A Comparative Study of Common Nature-Inspired Algorithms for Continuous Function Optimization
Abstract
:1. Introduction
1.1. Summary of the Current Survey Work
1.2. Motivations
1.3. Research Methodology
1.4. Scope of Discussion
1.5. Our Contributions
1.6. Structure of the Paper
2. Common NIOAs
2.1. The Common Process for the 11 NIOAs
2.2. The Principles of the 11 NIOAs
2.2.1. Genetic Algorithm (GA)
Algorithm 1 GA |
|
2.2.2. Particle Swarm Optimization (PSO) Algorithm
Algorithm 2 PSO |
|
2.2.3. Artificial Bee Colony (ABC) Algorithm
Algorithm 3 ABC |
|
2.2.4. Bat Algorithm (BA)
Algorithm 4 BA |
|
2.2.5. Immune Algorithm (IA)
Algorithm 5 IA |
|
2.2.6. Firefly Algorithm (FA)
Algorithm 6 FA |
|
2.2.7. Cuckoo Search (CS) Algorithm
Algorithm 7 CS |
|
2.2.8. Differential Evolution (DE) Algorithm
Algorithm 8 DE |
|
2.2.9. Gravitational Search Algorithm (GSA)
Algorithm 9 GSA |
|
2.2.10. Grey Wolf Optimizer (GWO)
Algorithm 10 GWO |
|
2.2.11. Harmony Search (HS) Optimization
Algorithm 11 HS |
|
3. Theoretical Comparison and Analysis of the 11 NIOAs
3.1. Common Characteristics
3.2. Variant Methods of Common NIOAs
3.3. Differences
4. Performance Comparison and Analysis for the 11 NIOAs
4.1. The Description of BBOB Test Functions
4.2. Performance Comparison and Analysis on Benchmark Functions
4.2.1. The Comparison and Analysis on the Accuracy, Stability and Parameter Sensitivity
4.2.2. The Efficiency Comparison and Analysis
4.2.3. The Comparison of Running Time
4.3. Statistical Tests for Algorithm Comparison
4.4. Performance Comparison on Engineering Optimization Problem
5. Challenges and Future Directions
6. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Conceptions | Symbols | Description |
---|---|---|
Space dimension | The problem space description | |
Population size | Individual quantity | |
Iteration times | Algorithm termination condition | |
Individual position | The expression of the ith solution on the tth iteration, also used to represent the ith individual | |
Local best solution | Local best solution of the ith individual on the tth iteration | |
Global best solution | Global best solution of the whole populationon the tth iteration | |
Fitness function | Unique standard to evaluate solutions | |
Precision threshold | Algorithm termination condition |
NIOAs | Multiple Objectives | Adaptive | Spatial Property | Hybridization | ||||
---|---|---|---|---|---|---|---|---|
Discrete | Continuous | Fuzzy Theory | Chaos Theory | Combination among NIOAs | Others | |||
GA | [37]3594 [38]44334 | [39]204 | [40]142 [41]73 | [42]1405 [43]831 | [44]305 | [45]195 | [46]420 [47]1373 | [43]831 [48]355 |
PSO | [49]800 [50]67 | [51]2363 [52]914 | [53]732 [54]640 | [55]296 | [52]914 | [56]252 | [57]381 [58]11 | [59]214 [60]154 [50]67 |
ABC | [61]334 | [62]47 | [63]235 [64]851 | [5]3932 | [65]42 | [66]197 | [67]122 | [68]428 |
BA | [69]433 | [70]204 | [71]560 [72]285 | [73]136 | [74]27 | [75]158 | [76]64 | [73]136 |
FA | [77]81 | [78]66 | [79]43 [80]165 | [81]142 | [82]45 | [83]140 | [84]99 | [85]56 |
IA | [86]166 | [87]97 | [88]157 | [89]141 | [90]205 [91]17 | [91]17 | [92]166 | [93]230 |
CS | [94]192 | [95]114 | [96]142 [97]438 | [7]2801 | [98]41 | [99]104 | [100]77 | [101]308 |
DE | [102]350 | [103]198 | [104]375 [105]219 | [4]1925 | [106]251 | [107]86 | [108]70 [109]257 | [110]81 |
GSA | [111]135 | [112]216 | [113]133 [114]114 | [12]5909 | [115]154 | [116]145 | [117]253 | [118]152 |
GWO | [119]627 | [120]60 | [121]28 | [16]6135 | [122]225 | [123]188 | [124]29 | [125]105 |
HS | [126]221 | [127]186 | [128]429 | [18]6808 | [129]38 | [130]345 | [131]194 | [132]133 |
NIOAs | Time Complexity | Comments |
---|---|---|
PSO | Tupd = Tvec + Tpos = D∙M + D∙M = 2∙D∙M; O(TPSO) = O(D∙M + (3∙D∙M + M)∙N) ≈ O(D∙M∙N) | Tupd denotes the cost of updating velocity (Tvec) and position (Tpos) |
GA | Tupd = Tcross + Tmut = D∙M + D∙M = 2∙D∙M; O(TGA) = O(D∙M + (2∙M∙D + M)∙N) ≈ O(D∙M∙N) | Tupd denotes the cost of crossover (Tcross) and mutation (Tmut) operations |
ABC | Tupd = Temp + Tsct + Tonk = D∙M/2 + D∙M/2 + M = D∙M + M; O(TABC) =O(D∙M + (2∙M∙D + 2M)∙N) ≈ O(D∙M∙N) | Tupd denotes the cost of updating the positions of employed foragers (Temp), scouts (Tsct) and onlookers (Tonk) |
BA | Tupd = Tfreq + Tvec + Tpos = D∙M+ D∙M+ D∙M = 3∙D∙M; O(TBA) = O(D∙M + (4∙M∙D + M)∙N) ≈ O(D∙M∙N) | Tupd denotes the cost of updating the frequency (Tfreq), velocity (Tvec) and positions (Tpos) |
IA | Tupd = Tden + Tact + Tcross+ Tmut =M∙M + M+ D∙M + D∙M = M(M + 1) + 2 D∙M; O(TIA) = O(D∙M +(M∙(M +4) + 3∙M∙D)∙N) ≈ O(D∙M∙N + M∙M∙N) =O((D + M)∙M∙N) | Tupd is the cost of updating the density (Tden), activity (Tact), crossover (Tcross) and mutation (Tmut) operations |
FA | Tupd = M∙M∙D; O(TFA) = O(D∙M +(M∙M∙D + M∙D + M)∙N) ≈ O(D∙M2∙N) | Tupd is the cost of updating the positions of fireflies |
CS | Tupd = 2∙M∙D; O(TCS) = O(D∙M + (3∙M∙D + M)∙N) ≈ O(D∙M∙N) | Tupd is the cost of updating the host nests of cuckoos |
DE | Tupd = M∙D+ M∙D+ M; O(TDE) = O(D∙M +(4M∙D + 2∙M)∙N) ≈ O(D∙M∙N) | Tupd is the cost of crossover mutation and selection operations |
GSA | Tupd = Tgrav + Tvec + Tpos = M∙M+ M∙D+ M∙D; O(TGSA) = O(D∙M + (M∙M + 3∙M∙D + M)∙N) ≈ O(D∙M∙N + M∙M∙N) = O((D + M)∙M∙N) | Tupd is the cost of updating gravitational acceleration (Tgrav), velocity (Tvec) and position (Tpos) |
GWO | Tupd = M∙D; O(TGWO) = O(D∙M + (2M∙D + M)∙N) ≈ O(D∙M∙N) | Tupd denotes the cost of updating the positions of wolves |
HS | Tupd = M∙D; O(THS) = O(D∙M + (2M∙D + M)∙N) ≈ O(D∙M∙N) | Tupd is the cost of updating harmony vectors |
Algorithms | Parameters I | Parameters II |
---|---|---|
GA | = 1, = 0.8, = 0.2 | = 1, = 0.75, = 0.25 |
PSO | = 2, = 2 | = 1.5, = 1.5 |
ABC | , Limit = 20 | , Limit = 30 |
BA | = 0.9, = 0.9 = 100, = 1, = 100, = 1 | = 0.8, = 0.8 = 150, = 1, = 150, = 1 |
IA | = 0.8, = 0.2 | = 0.75, = 0.25 |
FA | = 0.6, step = 0.4, = 1 | = 0.5, step = 0.5, = 1.1 |
CS | = 1, = 0.25 | = 1.1, = 0.15 |
DE | F = 0.5, CR = 0.1 | F = 0.6, CR = 0.2 |
GSA | = 100, = 20 | = 90, = 15 |
GWO | None | None |
HS | HMCR = 0.995, PAR = 0.4, BW = 1 | HMCR = 0.85, PAR = 0.5, BW = 0.9 |
Criteria | WORST | AVERGAE | BEST | STD | |
---|---|---|---|---|---|
NIOAs | |||||
DE | D = 10 | 10 (F5, F6, F8, F9, F10, F17, F20, F27, F29, F30) | 10 (F5, F6, F8, F9, F10, F17, F19, F20, F27, F30) | 13 (F6, F9, F11, F14, F15, F16, F17, F18, F19, F20, F21, F27, F30) | 7 (F5, F6, F8, F9, F23, F28, F30) |
D = 50 | 9 (F4, F6, F8, F16, F20, F25, F27, F28, F30) | 8 (F6, F9, F11, F20, F25, F27, F29, F30) | 8 (F6, F9, F11, F20, F22, F25, F27, F30) | 7 (F4, F6, F21, F25, F27, F28, F30) | |
CS | D = 10 | 17 (F2, F3, F4, F11, F12, F13, F14, F15, F16,F18, F19, F21, F23, F24, F25, F26, F28) | 16 (F2, F3, F4, F11, F12, F13, F14, F15, F16, F18, F21, F22, F24, F25, F26, F28) | 7 (F2, F3, F4, F12, F13, F25, F26) | 17 (F2, F3, F4, F10, F11, F12, F13, F14, F15, F16, F17, F18, F19, F20, F21, F27, F29) |
D = 50 | 8 (F3, F12, F14, F15, F18, F19, F24, F29) | 10 (F1, F3, F4, F12, F13, F14, F15, F18, F19, F28) | 8 (F3, F4, F12, F13, F14, F18, F19, F28) | 9 (F3, F10, F14, F15, F16, F18, F19, F22, F29) | |
HS | D = 10 | - | - | - | - |
D = 50 | 2 (F9, F23) | 2 (F8, F23) | 5 (F5, F8, F21, F23, F29) | 3 (F9, F23, F24) | |
GSA | D = 10 | 4 (F1, F7, F9, F22) | 3 (F1, F7, F9) | 3 (F1, F7, F9) | 5 (F1, F7, F9, F22, F25) |
D = 50 | 6 (F1, F2, F7, F11, F13, F26) | 3 (F2, F7, F26) | 4 (F1, F2, F7, F26) | 9 (F1, F2, F5, F7, F8, F11, F12, F13, F26) | |
GWO | D = 10 | - | 1 (F29) | 3 (F5, F8, F22) | 2 (F25, F26) |
D = 50 | 3 (F5, F17, F21) | 7 (F5, F10, F16, F17, F21, F22, F24) | 3 (F16, F17, F24) | - | |
ABC | D = 10 | - | 1 (F23) | 5 (F10, F23, F24, F28, F29) | - |
D = 50 | 2 (F10, F22) | - | - | 1 (F17) | |
PSO | D = 10 | - | - | 1 (F6) | - |
D = 50 | - | - | 2 (F10, F15) | - | |
FA | D = 10 | - | - | - | - |
D = 50 | - | - | - | 1 (F20) | |
BA | - | - | - | - | |
GA | - | - | - | - | |
IA | - | - | - | - |
Criteria | WORST | AVERGAE | BEST | STD | |
---|---|---|---|---|---|
NIOAs | |||||
DE | D = 10 | 11 (F5, F6, F8, F10, F14, F15, F17, F18, F19, F20, F30) | 12 (F5, F6, F8, F15, F17, F18, F19, F20, F23, F27, F29, F30) | 14 (F5, F6, F8, F9, F14, F15, F16, F17, F18, F19, F20, F23, F27, F30) | 9 (F5, F6, F8, F15, F18, F19, F25, F28, F30) |
D = 50 | 7 (F4, F6, F20, F25, F26, F27, F28) | 4 (F6, F9,F25, F27) | 5 (F6, F9, F25, F27, F30) | 5 (F4, F6, F25, F27, F28) | |
CS | D = 10 | 16 (F1, F2, F3, F4, F11, F12, F13, F16, F21, F23, F24, F25, F26, F28, F29) | 14 (F2, F3, F4, F11, F12, F13, F14,F16, F21, F22, F24, F25, F26, F28) | 11 (F2, F3, F4, F11, F12, F13, F21, F22, F25, F26, F28) | 15 (F1, F2, F3, F4, F11, F12, F13, F14, F16, F17, F20, F21, F23, F27, F29) |
D = 50 | 12 (F1, F2, F11, F12, F13, F14, F15, F18, F19, F24, F29, F30) | 13 (F1, F2, F4, F11, F12, F13, F14, F15, F18, F19, F28, F29, F30) | 11 (F2, F4, F11, F12, F13, F14,F15, F18, F19, F28, F29) | 12 (F1, F2, F11, F12, F13, F14, F15, F17, F18, F19, F29, F30) | |
HS | D = 10 | - | - | - | 1 (F24) |
D = 50 | - | - | - | 7 (F5, F8, F16, F21, F23, F24, F26) | |
GSA | D = 10 | 3 (F7, F9, F22) | 3 (F1, F7, F9) | 3 (F1, F7, F9) | 3 (F7, F9, F22) |
D = 50 | 2 (F7, F10) | 3 (F7, F10, F26) | 4 (F1, F7, F10, F26) | - | |
GWO | D = 10 | - | 1 (F10) | 2 (F10, F29) | - |
D = 50 | 7 (F5, F8, F16, F17, F21, F22, F23) | 9 (F5, F8, F16, F17, F20, F21, F22, F23, F24) | 8 (F5, F8, F16, F17, F20, F21, F23, F24) | - | |
ABC | D = 10 | - | - | - | 1 (F26) |
D = 50 | - | - | 1 (F22) | - | |
PSO | D = 10 | - | - | 1 (F24) | - |
D = 50 | 1 (F3) | 1 (F3) | 1 (F3) | 3 (F3, F7) | |
FA | D = 10 | - | - | - | 1 (F10) |
D = 50 | - | - | - | 2 (F20, F22) | |
GA | D = 10 | - | - | - | - |
D = 50 | 1 (F9) | - | - | 2 (F9, F10) | |
BA | - | - | - | - | |
IA | - | - | - | - |
Criteria | WORST | AVERGAE | BEST | STD | |
---|---|---|---|---|---|
NIOAs | |||||
DE | D = 10 | - | - | - | - |
D = 50 | 7 (F1 **, F2, F9, F12, F13 ***, F15 **, F30 **) | 8 (F1 **, F2, F7, F10, F12, F13 **, F15, F30) | 7 (F2, F3, F7, F8, F10, F18, F22) | 11 (F1 **, F2, F3, F4, F9, F12, F13 **, F15, F18, F22, F30 **) | |
CS | D = 10 | - | - | - | - |
D = 50 | 2 (F1, F18) | 1 (F18) | 2 (F8, F12) | 2 (F14, F29) | |
HS | D = 10 | - | - | - | - |
D = 50 | 8 (F1 ***, F2, F4, F5, F9, F12 **, F13, F30) | 8 (F1 ***, F2 **, F4, F8, F12 **, F13, F19, F30) | 8 (F1 **, F2, F4, F8, F12 **, F13, F19, F30) | 10 (F1 ***, F2, F4, F9, F12 **, F13, F15, F18, F25, F30) | |
GSA | D = 10 | - | - | - | - |
D = 50 | 6 (F1, F2, F9, F12 **, F13, F14 **) | 3 (F12 ***, F14, F22) | 4 (F8, F14, F19, F22) | 7 (F2 **, F9, F12 ***, F13, F14 **, F18, F19) | |
GWO | D = 10 | - | - | - | - |
D = 50 | 4 (F1, F9, F18, F19) | 3 (F4, F7, F13) | 3 (F1, F2, F19) | 3 (F13, F19, F24) | |
ABC | D = 10 | 2 (F18, F30) | 2 (F18, F30) | 2 (F18, F30) | 2 (F18, F30) |
D = 50 | 6 (F1, F11, F12, F13, F18, F19) | 2 (F1, F15) | 1 (F19) | 4 (F12, F13, F14, F19) | |
PSO | D = 10 | 12 (F1 **, F2, F3, F7, F9, F12 **, F13, F14, F15, F18, F19, F30) | 11 (F1 **, F2, F3, F9, F12, F13, F14, F15, F18, F19, F30) | 8 (F1 **, F3, F9, F12, F13, F14, F18, F30) | 12 (F1 ***, F2 **, F3 **, F7, F9, F12, F13, F14, F15 **, F18, F19, F30) |
D = 50 | 12 (F1 **, F2, F3 **, F4 **, F11, F12, F13 **, F14 **, F15 **, F18 **, F19 **, F30) | 14 (F1 **, F2 **, F3 **, F4 **, F11, F12 **, F13 **, F14 ***, F15 **, F18, F19 **, F26, F28, F30 **) | 15 (F1 ***, F2 **, F3 **, F4 **, F9, F11, F12 **, F13 **, F14 **, F15 **, F18 **, F19 **, F26, F28, F30) | 12 (F1 **, F2 **, F3 **, F4 **, F11 ***, F12, F13 **, F14 ***, F15 ***, F18 **, F19 ***, F30 **) | |
FA | D = 10 | - | - | - | - |
D = 50 | 4 (F2, F14, F17, F29) | 4 (F12, F14, F18, F29) | 4 (F12, F14, F15, F17) | 4 (F1, F13, F14, F17) | |
BA | D = 10 | - | - | - | - |
D = 50 | 3 (F2, F11, F19) | 6 (F2, F3, F12, F14, F15, F26) | 2 (F3, F9) | 6 (F2, F4, F11, F13, F19, F28) | |
GA | D = 10 | 3 (F1, F18, F19) | 1 (F1) | 1 (F1) | 2 (F1, F18) |
D = 50 | 4 (F15, F19, F26, F30) | 6 (F1, F14, F15, F18, F19, F26) | 5 (F1 **, F13, F15, F18, F26) | 4 (F1, F11, F15, F30) | |
IA | D = 10 | - | - | - | - |
D = 50 | - | 1 (F14) | - | 2 (F1, F13) |
Dimensions | NIOAs Parameters | Criteria | ||
---|---|---|---|---|
10-dimensional space | Parameters I | WORST | 89.9707 | 1.8634 |
BEST | 79.0949 | |||
AVERAGE | 94.9530 | |||
STD | 34.6416 | |||
Parameters II | WORST | 80.1552 | ||
BEST | 78.3713 | |||
AVERAGE | 95.4905 | |||
STD | 27.9553 | |||
50-dimensional space | Parameters I | WORST | 68.9997 | |
BEST | 69.7277 | |||
AVERAGE | 71.4619 | |||
STD | 32.7366 | |||
Parameters II | WORST | 61.3683 | ||
BEST | 92.1188 | |||
AVERAGE | 75.6435 | |||
STD | 14.9259 |
Algorithm | WORST | AVERAGE | BEST | STD |
---|---|---|---|---|
GA | 0.029080 | 0.016709 | 0.012691 | 0.004163 |
PSO | 0.030457 | 0.015028 | 0.012746 | 0.005420 |
ABC | 0.016446 | 0.014202 | 0.012827 | 0.001062 |
BA | 0.044217 | 0.023193 | 0.013194 | 0.011202 |
IA | 0.031477 | 0.021735 | 0.013134 | 0.006702 |
FA | 0.012880 | 0.012733 | 0.012718 | 3.48 × 10−5 |
CS | 0.012670 | 0.012666 | 0.012665 | 1.27 × 10−6 |
DE | 0.013397 | 0.013007 | 0.012755 | 0.000201 |
GSA | 0.013073 | 0.012953 | 0.012740 | 9.06 × 10−5 |
GWO | 0.012821 | 0.012715 | 0.012672 | 3.00 × 10−5 |
HS | 0.032620 | 0.020375 | 0.012877 | 0.006328 |
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Wang, Z.; Qin, C.; Wan, B.; Song, W.W. A Comparative Study of Common Nature-Inspired Algorithms for Continuous Function Optimization. Entropy 2021, 23, 874. https://doi.org/10.3390/e23070874
Wang Z, Qin C, Wan B, Song WW. A Comparative Study of Common Nature-Inspired Algorithms for Continuous Function Optimization. Entropy. 2021; 23(7):874. https://doi.org/10.3390/e23070874
Chicago/Turabian StyleWang, Zhenwu, Chao Qin, Benting Wan, and William Wei Song. 2021. "A Comparative Study of Common Nature-Inspired Algorithms for Continuous Function Optimization" Entropy 23, no. 7: 874. https://doi.org/10.3390/e23070874