电子学报 ›› 2022, Vol. 50 ›› Issue (7): 1664-1673.DOI: 10.12263/DZXB.20201305
穆磊1,2, 王鹏1
收稿日期:
2020-11-18
修回日期:
2021-03-01
出版日期:
2022-07-25
作者简介:
基金资助:
MU Lei1,2, WANG Peng1
Received:
2020-11-18
Revised:
2021-03-01
Online:
2022-07-25
Published:
2022-07-30
Supported by:
摘要:
尺度在量子启发谐振子优化算法中起着重要作用,反映了解空间中搜索探针的分辨率.当前研究中以固定速度调整尺度并未合理利用尺度资源.此外,候选解可能会因高斯采样的聚集效应而陷入边界.本文提出了一种在一定程度上反映了适应度利用效率的指标,称为适应度进化利用率.在此基础上,本文提出了一种具有尺度动态调速机制和边界映射反弹策略的量子启发式优化方法.该算法通过与尺度调整因子相关的适应度进化利用率动态调节尺度调整速度,通过2种不同的边界映射反弹策略增加可行解的多样性.将本算法与多种流行优化算法在基准测试函数集上进行对比实验,采用了一种带有动态可接受误差的成功率评估机制保证公平性,实验结果表明该算法具有较强的竞争性.
中图分类号:
穆磊, 王鹏. 量子启发式优化算法的尺度动态调速机制[J]. 电子学报, 2022, 50(7): 1664-1673.
MU Lei, WANG Peng. Speed Regulation of Scale Adjustment in Quantum-Inspired Optimization Algorithm[J]. Acta Electronica Sinica, 2022, 50(7): 1664-1673.
函数名称 | 函数表达式 | 维度 | 搜索空间 | 全局最优位置 |
---|---|---|---|---|
Griewank | n | [-100,100] | f(0,…,0)=0 | |
Rastrigin | n | [-5.12,5.12] | f (0,…,0)=0 | |
Ackley | n | [-32.77,32.77] | f (0,…,0)=0 | |
Levy | where | n | [-10,10] | f (1,…,1)=0 |
Alpine | n | [0,10] | f (0,…,0)=0 | |
Schwefel | n | [-500,500] | f (420.97…)=0 | |
Sphere | n | [-5.12,5.12] | f (0,…,0)=0 | |
Sum Squares | n | [-10,10] | f (0,…,0)=0 | |
Botated Hyper- Ellipsoid | n | [-65.54,65.54] | f (0,…,0)=0 | |
Ellipsoidal | n | [-100,100] | f (1,2,…n)=0 | |
Sum of Different Power | n | [-100,100] | f (0,…,0)=0 | |
Zakharov | n | [ | f (0,…,0)=0 | |
Rosenbrock | n | [ | f (1,…,1)=0 | |
Dixson-Price | n | [-10,10] |
表1 基准测试函数
函数名称 | 函数表达式 | 维度 | 搜索空间 | 全局最优位置 |
---|---|---|---|---|
Griewank | n | [-100,100] | f(0,…,0)=0 | |
Rastrigin | n | [-5.12,5.12] | f (0,…,0)=0 | |
Ackley | n | [-32.77,32.77] | f (0,…,0)=0 | |
Levy | where | n | [-10,10] | f (1,…,1)=0 |
Alpine | n | [0,10] | f (0,…,0)=0 | |
Schwefel | n | [-500,500] | f (420.97…)=0 | |
Sphere | n | [-5.12,5.12] | f (0,…,0)=0 | |
Sum Squares | n | [-10,10] | f (0,…,0)=0 | |
Botated Hyper- Ellipsoid | n | [-65.54,65.54] | f (0,…,0)=0 | |
Ellipsoidal | n | [-100,100] | f (1,2,…n)=0 | |
Sum of Different Power | n | [-100,100] | f (0,…,0)=0 | |
Zakharov | n | [ | f (0,…,0)=0 | |
Rosenbrock | n | [ | f (1,…,1)=0 | |
Dixson-Price | n | [-10,10] |
No. | SPSO2011 | MQHOA | NSR | PSR | BBFWA | QPSO |
---|---|---|---|---|---|---|
1 | 7.48E-03(7.73E-03)(3) | 1.02E-02(3.98E-02)(4) | 2.22E-03(4.65E-03)(1) | 4.25E-03(6.72E-03)(2) | 1.08E-02(9.11E-03)(5) | 4.31E-01(2.28E-01)(6) |
2 | 2.35E+01(6.62E+00)(1) | 1.32E+02(6.91E+01)(5) | 1.16E+02(7.83E+01)(3) | 1.21E+02(7.28E+01)(4) | 1.14E+02(3.15E+01)(2) | 1.63E+02(2.00E+01)(6) |
3 | 8.89E-01(7.77E-01)(5) | 1.96E-01(4.70E-01)(4) | 5.05E-02(2.63E-01)(2) | 1.37E-01(4.77E-01)(3) | 1.10E+00(1.05E+00)(6) | 4.33E-06(6.08E-06)(1) |
4 | 6.89E-01(6.04E-01)(5) | 7.11E-02(2.67E-01)(4) | 3.69E-02(6.25E-02)(2) | 4.08E-02(1.11E-01)(3) | 1.84E+01(1.05E+01)(6) | 7.57E-10(1.94E-09)(1) |
5 | 6.78E+00(5.82E+00)(5) | 6.68E+00(1.07E+00)(4) | 3.37E+00(6.45E+00)(2) | 1.31E+01(2.95E+00)(6) | 4.52E+00(2.18E+00)(3) | 5.89E-16(1.77E-15)(1) |
6 | 4.62E+03(5.34E+02)(1) | 7.84E+03(2.96E+02)(5) | 7.95E+03(2.55E+02)(6) | 7.13E+03(2.60E+02)(4) | 5.26E+03(6.36E+02)(2) | 6.05E+03(7.66E+02)(3) |
7 | 9.33E-101(5.77E-102)(2) | 4.83E-84(7.24E-84)(5) | 6.59E-86(8.80E-86)(4) | 4.03E-88(7.11E-88)(3) | 2.53E-107(1.55E-106)(1) | 1.64E-12(3.36E-12)(6) |
8 | 7.68E-30(1.83E-29)(2) | 1.37E-05(4.35E-05)(6) | 5.88E-06(8.74E-06)(4) | 8.03E-06(1.62E-05)(5) | 2.47E-55(1.40E-54)(1) | 1.41E-11(1.94E-11)(3) |
9 | 9.24E-101(5.34E-102)(2) | 3.44E-79(4.62E-79)(5) | 6.35E-81(1.69E-80)(4) | 2.72E-83(3.04E-83)(3) | 1.27E-103(5.92E-103)(1) | 1.06E-07(1.92E-07)(6) |
10 | 7.08E-29(1.21E-29)(5) | 0.00E+00(0.00E+00)(1) | 0.00E+00(0.00E+00)(1) | 0.00E+00(0.00E+00)(1) | 1.22E-29(1.53E-29)(4) | 9.33E-10(3.78E-09)(6) |
11 | 1.73E-09(1.11E-09)(5) | 1.68E-09(1.42E-09)(4) | 7.20E-10(7.23E-10)(2) | 1.66E-09(1.49E-09)(3) | 1.84E-09(4.63E-10)(6) | 4.83E-23(2.69E-22)(1) |
12 | 1.14E-37(2.78E-37)(2) | 4.38E-14(2.38E-14)(4) | 3.99E-14(3.18E-14)(3) | 6.62E-14(5.18E-14)(5) | 9.29E-83(5.03E-82)(1) | 6.42E+00(2.47E+00)(6) |
13 | 2.17E+01(2.56E+01)(5) | 4.85E-01(3.21E-01)(3) | 4.78E-01(2.59E-01)(2) | 4.31E-01(3.28E-01)(1) | 1.27E+01(1.14E+01)(4) | 3.54E+01(2.26E+01)(6) |
14 | 6.67E-01(3.46E-08)(2) | 6.79E-01(3.89E-02)(6) | 6.69E-01(7.55E-03)(4) | 6.71E-01(1.98E-02)(5) | 6.67E-01(5.48E-10)(1) | 6.67E-01(6.07E-05)(3) |
表2 基准测试函数上的误差结果
No. | SPSO2011 | MQHOA | NSR | PSR | BBFWA | QPSO |
---|---|---|---|---|---|---|
1 | 7.48E-03(7.73E-03)(3) | 1.02E-02(3.98E-02)(4) | 2.22E-03(4.65E-03)(1) | 4.25E-03(6.72E-03)(2) | 1.08E-02(9.11E-03)(5) | 4.31E-01(2.28E-01)(6) |
2 | 2.35E+01(6.62E+00)(1) | 1.32E+02(6.91E+01)(5) | 1.16E+02(7.83E+01)(3) | 1.21E+02(7.28E+01)(4) | 1.14E+02(3.15E+01)(2) | 1.63E+02(2.00E+01)(6) |
3 | 8.89E-01(7.77E-01)(5) | 1.96E-01(4.70E-01)(4) | 5.05E-02(2.63E-01)(2) | 1.37E-01(4.77E-01)(3) | 1.10E+00(1.05E+00)(6) | 4.33E-06(6.08E-06)(1) |
4 | 6.89E-01(6.04E-01)(5) | 7.11E-02(2.67E-01)(4) | 3.69E-02(6.25E-02)(2) | 4.08E-02(1.11E-01)(3) | 1.84E+01(1.05E+01)(6) | 7.57E-10(1.94E-09)(1) |
5 | 6.78E+00(5.82E+00)(5) | 6.68E+00(1.07E+00)(4) | 3.37E+00(6.45E+00)(2) | 1.31E+01(2.95E+00)(6) | 4.52E+00(2.18E+00)(3) | 5.89E-16(1.77E-15)(1) |
6 | 4.62E+03(5.34E+02)(1) | 7.84E+03(2.96E+02)(5) | 7.95E+03(2.55E+02)(6) | 7.13E+03(2.60E+02)(4) | 5.26E+03(6.36E+02)(2) | 6.05E+03(7.66E+02)(3) |
7 | 9.33E-101(5.77E-102)(2) | 4.83E-84(7.24E-84)(5) | 6.59E-86(8.80E-86)(4) | 4.03E-88(7.11E-88)(3) | 2.53E-107(1.55E-106)(1) | 1.64E-12(3.36E-12)(6) |
8 | 7.68E-30(1.83E-29)(2) | 1.37E-05(4.35E-05)(6) | 5.88E-06(8.74E-06)(4) | 8.03E-06(1.62E-05)(5) | 2.47E-55(1.40E-54)(1) | 1.41E-11(1.94E-11)(3) |
9 | 9.24E-101(5.34E-102)(2) | 3.44E-79(4.62E-79)(5) | 6.35E-81(1.69E-80)(4) | 2.72E-83(3.04E-83)(3) | 1.27E-103(5.92E-103)(1) | 1.06E-07(1.92E-07)(6) |
10 | 7.08E-29(1.21E-29)(5) | 0.00E+00(0.00E+00)(1) | 0.00E+00(0.00E+00)(1) | 0.00E+00(0.00E+00)(1) | 1.22E-29(1.53E-29)(4) | 9.33E-10(3.78E-09)(6) |
11 | 1.73E-09(1.11E-09)(5) | 1.68E-09(1.42E-09)(4) | 7.20E-10(7.23E-10)(2) | 1.66E-09(1.49E-09)(3) | 1.84E-09(4.63E-10)(6) | 4.83E-23(2.69E-22)(1) |
12 | 1.14E-37(2.78E-37)(2) | 4.38E-14(2.38E-14)(4) | 3.99E-14(3.18E-14)(3) | 6.62E-14(5.18E-14)(5) | 9.29E-83(5.03E-82)(1) | 6.42E+00(2.47E+00)(6) |
13 | 2.17E+01(2.56E+01)(5) | 4.85E-01(3.21E-01)(3) | 4.78E-01(2.59E-01)(2) | 4.31E-01(3.28E-01)(1) | 1.27E+01(1.14E+01)(4) | 3.54E+01(2.26E+01)(6) |
14 | 6.67E-01(3.46E-08)(2) | 6.79E-01(3.89E-02)(6) | 6.69E-01(7.55E-03)(4) | 6.71E-01(1.98E-02)(5) | 6.67E-01(5.48E-10)(1) | 6.67E-01(6.07E-05)(3) |
No. | SPSO2011 | MQHOA | NSR | PSR | BBFWA | QPSO | eaccept |
---|---|---|---|---|---|---|---|
1 | 1.00(1) | 0.98(5) | 1.00(1) | 1.00(1) | 1.00(1) | 0.10(6) | 7.77E-02 |
2 | 1.00(1) | 0.22(5) | 0.31(3) | 0.27(4) | 0.47(2) | 0.00(6) | 1.11E+02 |
3 | 0.39(6) | 0.84(4) | 0.96(2) | 0.92(3) | 0.43(5) | 1.00(1) | 3.95E-01 |
4 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.00(6) | 1.00(1) | 3.21E+00 |
5 | 0.53(4) | 0.20(5) | 0.92(2) | 0.00(6) | 0.67(3) | 1.00(1) | 5.75E+00 |
6 | 1.00(1) | 0.00(5) | 0.00(5) | 0.02(4) | 0.96(2) | 0.69(3) | 6.47E+03 |
7 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.35(6) | 2.73E-13 |
8 | 1.00(1) | 0.61(5) | 0.67(4) | 0.61(5) | 1.00(1) | 1.00(1) | 4.61E-06 |
9 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.35(6) | 1.76E-08 |
10 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.59(6) | 1.55E-10 |
11 | 0.37(5) | 0.57(3) | 0.82(2) | 0.53(4) | 0.10(6) | 1.00(1) | 1.27E-09 |
12 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.00(6) | 1.07E+00 |
13 | 0.41(5) | 1.00(1) | 1.00(1) | 1.00(1) | 0.73(4) | 0.00(6) | 1.19E+01 |
表3 基准测试函数成功率
No. | SPSO2011 | MQHOA | NSR | PSR | BBFWA | QPSO | eaccept |
---|---|---|---|---|---|---|---|
1 | 1.00(1) | 0.98(5) | 1.00(1) | 1.00(1) | 1.00(1) | 0.10(6) | 7.77E-02 |
2 | 1.00(1) | 0.22(5) | 0.31(3) | 0.27(4) | 0.47(2) | 0.00(6) | 1.11E+02 |
3 | 0.39(6) | 0.84(4) | 0.96(2) | 0.92(3) | 0.43(5) | 1.00(1) | 3.95E-01 |
4 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.00(6) | 1.00(1) | 3.21E+00 |
5 | 0.53(4) | 0.20(5) | 0.92(2) | 0.00(6) | 0.67(3) | 1.00(1) | 5.75E+00 |
6 | 1.00(1) | 0.00(5) | 0.00(5) | 0.02(4) | 0.96(2) | 0.69(3) | 6.47E+03 |
7 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.35(6) | 2.73E-13 |
8 | 1.00(1) | 0.61(5) | 0.67(4) | 0.61(5) | 1.00(1) | 1.00(1) | 4.61E-06 |
9 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.35(6) | 1.76E-08 |
10 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.59(6) | 1.55E-10 |
11 | 0.37(5) | 0.57(3) | 0.82(2) | 0.53(4) | 0.10(6) | 1.00(1) | 1.27E-09 |
12 | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 1.00(1) | 0.00(6) | 1.07E+00 |
13 | 0.41(5) | 1.00(1) | 1.00(1) | 1.00(1) | 0.73(4) | 0.00(6) | 1.19E+01 |
No. | SPSO2011 | MQHOA | NSR | PSR | BBFWA | QPSO |
---|---|---|---|---|---|---|
1 | 4.64(4) | 17.06(6) | 1.59(2) | 16.02(5) | 0.73(1) | 4.51(3) |
2 | 5.87(6) | 2.87(4) | 1.32(2) | 1.34(3) | 0.73(1) | 4.42(5) |
3 | 5.85(6) | 3.03(3) | 1.45(2) | 4.47(5) | 0.76(1) | 4.32(4) |
4 | 9.53(4) | 15.50(6) | 3.79(2) | 13.54(5) | 3.19(1) | 8.50(3) |
5 | 5.57(6) | 2.83(4) | 1.48(2) | 2.44(3) | 1.02(1) | 4.22(5) |
6 | 5.72(6) | 2.85(4) | 2.51(2) | 2.81(3) | 0.86(1) | 4.57(5) |
7 | 2.62(3) | 30.32(5) | 2.31(2) | 35.73(6) | 0.89(1) | 4.09(4) |
8 | 5.10(6) | 2.63(4) | 1.36(3) | 1.35(2) | 0.87(1) | 4.14(5) |
9 | 3.09(3) | 4.09(5) | 2.58(2) | 3.47(4) | 1.30(1) | 4.90(6) |
10 | 5.70(6) | 1.72(4) | 1.07(3) | 1.04(2) | 0.70(1) | 4.22(5) |
11 | 6.50(6) | 3.79(4) | 2.49(3) | 2.47(2) | 2.09(1) | 5.85(5) |
12 | 5.07(6) | 2.88(4) | 1.55(2) | 1.60(3) | 0.88(1) | 4.34(5) |
13 | 4.75(3) | 7.21(4) | 81.79(6) | 78.94(5) | 0.86(1) | 4.23(2) |
14 | 4.70(6) | 2.62(4) | 1.35(2) | 1.44(3) | 0.88(1) | 4.26(5) |
表4 基准测试函数平均执行时间 (s)
No. | SPSO2011 | MQHOA | NSR | PSR | BBFWA | QPSO |
---|---|---|---|---|---|---|
1 | 4.64(4) | 17.06(6) | 1.59(2) | 16.02(5) | 0.73(1) | 4.51(3) |
2 | 5.87(6) | 2.87(4) | 1.32(2) | 1.34(3) | 0.73(1) | 4.42(5) |
3 | 5.85(6) | 3.03(3) | 1.45(2) | 4.47(5) | 0.76(1) | 4.32(4) |
4 | 9.53(4) | 15.50(6) | 3.79(2) | 13.54(5) | 3.19(1) | 8.50(3) |
5 | 5.57(6) | 2.83(4) | 1.48(2) | 2.44(3) | 1.02(1) | 4.22(5) |
6 | 5.72(6) | 2.85(4) | 2.51(2) | 2.81(3) | 0.86(1) | 4.57(5) |
7 | 2.62(3) | 30.32(5) | 2.31(2) | 35.73(6) | 0.89(1) | 4.09(4) |
8 | 5.10(6) | 2.63(4) | 1.36(3) | 1.35(2) | 0.87(1) | 4.14(5) |
9 | 3.09(3) | 4.09(5) | 2.58(2) | 3.47(4) | 1.30(1) | 4.90(6) |
10 | 5.70(6) | 1.72(4) | 1.07(3) | 1.04(2) | 0.70(1) | 4.22(5) |
11 | 6.50(6) | 3.79(4) | 2.49(3) | 2.47(2) | 2.09(1) | 5.85(5) |
12 | 5.07(6) | 2.88(4) | 1.55(2) | 1.60(3) | 0.88(1) | 4.34(5) |
13 | 4.75(3) | 7.21(4) | 81.79(6) | 78.94(5) | 0.86(1) | 4.23(2) |
14 | 4.70(6) | 2.62(4) | 1.35(2) | 1.44(3) | 0.88(1) | 4.26(5) |
精度排名 | 成功率 排名 | 执行时间排名 | 平均综合排名 | |
---|---|---|---|---|
NSR | 1 | 1 | 2 | 1.33 |
PSR | 4 | 4 | 3 | 3.67 |
SPSO2011 | 3 | 1 | 6 | 3.33 |
QPSO | 5 | 6 | 5 | 5.33 |
MQHOA | 6 | 5 | 4 | 5.00 |
BBFWA | 2 | 3 | 1 | 2.00 |
表5 不同算法的平均综合排名
精度排名 | 成功率 排名 | 执行时间排名 | 平均综合排名 | |
---|---|---|---|---|
NSR | 1 | 1 | 2 | 1.33 |
PSR | 4 | 4 | 3 | 3.67 |
SPSO2011 | 3 | 1 | 6 | 3.33 |
QPSO | 5 | 6 | 5 | 5.33 |
MQHOA | 6 | 5 | 4 | 5.00 |
BBFWA | 2 | 3 | 1 | 2.00 |
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