Efficient Chaotic Imperialist Competitive Algorithm with Dropout Strategy for Global Optimization
Abstract
:1. Introduction
2. Literature Review
2.1. Imperialist Competitive Algorithm (ICA)
2.2. Chaotic Imperialist Competitive Algorithm (CICA)
2.3. Dropout
3. Chaotic Imperialist Competitive Algorithm with Dropout (CICA-D)
Algorithm 1 CICA-D |
|
4. Numerical Examples
4.1. Success Rate
4.2. Statistical Results
4.3. Computational Complexity
5. Application
5.1. Objective Function
5.2. Experimental Results
- The optimal solution passes through the obstacles.
- The optimal solution is worse than the median of all trails.
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Map | Definition | Parameters |
---|---|---|
Logistic map [19] | ||
ICMIC map [20] | ||
Sinusoidal map [19] | ||
Gauss map [21] | ||
Tent map [22] | ||
Circle map [23] | ||
Complex squaring map [24] |
Function | Definition | Interval | Optimum |
---|---|---|---|
Griewank | [−150, 150] | 0.0 | |
Ackley | [−32, 32] | 0.0 | |
Brown | [−1, 4] | 0.0 | |
Rastrigin | [−10, 10] | 0.0 | |
Schwefel’s 2.22 | [−100, 100] | 0.0 | |
Schwefel’s 2.23 | [−10, 10] | 0.0 | |
Qing | [−500, 500] | 0.0 | |
Rosenbrock | [−2.048, 2.048] | 0.0 | |
Schwefel | [−10, 10] | 0.0 | |
Weierstrass | [−0.5, 0.5] | 0.0 | |
Whitley | [−10.24, 10.24] | 0.0 | |
Zakharov | [−5, 10] | 0.0 |
CICA | CICA-D(0.1) | CICA-D(0.2) | CICA-D(0.3) | CICA-D(0.4) | CICA-D(0.5) | |
---|---|---|---|---|---|---|
Logistic map | 76 | 63 | 56 | 38 | 49 | 52 |
ICMIC map | 55 | 47 | 42 | 39 | 31 | 36 |
Sinusoidal map | 89 | 83 | 63 | 72 | 54 | 67 |
Gauss map | 23 | 21 | 19 | 23 | 14 | 16 |
Tent map | 32 | 29 | 22 | 24 | 19 | 26 |
Circle map | 25 | 22 | 18 | 8 | 13 | 10 |
Complex squaring map | 39 | 32 | 27 | 25 | 29 | 27 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 2.6990e-11 | 1.0341e-10 | 2.6780e-08 | 8.1404e-10 |
CICA | 1.1707e-16 | 3.4777e-14 | 2.5794e-12 | 5.0708e-15 |
CICA-D | 0 | 1.0767e-08 | 2.9765e-07 | 5.4310e-08 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 8.3538e-07 | 7.1169e-05 | 9.7544e-05 | 8.2014e-06 |
CICA | 5.7959e-08 | 1.0239e-07 | 5.1388e-06 | 1.2366e-07 |
CICA-D | 2.4248e-11 | 1.8701e-06 | 9.4602e-06 | 3.0191e-06 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 5.3185e-03 | 8.2913e-02 | 3.4768e-01 | 5.9350e-02 |
CICA | 0 | 3.6893e-04 | 7.6523e-04 | 2.3507e-04 |
CICA-D | 0 | 1.6878e-03 | 3.6494e-03 | 1.1136e-03 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 0 | 1.6667e-06 | 0.00005 | 9.1287e-06 |
CICA | 0 | 9.3427e-09 | 1.0685e-07 | 3.4296e-08 |
CICA-D | 0 | 1.0604e-07 | 1.9899e-06 | 3.9926e-07 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 2.1853e-06 | 3.6150e-05 | 4.3900e-05 | 1.0507e-05 |
CICA | 8.4180e-09 | 1.3307e-08 | 1.5124e-08 | 1.8661e-09 |
CICA-D | 7.6125e-10 | 7.7876e-07 | 3.5125e-05 | 3.1734e-06 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 6.2357e-11 | 2.4227e-09 | 8.1524e-09 | 1.9060e-09 |
CICA | 8.4576e-14 | 6.3189e-12 | 4.8157e-11 | 4.2206e-12 |
CICA-D | 3.6451e-15 | 1.2597e-09 | 7.6530e-08 | 6.9023e-09 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 3.7096e-04 | 1.2428e-03 | 5.8762e-02 | 5.2740e-03 |
CICA | 5.2507e-08 | 7.8159e-08 | 8.3169e-07 | 6.9295e-08 |
CICA-D | 1.8346e-11 | 6.8298e-07 | 7.4924e-07 | 1.9475e-07 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 0.001296 | 0.201608 | 1.217682 | 0.362075 |
CICA | 0.000182 | 0.024174 | 0.07179 | 0.021891 |
CICA-D | 0.000061 | 0.081672 | 0.36303 | 0.101833 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 3.2679e-06 | 2.9866e-05 | 7.2682e-05 | 8.4658e-06 |
CICA | 6.2612e-09 | 8.1053e-09 | 1.5335e-08 | 1.2321e-09 |
CICA-D | 5.3896e-10 | 6.4659e-08 | 4.2363e-06 | 3.8251e-07 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 2.7647e-03 | 3.2945e-02 | 6.3290e-02 | 1.8813e-02 |
CICA | 0 | 1.5156e-05 | 3.1263e-05 | 9.8972e-06 |
CICA-D | 0 | 3.9078e-04 | 7.9613e-04 | 2.5196e-04 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 5.3185e-09 | 3.0960e-08 | 3.4768e-08 | 7.4846e-09 |
CICA | 6.8309e-14 | 9.5997e-13 | 7.6523e-12 | 6.5952e-13 |
CICA-D | 4.2985e-15 | 7.3188e-11 | 3.6494e-10 | 3.3985e-11 |
Min(best) | Mean | Max(worst) | St.Dev. | |
---|---|---|---|---|
ICA | 8.3546e-11 | 4.3202e-09 | 3.7554e-08 | 4.2478e-09 |
CICA | 4.4263e-13 | 6.4897e-10 | 5.4924e-09 | 6.2908e-10 |
CICA-D | 3.8908e-15 | 8.2997e-09 | 8.8761e-08 | 7.7855e-09 |
ICA | CICA | CICA-D(0.1) | CICA-D(0.3) | CICA-D(0.5) | ||
---|---|---|---|---|---|---|
Fitness | Min. | 64.21 | 61.83 | 60.93 | 61.98 | 63.33 |
Mean | 66.50 | 63.49 | 64.60 | 66.13 | 67.76 | |
Max. | 69.76 | 67.91 | 70.13 | 70.97 | 72.41 | |
St. Dev. | 1.6335 | 1.1657 | 2.7274 | 2.5325 | 2.7108 | |
Execution | Mean | 28.67 | 34.91 | 12.87 | 19.81 | 36.03 |
Time (sec) | St. Dev. | 14.28 | 18.15 | 7.24 | 8.14 | 14.56 |
Success Rate | 86.67% | 93.33% | 87.50% | 82.50% | 75.83% |
ICA | CICA | CICA-D(0.1) | CICA-D(0.3) | CICA-D(0.5) | ||
---|---|---|---|---|---|---|
Fitness | Min. | 67.42 | 64.11 | 63.89 | 64.02 | 64.57 |
Mean | 69.68 | 66.18 | 67.57 | 70.28 | 72.62 | |
Max. | 75.65 | 73.71 | 79.71 | 79.52 | 83.55 | |
St. Dev. | 1.5567 | 1.3839 | 3.0785 | 4.0630 | 5.7615 | |
Execution | Mean | 52.18 | 57.89 | 17.98 | 46.57 | 69.05 |
Time (sec) | St. Dev. | 12.41 | 14.59 | 4.79 | 12.48 | 19.24 |
Success Rate | 80.83% | 88.33% | 82.50% | 76.67% | 70.83% |
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Wang, Z.-S.; Lee, J.; Song, C.G.; Kim, S.-J. Efficient Chaotic Imperialist Competitive Algorithm with Dropout Strategy for Global Optimization. Symmetry 2020, 12, 635. https://doi.org/10.3390/sym12040635
Wang Z-S, Lee J, Song CG, Kim S-J. Efficient Chaotic Imperialist Competitive Algorithm with Dropout Strategy for Global Optimization. Symmetry. 2020; 12(4):635. https://doi.org/10.3390/sym12040635
Chicago/Turabian StyleWang, Zong-Sheng, Jung Lee, Chang Geun Song, and Sun-Jeong Kim. 2020. "Efficient Chaotic Imperialist Competitive Algorithm with Dropout Strategy for Global Optimization" Symmetry 12, no. 4: 635. https://doi.org/10.3390/sym12040635