Improved adaptive artificial immune algorithm for solving function optimization problems
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摘要:
为克服经典人工免疫算法（AIA）在函数优化过程中存在的计算量大、收敛精度不高和收敛速度较慢等不足，引入多个自适应免疫算子，提出了一种改进自适应人工免疫算法（IAAIA）。在经典人工免疫算法中，引入迭代次数对抗体激励度计算算子进行自适应设计，引入种群抗体平均激励度与抗体激励度对免疫选择算子、克隆算子、变异算子与克隆抑制算子进行自适应设计，提升人工免疫算法的收敛速度、收敛精度和稳定性。选择9个典型测试函数作为实验对象，同时选择4种典型人工免疫算法作为对比算法优化实验函数，对比实验结果表明了改进的自适应人工免疫算法在求解函数优化问题的有效性和优越性。
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关键词:
- 人工免疫算法(AIA) /
- 免疫系统 /
- 自适应 /
- 免疫算子 /
- 函数优化
Abstract:In order to overcome the shortcomings of Artificial Immune Algorithm (AIA) used in the function optimization process, such as huge calculation amount, low convergence accuracy and slow convergence speed, multiple adaptive immune operators are introduced, and an Improved Adaptive Artificial Immune Algorithm (IAAIA) is proposed. In the classic AIA, antibody excitation calculation operator is adaptively designed by introducing the number of iterations, and immune selection operator, clone operator, mutation operator and clonal inhibitory operator are adaptively designed by introducing antibody population average excitation and antibody excitation, which can improve the convergence accuracy, convergence speed and stability of AIA. Nine kinds of typical and widely used functions are chosen as experiment function, and four kinds of typical AIAs are selected as comparative algorithms to optimize the experiment functions. The comparative experiment results indicate the effectiveness and superiority of the IAAIA for solving function optimization problems.
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Key words:
- Artificial Immune Algorithm (AIA) /
- immune system /
- adaptive /
- immune operator /
- function optimization
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图 2 测试函数f1(x)在5种人工免疫算法下的优化过程
Figure 2. Optimization process of test function f1(x)with 5 kinds of AIAs
图 3 测试函数f3(x)在5种人工免疫算法下的优化过程
Figure 3. Optimization process of test function f3(x) with 5 kinds of AIAs
图 4 测试函数f4(x)在5种人工免疫算法下的优化过程
Figure 4. Optimization process of test function f4(x) with 5 kinds of AIAs
图 5 测试函数f5(x)在5种人工免疫算法下的优化过程
Figure 5. Optimization process of test function f5(x) with 5 kinds of AIAs
图 6 测试函数f7(x)在5种人工免疫算法下的优化过程
Figure 6. Optimization process of test function f7(x) with 5 kinds of AIAs
表 1 人工免疫算法与生物免疫系统中生物概念间对应关系
Table 1. Biology concept corresponding relationship between AIA and biology immune system
生物免疫系统中生物概念 AIA 抗原 待优化问题 抗体 可行解 亲和度 可行解的质量 细胞活化 免疫选择 细胞分化 克隆 亲和度成熟 变异 克隆抑制 克隆抑制 维持动态平衡 种群刷新 
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表 2 九个典型测试函数的基本特性
Table 2. Basic characteristics of 9 typical test functions
函数 函数维数D 搜索空间 最大迭代次数Gmax 最优解 f1(x) 10 [-20, 20] 1 000 最小值为0 f2(x) 1 [1, 20] 1 000 最小值为-1.293 7 f3(x) 10 [-100, 100] 1 000 最小值为0 f4(x) 2 [-20, 20] 1 000 最小值为0 f5(x) 10 [-5.12, 5.12] 1 000 最小值为0 f6(x) 10 [-600, 600] 1 000 最小值为0 f7(x) 2 [-100, 100] 1 000 最小值为0 f8(x) 10 [-500, 500] 1 500 最小值为-4.189 8×103 f9(x) 10 [-5, 5] 1 500 最小值为-78.332 3
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表 3 9个测试函数在5种人工免疫算法下的优化结果分析
Table 3. Optimization results analysis of 9 test functions with 5 kinds of AIAs
函数 对比项目 AIA GMAIA GAIA DQABCIA IAAIA f1(x) 最优解 1.540 5×10-6 0.027 2 0.029 8 0 0 最差解 8.287 9×10-6 0.221 4 0.960 1 0 0 平均最优解 5.447 4×10-6 0.069 6 0.237 7 0 0 最优解方差 8.672 2×10-6 0.247 2 1.101 6 0 0 平均迭代次数 919.7 341.857 1 11.871 0 915.9 715.2 迭代次数方差 354.756 7 900.238 1 19.169 9 210.662 5 179.846 6 平均优化时间/s 43.456 8 26.525 7 0.637 7 37.647 2 29.609 3 收敛到全局最优解成功率/% 0 0 0 100 100 f2(x) 最优解 -1.293 7 -1.293 7 -1.293 7 -1.293 7 -1.293 7 最差解 -1.293 7 -1.293 7 -1.293 7 -1.293 7 -1.293 7 平均最优解 -1.293 7 -1.293 7 -1.293 7 -1.293 7 -1.293 7 最优解方差 0 0 0 0 0 平均迭代次数 749.575 8 390.700 0 548.500 0 573.200 0 176 迭代次数方差 3.756 8×104 3.152 3×104 4.501 1×104 8.782 6×104 2.064 9×104 平均优化时间/s 3.486 9 5.828 8 5.861 1 3.760 5 0.578 1 收敛到全局最优解成功率/% 100 100 100 100 100 f3(x) 最优解 0 0 0 0 0 最差解 0 0 0 0 0 平均最优解 0 0 0 0 0 最优解方差 0 0 0 0 0 平均迭代次数 11.562 5 32.857 1 8.529 4 5.564 1 4.625 0 迭代次数方差 6.623 8 41.452 2 3.804 0 5.439 6 3.391 2 平均优化时间/s 0.551 4 2.262 4 0.492 0 0.228 6 0.192 8 收敛到全局最优解成功率/% 100 100 100 100 100 f4(x) 最优解 4.266 1×10-4 0.031 7 0.179 3 0 0 最差解 0.001 2 1.999 5 2.062 7 0 0 平均最优解 8.449 9×10-4 0.238 2 0.989 7 0 0 最优解方差 8.312 5×10-4 2.477 4 2.913 2 0 0 平均迭代次数 785.303 0 22.030 3 2.135 1 342 214.393 9 迭代次数方差 917.073 0 96.555 5 28.431 0 116.944 4 66.857 2 平均优化时间/s 20.035 4 1.307 2 0.075 4 7.880 7 6.363 4 收敛到全局最优解成功率/% 0 0 0 100 100 f5(x) 最优解 2.985 1 9.220 9 4.482 5 0 0 最差解 16.914 4 25.479 4 20.679 9 0 0 平均最优解 9.628 7 17.891 4 13.115 9 0 0 最优解方差 15.756 6 23.313 1 23.370 3 0 0 平均迭代次数 932.787 9 44.888 9 6.062 5 62.055 6 48.289 5 迭代次数方差 414.525 7 138.483 1 8.824 7 94.445 2 28.457 3 平均优化时间/s 44.309 3 3.124 4 0.255 4 2.647 4 1.689 5 收敛到全局最优解成功率/% 0 0 0 100 100 f6(x) 最优解 0.019 2 0.605 6 0.257 6 0 0 最差解 0.095 5 3.160 1 2.169 4 0 0 平均最优解 0.057 7 1.635 9 0.968 1 0 0 最优解方差 0.109 7 3.121 7 2.371 4 0 0 平均迭代次数 933.969 7 132.727 3 9.925 0 212.789 5 44.500 0 迭代次数方差 330.649 3 145.425 4 15.125 3 183.222 0 19.912 3 平均优化时间/s 46.585 5 9.960 1 0.522 0 8.799 5 2.152 7 收敛到全局最优解成功率/% 0 0 0 100 100 f7(x) 最优解 0.009 7 0.009 7 0.009 7 0 0 最差解 0.078 2 0.498 9 0.154 4 0 0 平均最优解 0.067 8 0.074 4 0.043 9 0 0 最优解方差 0.213 5 0.336 6 0.232 2 0 0 平均迭代次数 661.647 1 2.657 9 1.382 4 22.947 4 9.542 9 迭代次数方差 1.548 4×103 13.287 3 3.745 6 22.757 3 11.344 0 平均优化时间/s 16.399 6 0.166 4 0.053 1 0.496 3 0.351 8 收敛到全局最优解成功率/% 0 0 0 100 100 f8(x) 最优解 -3.953 0×103 -4.071 4×103 -4.189 8×103 -4.189 8×103 -4.189 8×103 最差解 -3.299 6×103 -3.558 0×103 -4.189 8×103 -4.189 8×103 -4.189 8×103 平均最优解 -3.607 2×103 -3.755 5×103 -4.189 8×103 -4.189 8×103 -4.189 8×103 最优解方差 0.026 3 0.021 2 0 0 0 平均迭代次数 1 416.8 868.363 6 855 904.300 0 404.500 0 迭代次数方差 5.960×103 1.170 0×104 3.403 7×104 1.313 4×105 6.685 4×104 平均优化时间/s 44.445 8 47.129 6 34.009 0 32.291 4 16.743 5 收敛到全局最优解成功率/% 0 0 100 100 100 f9(x) 最优解 -78.332 3 -78.329 2 -78.332 0 -78.332 3 -78.332 3 最差解 -72.677 6 -75.500 8 -78.305 4 -78.332 3 -78.332 3 平均最优解 -76.752 9 -76.208 9 -78.317 7 -78.332 3 -78.332 3 最优解方差 2.453 2 1.498 3 1.022 2×10-4 0 0 平均迭代次数 1 375.4 752.875 0 294.900 0 558 421.400 0 迭代次数方差 1.273 5×104 5.940 7×104 5.090 3×104 7.992 3×104 9.140 5×104 平均优化时间/s 42.202 8 71.875 4 12.189 1 15.948 5 7.028 1 收敛到全局最优解成功率/% 43.33 0 0 100 100
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