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
APA StyleWang, Z.-S., Lee, J., Song, C. G., & Kim, S.-J. (2020). Efficient Chaotic Imperialist Competitive Algorithm with Dropout Strategy for Global Optimization. Symmetry, 12(4), 635. https://doi.org/10.3390/sym12040635