 2.11E+01 9.03E+02 9.65E+02 + CF6 PSO 9.02E+02 8.42E-01 9.00E+02 9.03E+02 + FPA 7.30E+02 1.76E+02 4.66E+02 9.07E+02 + GSFPA 6.63E+02 1.64E+02 5.26+02 9.06E+02
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表 8 6种不同算法求解弹簧张力设计问题的最优解比较
Table 8 Comparison of the optimal solution for the spring tension design problem by 6 different algorithms
算法 x1(d) x2(D) x3(P) f(x) CDE 0.051609 0.354714 11.410831 0.126702 HPSO 0.051706 0.357126 11.265083 0.0126652 AATM 0.051813 0.359690 11.119252 0.0126682 SCA 0.052160 0.368158 10.648442 0.0126692 FPA 0.05227727802 0.3712035248 10.49087566498 0.012673971099 GSFPA 0.05141361727 0.0350365659 11.65914644838 0.012659429146
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表 9 6种不同算法求解压力管设计问题的最优解对比
Table 9 Comparison of the optimal solution for the pressure vessle design problem by 6 different algorithms
算法 x1(Ts) x2(Th) x3(R) x4(L) f(x) CPSO 0.8125 0.4375 42.0913 176.7465 6 061.0777 GenAS 1.1250 0.6250 47.7000 117.7010 8 129.8000 HPSO 0.8125 0.4375 42.0984 176.6366 6 059.7143 SPA 0.8125 0.4375 40.3239 200.0000 6 288.7445 FPA 0.7956663 0.3975829 41.22425 187.8048 5 929.6933 GSFPA 0.7862954 0.3924454 40.72142 194.6374 5 916.4771
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