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摘要: 针对黑猩猩优化算法(Chimp optimization algorithm, ChOA)存在收敛速度慢、精度低和易陷入局部最优值的问题, 提出一种融合多策略的黄金正弦黑猩猩优化算法(IChOA). 引入Halton序列初始化种群, 提高初始化种群的多样性, 加快算法收敛, 提高收敛精度; 考虑到收敛因子和权重因子对于平衡算法勘探和开发能力的重要作用, 引入改进的非线性收敛因子和自适应权重因子, 平衡算法的搜索能力; 结合黄金正弦算法相关思想更新个体位置, 提高算法对于局部极值的处理能力. 通过对23个基准测试函数的寻优对比分析和Wilcoxon秩和统计检验以及部分CEC2014测试函数寻优结果对比可知, 改进的算法具有更好的鲁棒性, 最后, 通过2个实际工程优化问题的实验对比分析, 进一步验证了IChOA在处理现实优化问题上的优越性.Abstract: Chimp optimization algorithm (ChOA) has the problems of slow convergence speed, low accuracy and easy to fall into local optimal value. A multi-strategy golden sinusoidal chimp optimization algorithm (IChOA) was proposed. Introducing Halton sequence to initialize the population, improving the diversity of initialized population, accelerating the convergence of the algorithm, and improving the convergence accuracy. Considering the important role of convergence factor and weight factor on the exploration and exploitation of the balance algorithm, the improved nonlinear convergence factor and adaptive weight factor are introduced to balance the search ability of the algorithm. The gold sine algorithm is used to update the individual position and improve the ability of the algorithm to deal with the local extreme value. Through comparative analysis of the optimization of 23 benchmark test functions, Wilcoxon rank sum statistical test and optimization results of some CEC2014 test functions, it can be seen that the improved algorithm has better robustness. Finally, through comparative analysis of two practical engineering optimization problems, The superiority of IChOA in dealing with realistic optimization problem is further verified.
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表 1 算法参数设置
Table 1 Algorithm Parameter Setting
算法 参数 ChOA m = chaos (3, 1, 1) PSO c1 = 1.5, c2 = 2, w = 1, wdamp = 0.99 GWO — WOA b = 1 IChOA m = chaos(3, 1, 1), δ1 = 0.3, δ2 = 300, δ3 = 1.8,
ρ1 = 0.1, ρ2 = 0.05, ρ3 = 0.3, ε = 300表 2 基准测试函数介绍
Table 2 Introduction to Benchmark Functions
编号 函数名 定义域 维度 最优值 绝对精度误差ε f1 Sphere [−100, 100] 30 0 1.00×10−03 f2 Schwefel’problem 2.22 [−10, 10] 30 0 1.00×10−03 f3 Schwefel’problem 1.2 [−100, 100] 30 0 1.00×10−03 f4 Schwefel’problem 2.21 [−100, 100] 30 0 1.00×10−03 f5 Generalized Rosenbrock's Function [−30, 30] 30 0 1.00×10−02 f6 Step Function [−100, 100] 30 0 1.00×10−02 f7 Quartic Function [−1.28, 1.28] 30 0 1.00×10−02 f8 Generalized Schwefel’s problem 2.26 [−500, 500] 30 −12569.5 1.00×10+02 f9 Generalized Rastrigin’s Function [−5.12, 5.12] 30 0 1.00×10−02 f10 Ackley’s Function [−32, 32] 30 0 1.00×10−02 f11 Ceneralized Criewank Function [−600, 600] 30 0 1.00×10−02 f12 Ceneralized Penalized Function 1 [−50, 50] 30 0 1.00×10−02 f13 Ceneralized Penalized Function 2 [−50, 50] 30 0 1.00×10−02 f14 Shekell’s Foxholes Function [−65, 65] 2 1 1.00×10−02 f15 Kowalik's Function [−5, 5] 4 0.0003 1.00×10−02 f16 Six-Hump Camel-Back Function [−5, 5] 2 −1.03 1.00×10−02 f17 Branin Function [−5, 5] 2 0.398 1.00×10−02 f18 Goldstein-Price Function [−2, 2] 2 3 1.00×10−02 f19 Hatman’s Function1 [0, 1] 3 −3.86 1.00×10−02 f20 Hatman’s Function2 [0, 1] 6 −3.32 1.00×10−02 f21 Shekel's Family 1 [0, 10] 4 −10 1.00×10−02 f22 Shekel's Family 2 [0, 10] 4 −10 1.00×10−02 f23 Shekel's Family 3 [0, 10] 4 −10 1.00×10−02 表 3 各算法寻优结果对比(30维)
Table 3 Comparison of optimization results of each algorithm (30dim)
函数 ChOA算法 PSO算法[19] GWO算法[20] SChOA算法[21] IChOA算法 平均值 标准差 平均值 标准差 平均值 标准差 平均值 标准差 平均值 标准差 f1 1.34×10−5 1.19×10−20 1.40×10−4 2.11×10−4 5.95×10−28 6.85×10−28 5.66×10−33 5.68×103 0 0 f2 1.42×10−5 8.55×10−21 4.21×10−2 4.54×10−2 7.95×10−17 4.97×10−17 1.72×10−20 1.91×1010 0 0 f3 6.31×100 1.40×10−17 7.01×101 2.21×101 2.83×10−5 1.12×10−4 6.19×10−8 2.25×104 0 0 f4 2.75×10−2 5.95×10−28 1.08×100 3.17×10−1 5.69×10−7 5.55×10−7 2.75×10−10 1.26×100 0 0 f5 2.87×101 2.51×10−14 9.67×101 6.01×101 2.70×101 8.26×10−1 2.85×102 1.00×107 3.13×10−4 5.26×10−17 f6 3.72×100 4.48×10−15 1.10×10−4 8.28×10−5 7.64×10−1 3.58×10−1 3.01×100 5.62×103 6.51×10−3 3.25×10−4 f7 1.72×10−3 1.09×10−18 1.22×10−1 4.49×10−2 1.72×10−3 7.51×10−4 1.00×10−3 5.77×101 7.81×10−7 1.02×10−12 f8 −5648.58 2.75×10−12 −4841.28 1152.84 −6084.02 1.02×103 −9867.01 180.01 −12555.73 1.83×10−11 f9 1.41×101 0 4.67×101 1.16×101 3.22×100 4.16×100 0 7.77×101 0 0 f10 1.96×101 1.79×10−14 2.76×10−1 5.09×10−1 1.05×10−13 2.39×10−14 1.50×10−14 1.76×101 8.88×10−16 0 f11 4.79×10−2 7.00×10−17 9.21×10−3 7.74×10−3 5.14×10−3 9.98×10−3 0 8.30×101 0 0 f12 3.98×10−1 5.60×10−17 6.92×10−3 1.19×10−2 5.99×10−2 9.78×10−2 1.62×10−1 3.31×107 6.46×10−4 4.39×10−18 f13 2.82×100 1.76×10−15 6.68×10−3 8.91×10−3 6.27×10−1 3.06×10−1 6.76×10−1 5.16×107 2.97×10−5 4.56×10−16 f14 0.99701 1.12×10−15 3.62717 2.5×100 5.08661 4.34×100 0.9980 1.04×101 0.988 4.48×10−16 f15 0.00136 4.39×10−19 0.00058 2.21×10−4 0.00573 8.98×10−3 0.00068 1.70×10−3 0.00023 5.46×10−19 f16 −1.03160 6.72×10−15 −1.03162 6.25×10−16 −1.0316 2.42×10−8 −1.0316 2.21×10−1 −1.0316 5.60×10−16 f17 0.39805 3.36×10−16 0.39789 0 0.39789 0 0.3979 4.90×10−3 0.39789 8.98×10−16 f18 3.00010 0 3.18240 1.33×10−15 5.70003 1.47×101 3.0000 1.92×10−1 3.0000 0 f19 −3.85310 2.69×10−15 −3.86278 2.58×10−15 −3.86168 2.17×10−3 −3.8649 9.16×10−2 −3.7232 1.34×10−18 f20 −1.92074 1.12×10−15 −3.26134 6.05×10−2 −3.23036 8.433×10−2 −3.3219 1.24×10−1 −2.8954 5.23×10−15 f21 −4.92478 2.69×10−15 −6.86511 3.01×100 −8.79956 2.2×100 −10.0944 3.34×10−1 −10.1237 2.38×10−15 f22 −4.99454 2.69×10−15 −8.45653 3.08×100 −10.22384 9.70×10−1 −5.1759 5.74×10−2 −9.7833 8.79×10−15 f23 −5.02530 4.48×10−16 −8.95291 1.78×100 −9.90500 1.96×100 −10.5139 4.95×10−2 −9.9335 8.97×10−16 表 4 Wilcoxon秩和检验结果
Table 4 Wilcoxon rank sum test results
编号 PSO(p1) GWO(p2) WOA(p3) ChOA(p4) GChOA(p5) f1 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 f2 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 3.25×10−20 f3 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 f4 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 3.31×10−20 f5 1.01×10−17 2.47×10−17 1.04×10−15 2.29×10−15 7.96×10−18 f6 7.06×10−18 1.28×10−17 1.38×10−15 2.13×10−16 7.06×10−18 f7 4.20×10−17 7.06×10−18 6.88×10−14 1.36×10−17 1.27×10−16 f8 7.06×10−18 7.06×10−18 2.21×10−10 7.06×10−18 7.06×10−18 f9 3.31×10−20 3.31×10−20 NaN 1.17×10−19 3.31×10−20 f10 3.31×10−20 3.31×10−20 2.39×10−16 2.91×10−20 2.62×10−23 f11 3.31×10−20 3.31×10−20 3.27×10−01 2.50×10−04 3.31×10−20 f12 7.06×10−18 9.37×10−11 1.83×10−17 7.96×10−18 7.06×10−18 +/=/− 12/0/0 12/0/0 10/1/1 12/0/0 12/0/0 表 5 部分CEC2014函数介绍
Table 5 Part of the CEC2014 function
函数 维度 特征 定义域 最佳值 CEC03 30 UN [−100, 100] 300 CEC05 30 MF [−100, 100] 500 CEC06 30 MF [−100, 100] 600 CEC16 30 MF [−100, 100] 1600 CEC19 30 HF [−100, 100] 1900 CEC22 30 HF [−100, 100] 2200 CEC25 30 CF [−100, 100] 2500 CEC27 30 CF [−100, 100] 2700 表 6 CEC2014函数优化对比
Table 6 CEC2014 function optimization comparison
函数 PSO算法[24] SCA算法[25] L-SHADE算法[26] HChOA算法 GChOA算法 IChOA算法 平均值 标准差 平均值 标准差 平均值 标准差 平均值 标准差 平均值 标准差 平均值 标准差 CEC03 4.87×101 6.61×101 8.83×100 1.36×100 0 0 7.78×104 7.44×103 7.54×104 6.58×103 7.35×104 6.23×103 CEC05 2.09×101 8.52×10-2 2.21×100 2.72×100 2.01×101 1.70×10-2 5.22×102 6.67×10-2 5.26×102 4.21×10-2 5.20×102 1.02×10-2 CEC06 1.08×101 2.53×100 6.63×101 3.74×101 1.67×10-2 9.17×10-2 6.33×102 2.42×100 6.33×10-5 2.42×100 6.31×102 2.39×100 CEC16 1.13×101 7.05×10-1 2.27×101 1.66×10-1 8.48×100 2.97×10-1 1.62×103 2.81×10-1 1.61×103 1.88×10-1 1.61×103 1.24×10-1 CEC19 7.76×100 1.87×100 2.88×102 2.99×101 3.59×100 7.22×10-1 2.56×103 2.46×100 1.76×103 2.39×100 2.32×103 1.96×100 CEC22 2.31×102 1.04×102 2.43×101 3.03×101 3.69×101 3.36×101 3.57×103 9.65×101 4.21×103 1.48×102 3.56×103 7.48×101 CEC25 2.09×102 1.65×100 2.69×102 2.71×101 2.03×102 4.97×10-2 2.71×103 9.49×100 2.71×103 3.21×100 2.70×103 0 CEC27 5.36×102 8.15×103 2.08×102 1.89×101 3.00×102 1.34×10-13 2.93×103 5.36×100 2.91×103 8.12×100 2.90×103 0 表 7 基准函数寻优平均时间及成功率对比
Table 7 Comparison of average time and success rate for optimization of benchmark function
函数 ChOA HChOA WChOA GChOA IChOA 平均值 标准差 成功率 平均值 标准差 成功率 平均值 标准差 成功率 平均值 标准差 成功率 平均值 标准差 成功率 f1 1.9732 0.0136 100% 1.9824 0.0088 100% 1.9302 0.0155 100% 1.4113 0.0181 100% 1.3916 0.0101 100% f2 1.9546 0.0101 100% 1.9841 0.0181 100% 1.7962 0.0063 100% 1.4176 0.0101 100% 1.4086 0.0081 100% f3 2.2999 0.0083 0 2.2988 0.0077 100% 2.0616 0.0906 100% 2.1272 0.0128 100% 2.1150 0.0275 100% f4 2.0456 0.0107 0 1.9884 0.0309 100% 1.9955 0.0676 100% 1.4381 0.0330 100% 1.4208 0.0446 100% f5 2.0696 0.0663 0 2.0691 0.0649 0 1.9966 0.0326 33.3% 1.4670 0.0530 96.6% 1.4574 0.0244 100% f6 2.0127 0.0435 0 1.9617 0.0107 36.6% 1.9604 0.0138 16.6% 1.4122 0.0111 43.3% 1.4006 0.0121 90% f7 2.0567 0.0089 100% 2.0364 0.0190 100% 2.0520 0.0147 100% 1.5407 0.0070 100% 1.5356 0.0098 100% f8 2.0246 0.0127 0 2.0195 0.0310 0 1.9784 0.0294 0 1.4638 0.0086 73.3% 1.4625 0.0132 83.3% f9 2.0138 0.0343 0 2.0203 0.0112 96.6% 1.9855 0.0096 100% 1.4316 0.0184 100% 1.4169 0.0141 100% f10 2.0011 0.0113 0 2.0045 0.0143 73.3% 1.9999 0.0086 100% 1.4521 0.0113 100% 1.4395 0.0089 100% f11 2.0293 0.0081 63.3% 2.0301 0.0144 100% 2.0155 0.0095 100% 1.4776 0.0119 100% 1.4735 0.0116 100% f12 2.2040 0.0114 0 2.1930 0.0142 33.3% 2.1697 0.0169 86.6% 1.8354 0.0110 43.3% 1.8422 0.0530 100% f13 2.1873 0.0112 0 2.1792 0.0115 36.6% 2.1811 0.0184 56.6% 1.8180 0.0140 66.6% 1.8140 0.0084 100% f14 0.7893 0.0090 100% 0.7916 0.0068 100% 0.7898 0.0073 100% 1.3830 0.0100 100% 1.3571 0.0064 100% f15 0.3131 0.0039 0 0.3141 0.0034 50.0% 0.3174 0.0024 63.3% 0.2792 0.0027 90.0% 0.2788 0.0024 96.6% f16 0.1730 0.0038 100% 0.1743 0.0034 100% 0.1728 0.0037 100% 0.1647 0.0037 100% 0.1647 0.0013 100% f17 0.1690 0.0012 100% 0.1709 0.0043 100% 0.1689 0.0013 100% 0.1556 0.0061 100% 0.1524 0.0022 100% f18 0.1673 0.0025 100% 0.1699 0.0035 100% 0.1689 0.0021 100% 0.1516 0.0012 100% 0.1510 0.0020 100% f19 0.2804 0.0045 100% 0.2847 0.0031 100% 0.2849 0.0019 100% 0.3021 0.0064 100% 0.2580 0.0017 100% f20 0.4751 0.0042 46.6% 0.4737 0.0024 70.0% 0.4724 0.0028 63.3% 0.4304 0.0053 63.3% 0.4278 0.0033 76.6% f21 0.4046 0.0251 16.6% 0.4068 0.0035 20.0% 0.4012 0.0028 70.0% 0.4581 0.0029 76.6% 0.4035 0.0055 83.3% f22 0.4447 0.0043 0 0.4458 0.0030 36.6% 0.4403 0.0050 53.3% 0.5391 0.0048 73.3% 0.4349 0.0051 80.0% f23 0.5087 0.0029 0 0.5089 0.0036 43.3% 0.5080 0.0021 36.6% 0.6610 0.0034 76.6% 0.6347 0.0039 86.6% 表 8 焊接梁设计问题结果对比
Table 8 Comparison of welded beam design
算法 h l t b 平均值 GA 0.2455 6.1986 8.1264 0.2247 2.4412 PSO 0.2027 3.4705 9.0366 0.2057 1.7249 WOA 0.2024 3.4772 9.0435 0.2189 1.7299 GWO 0.2022 3.4893 9.0541 0.2155 1.7265 RO[21] 0.2036 3.5284 9.0042 0.2072 1.7353 MOV[21] 0.2054 3.4731 9.0445 0.2056 1.7246 HSSAHHO[21] 0.2057 3.4705 9.0367 0.2057 1.7248 ChOA 0.2214 3.5358 8.9115 0.2127 1.7737 SChOA[21] 0.2057 3.4705 9.0306 0.2056 1.7229 IChOA 0.2038 3.4713 9.0300 0.2060 1.7228 表 9 拉力/压力弹簧优化设计问题结果对比
Table 9 Comparison of tension/compression spring design
算法 d D P 平均值 GA 0.0528 0.3523 11.5980 0.0125 PSO 0.0500 0.3174 14.0278 0.0127 WOA 0.5119 0.3452 12.0052 0.0126 GWO 0.5156 0.3562 11.5560 0.0125 RO[21] 0.0413 0.3490 11.7620 0.0126 MFO[21] 0.0510 0.3641 10.8684 0.0126 HSSAHHO[21] 0.0514 0.3535 11.3546 0.0124 ChOA 0.0500 0.3159 14.2629 0.0128 SChOA[21] 0.0524 0.3489 10.6543 0.01187 IChOA 0.0510 0.3374 11.5068 0.01185 -
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