A Hybrid Intelligent Optimization Algorithm Based on a Learning Strategy
Abstract
:1. Introduction
1.1. Motivation and Incitement
1.2. Literature Review
1.3. Contribution and Paper Organization
2. Hybrid Intelligent Optimization Algorithm Based on Multiple Populations (SDIQ)
2.1. The Basic Principles and Procedures of Algorithms
2.2. Initial Optimization and Sorting of Population Particles
2.3. Optimization of the Subpopulations
- (1)
- Subpopulation Optimization Algorithm: DE algorithm
- (2)
- Subpopulation Optimization Algorithm: IPSO Algorithm
- (3)
- Subpopulation Optimization Algorithm: QPSO algorithm
2.4. Inter-Population Learning Strategies
3. Solving and Discussing Standard Test Functions
4. Solution of Classic Engineering Optimization Problems
4.1. Minimum Cost Welded Beam Design
4.2. Minimum Mass Design of Telescopic Springs
4.3. Design of Pressure Vessel
5. Conclusions
- Computation time issue: Although the SDIQ algorithm enhances global optimization capabilities through multiple population iterations and mutual learning strategies, this approach also incurs longer computation times due to increased complexity. This can be a significant drawback when dealing with large-scale optimization problems or real-time applications.
- Parameter setting issue: The SDIQ algorithm includes many parameters that control the algorithmic process. Achieving optimal performance requires fine-tuning these parameters, which can be time-consuming and may require domain-specific knowledge.
- Scalability issue: While the algorithm performs well on tested benchmark functions and specific engineering problems, designing a basic framework to integrate more excellent algorithms requires further validation.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Algorithm Name | Abbreviation |
---|---|
simulated annealing | SA |
differential evolution | DE |
quantum behaved particle swarm optimization | QPSO |
improved particle swarm optimization | IPSO |
genetic algorithm | GA |
ant colony optimization | ACO |
particle swarm optimization | PSO |
arithmetic optimization algorithm | AOA |
equilibrium optimizer | EO |
dwarf mongoose optimization algorithm | DMO |
starling murmuration optimizer | SMO |
reptile search algorithm | RSA |
gaze cues learning-based grey wolf optimizer | GGWO |
Standard Test Functions | Problem Dimensions | Variable Value Range | Theoretical Minimum Value |
---|---|---|---|
30 | [−100, 100] | 0 | |
30 | [−10, 10] | 0 | |
30 | [−100, 100] | 0 | |
30 | [−100, 100] | 0 | |
30 | [−30, 30] | 0 | |
30 | [−100, 100] | 0 | |
30 | [−1.28, 1.28] | 0 | |
30 | [−500, 500] | −12,569.5 | |
30 | [−5.12, 5.12] | 0 | |
30 | [−600, 600] | 0 |
Function | SDIQ | GWO [29] | ||||||
---|---|---|---|---|---|---|---|---|
Best | Worst | Ave | STD | Best | Worst | Ave | STD | |
f1 | 0.00000 | 3.84802 × 10−40 | 6.41584 × 10−43 | 1.57095 × 10−41 | 1.49549 × 10−40 | 6.55408 × 10−37 | 1.66953 × 10−38 | 3.63264 × 10−38 |
f2 | 0.00000 | 7.43841 × 10−24 | 1.53805 × 10−26 | 3.04406 × 10−25 | 7.97178 × 10−24 | 5.89467 × 10−22 | 9.84876 × 10−23 | 7.90054 × 10−23 |
f3 | 0.00000 | 5.56699 × 10−43 | 4.85527 × 10−45 | 3.57391 × 10−44 | 2.67978 × 10−40 | 2.39070 × 10−35 | 2.96431 × 10−37 | 1.08494 × 10−36 |
f4 | 0.00000 | 2.67335 × 10−20 | 3.51276 × 10−21 | 2.85925 × 10−21 | 2.09586 × 10−11 | 1.32510 × 10−8 | 9.14614 × 10−10 | 1.15566 × 10−9 |
f5 | 1.34298 × 101 | 1.84234 × 101 | 1.03031 × 101 | 3.11039 × 10−1 | 2.44374 × 101 | 2.87278 × 101 | 2.64815 × 101 | 6.91464 × 10−1 |
f6 | 0.00000 | 6.54408 × 10−21 | 1.09097 × 10−23 | 2.67161 × 10−22 | 1.46335 × 10−5 | 1.01275 × 100 | 2.73532 × 10−1 | 2.36999 × 10−1 |
f7 | 4.95241 × 10−1 | 8.91035 × 10−1 | 5.63591 × 10−1 | 2.68807 × 10−1 | 7.93151 × 10−5 | 2.76708 × 10−3 | 7.26837 × 10−4 | 3.97228 × 10−4 |
f8 | −1.12073 × 104 | −8.08828 × 103 | −9.72596 × 103 | 5.41181 × 102 | −8.21756 × 103 | −3.27530 × 103 | −6.35643 × 103 | 7.41731 × 102 |
f9 | 0.00000 | 1.19421 × 102 | 4.55210 × 10−1 | 1.30195 × 10−1 | 0.00000 | 1.46701 × 101 | 1.09811 × 100 | 2.47260 × 100 |
f10 | 0.00000 | 1.32210 × 10−3 | 2.16567 × 10−5 | 8.55356 × 10−5 | 7.69716 × 10−6 | 4.20950 × 10−1 | 1.09883 × 10−1 | 6.83938 × 10−2 |
Function | DE [13] | PSO [12] | ||||||
Best | Worst | Ave | STD | Best | Worst | Ave | STD | |
f1 | 2.11199 × 10−22 | 3.55143 × 10−15 | 5.99460 × 10−18 | 1.44984 × 10−16 | 8.99071 × 10−9 | 1.92746 × 10−5 | 6.05070 × 10−7 | 1.18895 × 10−6 |
f2 | 3.21178 × 10−12 | 1.00000 × 101 | 7.94848 × 10−2 | 8.71385 × 10−1 | 6.14204 × 10−5 | 2.32962 × 10−2 | 1.49228 × 10−3 | 2.41518 × 10−3 |
f3 | 1.58109 × 10−21 | 1.00000 × 104 | 3.33333 × 101 | 5.76868 × 102 | 1.23907 × 10−7 | 1.56786 × 10−4 | 7.94466 × 10−6 | 1.20159 × 10−5 |
f4 | 2.31328 × 10−1 | 1.27650 × 101 | 3.02171 × 100 | 1.89506 × 100 | 2.60656 × 10−1 | 1.32110 × 100 | 6.36342 × 10−1 | 1.59502 × 10−1 |
f5 | 9.94531 × 10−2 | 9.00236 × 104 | 2.09016 × 102 | 3.67529 × 103 | 4.93832 × 100 | 6.36935 × 102 | 6.86167 × 101 | 6.73738 × 101 |
f6 | 8.52562 × 10−23 | 1.06447 × 10−16 | 3.89201 × 10−19 | 5.70137 × 10−18 | 4.26701 × 10−9 | 9.18553 × 10−6 | 6.41544 × 10−7 | 9.04545 × 10−7 |
f7 | 5.80018 × 10−3 | 5.15659 × 10−2 | 1.85132 × 10−2 | 7.73621 × 10−3 | 1.95837 × 10−2 | 2.35668 × 10−1 | 8.21903 × 10−2 | 2.97511 × 10−2 |
f8 | −1.12819 × 104 | −7.81519 × 103 | −9.81169 × 103 | 5.56089 × 102 | −9.32654 × 103 | −2.75943 × 103 | −6.47824 × 103 | 9.85509 × 102 |
f9 | 1.39294 × 101 | 2.29943 × 102 | 5.07434 × 101 | 1.84017 × 101 | 1.99675 × 101 | 9.49988 × 101 | 4.37058 × 101 | 1.08193 × 101 |
f10 | 0.00000 | 2.12503 × 10−1 | 1.16214 × 102 | 1.66676 × 10−2 | 8.23491 × 10−10 | 4.91932 × 10−2 | 9.27812 × 10−3 | 9.65047 × 10−3 |
Function | SDIQ | GWO | DE | PSO |
---|---|---|---|---|
Mean Rank | Mean Rank | Mean Rank | Mean Rank | |
f1 | 1.00167 | 1.99833 | 3.00000 | 4.00000 |
f2 | 1.00000 | 2.00000 | 3.01167 | 3.98833 |
f3 | 1.00167 | 1.99833 | 3.01333 | 3.98667 |
f4 | 1.00000 | 2.00000 | 3.96500 | 3.03500 |
f5 | 1.00000 | 2.64333 | 2.95333 | 3.40333 |
f6 | 1.00000 | 4.00000 | 2.00000 | 3.00000 |
f7 | 3.87833 | 1.00167 | 2.03167 | 3.08833 |
f8 | 1.00167 | 2.57000 | 2.42833 | 4.00000 |
f9 | 1.20500 | 1.79500 | 3.62167 | 3.37833 |
f10 | 1.35833 | 3.92500 | 2.24500 | 2.47167 |
Algorithm | SDIQ | GWO | DE | PSO |
---|---|---|---|---|
Ave (s) | 18.33173 | 0.41582 | 0.58463 | 0.14015 |
STD | 0.37785 | 0.01899 | 0.02346 | 0.00629 |
Algorithm | Optimization Variables | Cost | |||
---|---|---|---|---|---|
SDIQ | 0.16800 | 4.06717 | 9.99978 | 0.16801 | 1.58713 |
RSA | 0.14468 | 3.51400 | 8.92510 | 0.21162 | 1.67260 |
DMO | 0.20557 | 3.25677 | 9.03618 | 0.20577 | 1.69530 |
AOA | 0.19448 | 2.57092 | 10.00000 | 0.20182 | 1.71640 |
GGWO | 0.20570 | 3.47050 | 9.03660 | 0.20570 | 1.72490 |
EO | 0.20570 | 3.47050 | 9.03664 | 0.20570 | 1.72490 |
SMO | 0.20573 | 3.47049 | 9.03662 | 0.20573 | 1.72485 |
Algorithm | Optimization Variables | Constraints | Objective Function | |||||
---|---|---|---|---|---|---|---|---|
SDIQ | 0.05167 | 0.35615 | 11.32242 | −1.50132 × 10−6 | −4.37484 × 10−6 | −4.05263 | −0.72812 | 1.26653 × 10−2 |
SMO | 0.05168 | 0.35640 | 11.30756 | 6.71223 × 10−7 | −5.22062 × 10−7 | −4.05316 | −0.72795 | 1.26653 × 10−2 |
GGWO | 0.05170 | 0.35670 | 11.28900 | 9.92097 × 10−4 | −6.33258 × 10−4 | −4.05534 | −0.72773 | 1.26701 × 10−2 |
EO | 0.05162 | 0.35505 | 11.38797 | −6.48728 × 10−5 | −4.62668 × 10−6 | −4.05014 | −0.72888 | 1.26663 × 10−2 |
RSA | 0.05781 | 0.58478 | 4.01670 | −1.56964 × 10−3 | 0.10112 | −4.91153 | −0.57160 | |
AOA | 0.05000 | 0.34980 | 11.86370 | −0.13190 | 0.08054 | −3.83737 | −0.73346 |
Algorithm | Optimization Variables | Constraints | Objective Function | ||||||
---|---|---|---|---|---|---|---|---|---|
SDIQ | 0.77817 | 0.38465 | 40.31962 | 199.99999 | −2.47000 × 10−10 | −5.70435 × 10−10 | −3.27370 × 10−4 | −40.00000 | 5885.33279 |
EO | 0.77817 | 0.38465 | 40.31962 | 200.00000 | −2.47000 × 10−10 | −5.70435 × 10−10 | −3.27370 × 10−4 | −40.00000 | 5885.33279 |
GGWO | 0.77817 | 0.38465 | 40.31962 | 200.00000 | −2.47000 × 10−10 | −5.70435 × 10−10 | −3.27370 × 10−4 | −40.00000 | 5885.33279 |
SMO | 0.77817 | 0.38465 | 40.31962 | 199.99993 | −7.60000 × 10−8 | −8.25200 × 10−7 | −2.66790 × 10−1 | −40.00007 | 5885.33279 |
RSA | 0.84007 | 0.41896 | 43.38117 | 161.55560 | −2.80000 × 10−3 | −5.10000 × 10−3 | −1.10000 × 103 | −78.44440 | 6034.75910 |
AOA | 0.83037 | 0.41620 | 42.75127 | 169.34540 | −5.30000 × 10−3 | −8.40000 × 10−3 | −3.60000 × 103 | −70.65460 | 6048.78440 |
DMO | 1.09367 | 1.04000 × 10−17 | 65.22523 | 10.00000 | 1.65300 × 10−1 | 6.22200 × 10−1 | 1.50400 × 10−1 | −230.00000 |
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Deng, W.; Ma, X.; Qiao, W. A Hybrid Intelligent Optimization Algorithm Based on a Learning Strategy. Mathematics 2024, 12, 2570. https://doi.org/10.3390/math12162570
Deng W, Ma X, Qiao W. A Hybrid Intelligent Optimization Algorithm Based on a Learning Strategy. Mathematics. 2024; 12(16):2570. https://doi.org/10.3390/math12162570
Chicago/Turabian StyleDeng, Wanyi, Xiaoxue Ma, and Weiliang Qiao. 2024. "A Hybrid Intelligent Optimization Algorithm Based on a Learning Strategy" Mathematics 12, no. 16: 2570. https://doi.org/10.3390/math12162570