A chicken swarm optimization algorithm based on improved X-best guided individual and dynamic hierarchy update mechanism
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摘要:
在群智能算法的改进中，常利用优秀个体加速算法收敛，但对其依赖过度会导致种群多样性和算法全局收敛性下降的现象。对此，提出一种改进X-best引导个体和动态等级更新机制的鸡群算法。首先，在个体更新阶段不仅引入优秀个体加速收敛，并且通过普通个体对优秀个体的影响进行适当平衡，因此，优秀个体与普通个体的信息都能得到利用，进而种群多样性和算法全局收敛性得到提升。其次，通过对等级更新参数进行动态优化，加强了种群等级更新机制对算法收敛的促进作用。最后，经过时间复杂度与收敛性分析，证明了改进算法仍具有简单性和全局收敛性。仿真结果表明：所提出的改进算法较其他对比算法在寻优精度、寻优成功率和收敛速度等方面都具有明显优势。
Abstract:In the improvement process of swarm intelligence algorithms, elite individuals are often used to accelerate the convergence, but excessive dependence on them will lead to the decline of population diversity and global convergence. In this regard, a chicken swarm optimization algorithm based on improved X-best guided individual and dynamic hierarchy update mechanism is proposed in this paper. Firstly, in the individual update stage, elite individuals are introduced into the search equation to accelerate the convergence, while the ordinary individuals are also introduced into the search equation to balance the influence of the elite individuals. Therefore, the information of elite and ordinary individuals can be fully used, and the population diversity and global convergence are improved. Secondly, by dynamically optimizing the hierarchy update parameter, the promotion effect of the population hierarchy update mechanism on the convergence is strengthened. Finally, through complexity and convergence analysis, the simplicity and global convergence of IDCSO are proved. The simulation results show that IDCSO has obvious advantages over other comparative algorithms in terms of optimization accuracy, optimization success rate and convergence speed.
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图 6 不同改进机制算法的收敛曲线(D=50)
Figure 6. Convergence curves of different improved mechanism algorithms(D=50)
表 1 测试函数及参数
Table 1. Test functions and parameters
函数序号 函数名 搜索空间 理论最优值 可接受精度 f1 Sphere [-100, 100] 0 1×1040 f2 Schwefel P2.22 [-10, 10] 0 1×1040 f3 Perm0, d, beta [-10, 10] 0 1×103 f4 Sum squares [-10, 10] 0 1×1040 f5 Rastrigin [-5.12, 5.12] 0 1×1020 f6 Griewank [-600, 600] 0 1×1020 f7 Ackley [-50, 50] 0 1×1020 f8 Levy [-10, 10] 0 1×103 f9 Powell [-5, 5] 0 1×103 f10 Alpine [-10, 10] 0 1×1040 f11 Rosenbrock [-10, 10] 0 1×103 f12 Three-hump Camel [-5, 5] 0 1×1040 f13 Zakharov [-10, 10] 0 1×1020 f14 Matyas [-10, 10] 0 1×1040 f15 Booth [-10, 10] 0 1×1040 
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表 2 算法参数设置
Table 2. Algorithm parameter setting
算法 参数设置 ABC[2] Npop=100，Limit=50 CSO Npop =100，Nr= Nc=0.2 Npop，Nh=0.6 Npop，Nm=0.1 Nh，G=10，0.4≤ F≤1 HCSO[8] Npop=100，Nr= Nc=0.2 Npop，Nh=0.6 Npop，Nm=0.1 Nh，G=10，0.4≤ F≤1，0≤ r≤0.2 OBSA-CSO[9] Npop =100，Nr= Nc=0.2 Npop，Nh=0.6 Npop, Nm=0.1 Nh，G=10，0.4≤ F≤1，T0=100 IDCSO Npop=100，Nr= Nc=0.2 Npop，Nh=0.6 Npop，Nm=0.1 Nh，0.4≤ F≤1，Gmin=7，Gmax=14
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表 3 不同改进机制算法的测试结果
Table 3. Test results of different improved mechanism algorithms
函数 维度 CSO ICSO Mean(Std) SR(AVEN)/% Mean(Std) SR(AVEN)/% f1 50 5.30×1066(6.73×1067) 100(3 743) 2.36×1075(2.45×1074) 100(3 287) f2 50 8.48×1040(6.76×1041) 80(3 451) 9.36×1058(3.64×1058) 100(2 964) f3 50 1.44×10(8.69×10) 0(Nan) 1.72×10-4(7.80×10-5) 83(2 894) f4 50 7.93×1040(9.98×1039) 37(2971) 6.52×10-46(2.12×10-45) 97(2 645) f5 50 0(0) 100(1 878) 0(0) 100(1093) f6 50 3.41×10-12(1.99×10-13) 0(Nan) 0(0) 100(2 364) f7 50 5.36×10-7(3.92×10-7) 0(Nan) 4.63×10-13(7.82×10-13) 0(Nan) f8 50 2.68×10-1(1.99×10-1) 0(Nan) 1.45×10-7(6.31×10-7) 100(2 959) f9 50 5.25×10-3(6.22×10-03) 43(3 054) 6.23×10-7(4.62×10-8) 100(2 491) f10 50 9.19×10-49(6.73×10-48) 100(3 182) 4.87×10-76(3.26×10-75) 100(2 153) f11 50 5.51(2.07×10) 0(Nan) 6.82×10-7(7.83×10-6) 100(2 095) f12 50 3.20×10-75(7.46×10-75) 100(1 563) 0(0) 100(893) f13 50 9.95×10-5(3.36×10-6) 0(Nan) 8.39×10-39(7.52×10-40) 100(2 122) f14 50 2.43×10-81(9.74×10-82) 100(2 943) 0(0) 100(1 406) f15 2 0(0) 97(436) 0(0) 100(316) 函数 维度 DCSO IDCSO Mean(Std) SR(AVEN)/% Mean(Std) SR(AVEN)/% f1 50 2.97×10-84(8.13×10-83) 100(2 396) 0(0) 100(2 247) f2 50 0(0) 100(2 184) 0(0) 100(2 434) f3 50 5.14×10-6(5.14×10-6) 100(2431) 1.11×10-9(2.09×10-10) 100(1 963) f4 50 8.46×10-51(8.46×10-51) 100(2 349) 7.20×10-61(1.21×10-62) 100(1 743) f5 50 0(0) 100(1 298) 0(0) 100(827) f6 50 0(0) 100(2 863) 0(0) 100(1 726) f7 50 6.22×10-11(6.22×10-11) 0(Nan) 8.88×10-22(3.28×10-23) 96(1 937) f8 50 3.63×10-4(3.63×10-4) 87(3 083) 4.83×10-9(1.17×10-10) 100(2 176) f9 50 5.74×10-8(5.74×10-8) 100(2 691) 6.08×10-10(2.58×10-10) 100(1 763) f10 50 8.86×10-70(8.86×10-70) 100(2 557) 3.33×10-81(1.31×10-82) 100(1 846) f11 50 5.76×10-5(5.76×10-5) 93(2 816) 5.26×10-12(1.52×10-12) 100(1 796) f12 50 6.24×10-96(6.24×10-96) 100(1 076) 0(0) 100(961) f13 50 8.39×10-35(8.39×10-35) 100(2 694) 1.96×10-46(7.83×10-45) 100(1 534) f14 50 6.39×10-95(6.39×10-95) 100(2 036) 0(0) 100(1376) f15 2 0(0) 100(339) 0(0) 100(223)
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表 4 各算法对30维与100维函数的测试结果
Table 4. Test results of each algorithm on 30- and 100-dimensional functions
函数 维数 指标 ABC[2] CSO HCSO OBSA-CSO[9] IDCSO f1 30 Mean(Std) 5.61×10-35(6.82×10-36) 5.82×10-86(5.41×10-86) 8.51×10-102(5.61×10-101) 9.30×10-114(6.97×10-115) 0(0) SR(AVEN) 100(2 393) 100(2 369) 100(1 988) 100(1 907) 100(1 654) 100 Mean(Std) 6.61×10-21(5.19×10-22) 5.45×10-55(6.47×10-55) 6.13×10-64(9.90×10-65) 5.28×10-66(4.80×10-67) 8.01×10-95(2.28×10-96) SR(AVEN) 37(6 145) 83(7875) 100(5 419) 100(4 409) 100(4 184) f2 30 Mean(Std) 1.73×10-40(3.91×10-40) 4.05×10-47(4.48×10-46) 7.39×10-68(5.86×10-67) 2.47×10-74(6.66×10-75) 0(0) SR(AVEN) 93(1 802) 100(2 101) 100(2 541) 100(2 121) 100(1 614) 100 Mean(Std) 2.92×10-22(4.32×10-23) 6.96×10-36(2.94×10-37) 7.69×10-43(5.81×10-43) 9.28×10-48(5.80×10-49) 4.17×10-56(1.21×10-57) SR(AVEN) 0(Nan) 87(4 452) 100(4 769) 100(4 216) 100(3 521) f3 30 Mean(Std) 3.44×10-2(9.20×10-2) 1.80×10-2(1.52×10-2) 2.18×10-4(6.32×10-5) 5.89×10-4(6.15×10-5) 3.62×10-11(2.50×10-11) SR(AVEN) 37(1 745) 73(804) 100(845) 100(1 198) 100(612) 100 Mean(Std) 3.01×102(7.01×103) 7.36×10(3.95×10) 1.47×10-2(1.89×10-3) 4.43×10-2(6.35×10-2) 2.82×10-5(5.39×10-5) SR(AVEN) 0(Nan) 0(Nan) 57(8 065) 63(6 463) 97(4 433) f4 30 Mean(Std) 4.33×10-42(5.61×10-41) 8.22×10-55(4.30×10-56) 1.24×10-72(4.92×10-73) 8.53×10-75(8.74×10-76) 0(0) SR(AVEN) 93(2 741) 100(2 415) 100(1 984) 100(1 556) 100(1 435) 100 Mean(Std) 6.45×10-23(3.76×10-24) 7.91×10-31(9.49×10-32) 9.48×10-33(2.82×10-34) 1.06×10-35(1.42×10-36) 1.66×10-42(6.21×10-42) SR(AVEN) 0(Nan) 0(Nan) 0(Nan) 27(4 568) 87(3 956) f5 30 Mean(Std) 0(0) 0(0) 0(0) 0(0) 0(0) SR(AVEN) 100(803) 100(754) 100(667) 100(541) 100(469) 100 Mean(Std) 9.06×10-5(8.82×10-6) 5.34×10-6(1.90×10-7) 3.41×10-12(4.31×10-13) 2.27×10-13(3.01×10-13) 0(0) SR(AVEN) 0(Nan) 0(Nan) 0(Nan) 0(Nan) 100(3 649) f6 30 Mean(Std) 0(0) 0(0) 0(0) 0(0) 0(0) SR(AVEN) 100(1 734) 100(1 487) 100(973) 100(1 246) 100(698) 100 Mean(Std) 1.49×10-6(9.01×10-6) 1.32×10-10(5.42×10-11) 0(0) 0(0) 0(0) SR(AVEN) 0(Nan) 0(Nan) 100(5 096) 100(3 845) 100(2 596) f7 30 Mean(Std) 6.61×10-16(7.30×10-17) 1.78×10-23(3.61×10-23) 4.44×10-28(5.22×10-29) 4.44×10-34(5.22×10-35) 2.09×10-41(9.05×10-41) SR(AVEN) 17(1 731) 87(1 951) 100(1 865) 100(1 465) 100(1 094) 100 Mean(Std) 8.63×10-10(4.84×10-10) 9.12×10-17(1.04×10-18) 7.46×10-20(7.36×10-21) 5.62×10-25(1.84×10-25) 5.97×10-32(3.06×10-33) SR(AVEN) 0(Nan) 20(4 696) 67(4 367) 93(4 117) 100(3 714) f8 30 Mean(Std) 7.87×10-1(1.93×10-2) 1.94×10-2(4.71×10-2) 2.95×10-4(7.54×10-5) 4.11×10-6(3.13×10-7) 2.40×10-11(1.97×10-12) SR(AVEN) 0(Nan) 37(696) 83(724) 100(678) 100(684) 100 Mean(Std) 6.95×102(4.99×103) 4.56×10(1.02×10) 9.95×10-2(3.32×10-2) 2.97×10-3(4.62×10-3) 2.98×10-5(4.46×10-6) SR(AVEN) 0(Nan) 0(Nan) 43(6 741) 77(5 863) 97(4 213) f9 30 Mean(Std) 2.32×10-2(6.47×10-1) 6.63×10-4(1.92×10-3) 2.03×10-7(7.77×10-8) 6.22×10-9(5.34×10-9) 5.31×10-20(8.26×10-21) SR(AVEN) 33(2 134) 83(1 164) 100(1 042) 100(894) 100(864) 100 Mean(Std) 5.74(4.52) 7.46×10-1(4.12×10-2) 3.48×10-3(6.68×10-4) 6.03×10-5(5.26×10-6) 7.34×10-8(7.07×10-7) SR(AVEN) 0(Nan) 0(Nan) 53(3 914) 97(3 854) 100(3 421) f10 30 Mean(Std) 7.86×10-42(5.13×10-43) 6.93×10-58(5.57×10-57) 3.97×10-71(5.62×10-72) 7.87×10-79(3.38×10-80) 6.08×10-94(7.41×10-95) SR(AVEN) 90(1 731) 100(1 826) 100(2 145) 100(1 945) 100(1 696) 100 Mean(Std) 4.67×10-23(6.48×10-22) 5.54×10-31(4.85×10-32) 8.94×10-36(7.99×10-37) 7.34×10-38(2.51×10-39) 4.73×10-51(2.89×10-52) SR(AVEN) 0(Nan) 0(Nan) 17(5 078) 63(4 974) 100(3 974) f11 30 Mean(Std) 3.62(9.06) 5.03×10-1(9.13×10-2) 9.78×10-2(1.98×10-2) 2.14×10-3(5.47×10-3) 4.45×10-19(9.65×10-19) SR(AVEN) 0(Nan) 0(Nan) 50(1 379) 83(2 345) 100(943) 100 Mean(Std) 1.58×103(9.71×102) 9.57×102(4.85×102) 8.25(1.42) 4.22×10-2(9.16×10-1) 7.92×10-8(9.59×10-8) SR(AVEN) 0(Nan) 0(Nan) 0(Nan) 43(5 274) 100(4 173) f12 30 Mean(Std) 6.56×10-84(3.36×10-84) 8.49×10-92(9.34×10-93) 8.89×10-101(1.54×10-102) 1.26×10-105(2.36×10-105) 0(0) SR(AVEN) 100(946) 100(894) 100(793) 100(641) 100(593) 100 Mean(Std) 7.06×10-42(2.32×10-43) 2.77×10-52(1.46×10-52) 2.97×10-72(8.23×10-73) 6.95×10-81(3.17×10-81) 9.59×10-96(2.34×10-97) SR(AVEN) 87(3 641) 100(3 096) 100(2 784) 100(2 263) 100(1 643) f13 30 Mean(Std) 1.03×10-4(3.82×10-4) 7.66×10-6(7.95×10-7) 1.87×10-36(4.92×10-36) 4.46×10-42(6.49×10-41) 7.09×10-60(7.55×10-61) SR(AVEN) 0(Nan) 0(Nan) 100(2 374) 100(1 578) 100(1 076) 100 Mean(Std) 1.04×10-2(6.82×10-1) 1.03×10-4(1.63×10-5) 2.52×10-7(4.98×10-8) 1.32×10-9(3.43×10-9) 1.06×10-21(2.24×10-22) SR(AVEN) 0(Nan) 0(Nan) 0(Nan) 0(Nan) 80(4 566) f14 30 Mean(Std) 7.51×10-85(2.55×10-85) 5.06×10-96(6.99×10-97) 4.36×10-105(6.12×10-106) 1.94×10-108(8.25×10-109) 0(0) SR(AVEN) 100(1 294) 100(984) 100(841) 100(793) 100(624) 100 Mean(Std) 8.41×10-51(2.54×10-52) 8.14×10-64(2.44×10-65) 9.29×10-75(3.53×10-75) 1.97×10-79(2.51×10-80) 6.16×10-93(4.73×10-93) SR(AVEN) 100(4 836) 100(4 214) 100(3 935) 100(3 647) 100(3 056) f15 2 Mean(Std) 0(0) 0(0) 0(0) 0(0) 0(0) SR(AVEN) 100(475) 100(436) 100(397) 100(364) 100(223)
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表 5 IDCSO算法与其他改进CSO算法的比较结果
Table 5. Comparison results between IDCSO and other improved chicken swarm algorithms
函数 指标 T=1 000, D=30 T=200, D=30 T=1 000, D=100 EOCSO[25] IDCSO PRCSO[10] IDCSO DMCSO[26] GCSO[27] IDCSO f1 Mean 0 0 1.930×10-11 2.632×10-32 5.863×10-4 3.440×10-22 2.739×10-84 Std 0 0 7.556×10-22 5.486×10-33 2.093×10-3 1.490×10-22 3.002×10-90 f2 Mean 2.130×10-112 0 — — — — — Std 2.868×10-112 0 — — — — — f5 Mean 0 0 1.144×10-8 0 6.663×10-7 2.750×10-15 0 Std 0 0 3.584×10-16 0 2.008×10-6 5.040×10-15 0 f6 Mean 0 0 2.573×10-11 0 1.516×10-5 2.780×10-17 0 Std 0 0 9.663×10-22 0 3.803×10-5 6.160×10-17 0 f7 Mean 4.705×10-15 8.881×10-16 5.007×10-7 1.146×10-13 9.541×10-3 9.080×10-24 8.881×10-16 Std 7.999×10-16 1.610×10-16 2.919×10-14 4.165×10-15 2.248×10-2 2.440×10-24 1.61×10-16 f9 Mean — — — — 1.152×10-2 — 3.101×10-8 Std — — — — 1.962×10-2 — 3.458×10-8 f10 Mean — — — — 9.238×10-3 — 4.812×10-56 Std — — — — 1.949×10-2 — 2.599×10-57 f11 Mean 1.610×10-4 9.521×10-20 1.521×10-8 4.327×10-9 — — — Std 3.590×10-4 1.010×10-20 2.464×10-21 3.578×10-10 — — — f13 Mean — — — — 1.212×102 — 4.059×10-20 Std — — — — 1.147×102 — 1.135×10-22
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