An Optimization Algorithm Inspired by the Phase Transition Phenomenon for Global Optimization Problems with Continuous Variables
<p>Three possible positions of elements in a system.</p> "> Figure 2
<p>The critical intervals of the three phases.</p> "> Figure 3
<p>A phase transition from an unstable phase to a stable phase.</p> "> Figure 4
<p>Two-dimensional example showing the process of the free walking path of elements.</p> "> Figure 5
<p>The shrinkage trend of elements towards the optimal point.</p> "> Figure 6
<p>The vibration of elements in an equilibrium position.</p> "> Figure 7
<p>The main flowchart of PTBO.</p> "> Figure 8
<p>Population distribution at various generations in an evolutionary process of PTBO.</p> "> Figure 8 Cont.
<p>Population distribution at various generations in an evolutionary process of PTBO.</p> "> Figure 9
<p>Convergence performance of the compared algorithms on parts of functions.</p> "> Figure 10
<p>The mean central processing unit (CPU) time of the nine compared algorithms. 30D: 30 dimensions.</p> ">
Abstract
:1. Introduction
2. Prerequisite Preparation
2.1. Fundamental Concepts
2.2. The Determination of Critical Interval about Three Phases
3. Phase Transition-Based Optimization Algorithm
3.1. Basic Idea of the PTBO Algorithm
3.2. The Correspondence of PTBO and the Phase Transition Process
3.3. The Overall Design of the PTBO Algorithm
3.3.1. Population Initialization
3.3.2. Iterations of the Three Operators
3.3.3. Individual Selection
3.4. Flowchart and Implementation Steps of PTBO
4. The Analysis of PTBO and a Comparative Study
4.1. The Analysis of Time Complexity
- 1
- Population initialization operation: .
- 2
- Stochastic operator: .
- 3
- Shrinkage operator: .
- 4
- Vibration operator: .
4.2. The Dynamic Implementation Analysis of PTBO
4.3. The Differences between PTBO and Other Algorithms
4.3.1. The Differences between PTBO and PSO
4.3.2. The Differences between PTBO and SMS
5. Experimental Results
5.1. Benchmark Functions
5.2. Parameters Determination of the Interval Ratio of PTBO
5.3. Experimental Platform and Algorithms’ Parameter Settings
5.4. The Compared Experimental Results
5.4.1. Comparisons on Solution Accuracy
5.4.2. The Comparison Results of Convergence Speed
5.4.3. The Comparison Results of Wilcoxon Signed-Rank Test
5.4.4. The Comparison Results of Time Complexity
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Phase | Main Property | Motion Tendency | Example |
---|---|---|---|
UP | disorder, moving towards an arbitrary direction | stochastic motion | gaseous molecule such as water, atom, or electron in normal temperature |
MP | between disorder and order, moving according to a certain law | shrinkage motion | liquid molecule such as water or rapid freezing alloy crystal |
SP | order, moving in a very regular mode | vibration motion | solid molecule such as water or an atom of paramagnetic phase in a nail |
Phase | Interval |
---|---|
UP | [Fmax, F2] |
MP | (F2, F1) |
SP | [F1, Fmin] |
PTBO Algorithm | Phase Transition Process |
---|---|
Individual | Element |
Population size | The number of elements |
Fitness function | Stability degree of an element |
Global optimal solution | The lowest stability degree of an element |
Stochastic operator | The stochastic motion of an element in UP |
Shrinkage operator | The shrinkage motion of an element in a MP |
Vibration operator | The vibration and fine tuning of an element in SP |
Type | Function ID | Functions Name |
---|---|---|
Uni-modal Functions | F01 | Sphere Function |
F02 | Rotated High Conditioned Elliptic Function | |
F03 | Rotated Bent Cigar Function | |
F04 | Rotated Discus Function | |
F05 | Different Powers Function | |
Multimodal Functions | F06 | Rotated Rosenbrock’s Function |
F07 | Rotated Schaffers F7 Function | |
F08 | Rotated Ackley’s Function | |
F09 | Rotated Weierstrass Function | |
F10 | Rotated Griewank’s Function | |
F11 | Rastrigin’s Function | |
F12 | Rotated Rastrigin’s Function | |
F13 | Non-Continuous Rotated Rastrigin’s Function | |
F14 | Schwefel’s Function | |
F15 | Rotated Schwefel’s Function | |
F16 | Rotated Katsuura Function | |
F17 | Lunacek Bi_Rastrigin Function | |
F18 | Rotated Lunacek Bi_Rastrigin Function | |
F19 | Expanded Griewank’s plus Rosenbrock’s Function | |
F20 | Expanded Scaffer’s F6 Function | |
Composition Functions | F21 | Composition Function 1 (n = 5, Rotated) |
F22 | Composition Function 2 (n = 3, Unrotated) | |
F23 | Composition Function 3 (n = 3, Rotated) | |
F24 | Composition Function 4 (n = 3, Rotated) | |
F25 | Composition Function 5 (n = 3, Rotated) | |
F26 | Composition Function 6 (n = 5, Rotated) | |
F27 | Composition Function 7 (n = 5, Rotated) | |
F28 | Composition Function 8 (n = 5, Rotated) |
Ratio Method | No. | Stable Phase | Meta-Stable Phase | Unstable Phase |
---|---|---|---|---|
Fixed ratio | Prop1 | 0.05 | 0.8 | 0.15 |
Prop2 | 0.1 | 0.8 | 0.1 | |
Prop3 | 0.15 | 0.8 | 0.05 | |
Random ratio | Prop4 | 0.2 * rand | 0.8 | 0.2 * (1 − rand) |
Func. | Prop1 | Prop2 | Prop3 | Prop4 | Func. | Prop1 | Prop2 | Prop3 | Prop4 |
---|---|---|---|---|---|---|---|---|---|
F01 | 6.16 × 10−27 | 6.13 × 10−27 | 3.82 × 10−28 | 4.31 × 10−28 | F15 | 4.45 × 103 | 4.57 × 103 | 4.27 × 103 | 4.23 × 103 |
2.08 × 10−26 | 3.70 × 10−26 | 6.04 × 10−28 | 5.96 × 10−28 | 5.99 × 102 | 6.55 × 102 | 7.24 × 102 | 7.64 × 102 | ||
F02 | 7.90 × 105 | 8.73 × 105 | 1.02 × 106 | 8.63 × 105 | F16 | 4.10 × 10−1 | 4.71 × 10−1 | 4.11 × 10−1 | 3.94 × 10−1 |
2.89 × 105 | 3.25 × 105 | 3.66 × 105 | 6.60 × 105 | 2.01 × 10−1 | 2.59 × 10−1 | 2.07 × 10−1 | 1.96 × 10−1 | ||
F03 | 1.95 × 108 | 1.12 × 108 | 8.01 × 107 | 3.84 × 107 | F17 | 7.58 × 101 | 7.13 × 101 | 6.68 × 101 | 6.72 × 101 |
2.91 × 108 | 1.49 × 108 | 1.22 × 108 | 5.22 × 107 | 1.45 × 101 | 1.25 × 101 | 9.94 | 1.63 × 101 | ||
F04 | 8.08 × 103 | 9.03 × 103 | 1.30 × 104 | 8.26 × 103 | F18 | 7.96 × 101 | 7.57 × 101 | 6.99 × 101 | 6.66 × 101 |
2.59 × 103 | 2.35 × 103 | 3.43 × 103 | 3.71 × 103 | 1.47 × 101 | 1.15 × 101 | 1.14 × 101 | 1.23 × 101 | ||
F05 | 1.36 × 10−14 | 3.00 × 10−15 | 6.38 × 10−22 | 2.64 × 10−15 | F19 | 5.05 × 102 | 5.05 × 102 | 5.04 × 102 | 5.04 × 102 |
2.19 × 10−14 | 4.32 × 10−15 | 1.18 × 10−21 | 4.44 × 10−15 | 1.61 | 1.45 | 1.06 | 8.36 × 10−1 | ||
F06 | 2.32 × 101 | 3.09 × 101 | 2.74 × 101 | 2.53 × 101 | F20 | 1.21 × 101 | 1.19 × 101 | 1.26 × 101 | 1.15 × 101 |
2.57 × 101 | 2.88 × 101 | 2.58 × 101 | 2.67 × 101 | 1.68 | 2.01 | 2.07 | 2.12 | ||
F07 | 5.30 × 101 | 5.01 × 101 | 4.13 × 101 | 3.04 × 101 | F21 | 3.01 × 102 | 3.30 × 102 | 3.22 × 102 | 2.97 × 102 |
2.31 × 101 | 1.90 × 101 | 1.93 × 101 | 1.73 × 101 | 7.76 × 101 | 8.86 × 101 | 7.19 × 101 | 8.01 × 101 | ||
F08 | 2.09 × 101 | 2.09 × 101 | 2.09 × 101 | 2.09 × 101 | F22 | 1.01 × 103 | 8.03 × 102 | 9.74 × 102 | 8.61 × 102 |
6.18 × 10−2 | 6.57 × 10−2 | 6.50 × 10−2 | 6.50 × 10−2 | 4.92 × 102 | 2.79 × 102 | 4.68 × 102 | 3.11 × 102 | ||
F09 | 2.49 × 101 | 2.39 × 101 | 2.43 × 101 | 1.90 × 101 | F23 | 5.25 × 103 | 5.16 × 103 | 5.07 × 103 | 4.15 × 103 |
5.33 | 5.24 | 5.54 | 5.26 | 8.49 × 102 | 6.32 × 102 | 7.01 × 102 | 1.40 × 103 | ||
F10 | 3.50 × 10−1 | 3.39 × 10−1 | 3.43 × 10−1 | 3.27 × 10−1 | F24 | 2.42 × 102 | 2.35 × 102 | 2.33 × 102 | 2.33 × 102 |
2.03 × 10−1 | 1.85 × 10−1 | 1.73 × 10−1 | 1.77 × 10−1 | 8.21 | 8.66 | 8.46 | 1.04 × 101 | ||
F11 | 4.85 × 101 | 4.05 × 101 | 3.89 × 101 | 3.92 × 101 | F25 | 2.81 × 102 | 2.79 × 102 | 2.78 × 102 | 2.74 × 102 |
1.60 × 101 | 1.38 × 101 | 1.12 × 101 | 1.21 × 101 | 1.34 × 101 | 1.46 × 101 | 1.32 × 101 | 1.02 × 101 | ||
F12 | 1.40 × 102 | 1.28 × 102 | 1.21 × 102 | 6.32 × 101 | F26 | 2.53 × 102 | 2.77 × 102 | 2.48 × 102 | 2.51 × 102 |
2.82 × 101 | 3.41 × 101 | 3.72 × 101 | 2.51 × 101 | 6.85 × 101 | 6.93 × 101 | 6.71 × 101 | 6.57 × 101 | ||
F13 | 1.61 × 102 | 1.53 × 102 | 1.47 × 102 | 1.18 × 102 | F27 | 7.50 × 102 | 7.41 × 102 | 7.23 × 102 | 6.38 × 102 |
3.13 × 101 | 2.76 × 101 | 2.67 × 101 | 3.07 × 101 | 9.33 × 101 | 1.21 × 102 | 1.34 × 102 | 1.16 × 102 | ||
F14 | 9.52 × 102 | 8.63 × 102 | 9.35 × 102 | 9.47 × 102 | F28 | 3.93 × 102 | 3.46 × 102 | 3.51 × 102 | 2.96 × 102 |
3.28 × 102 | 3.31 × 102 | 3.49 × 102 | 2.81 × 102 | 4.61 × 102 | 2.74 × 102 | 3.92 × 102 | 2.83 × 101 |
No. | Algorithm | Parameter Setting |
---|---|---|
1 | PSO | |
2 | DE | |
3 | BA | |
4 | CS | |
5 | BSD | |
6 | WWO | |
7 | WCA | |
8 | SMS | Gas state: |
Liquid state: | ||
Solid state: | ||
9 | PTBO | alpha is random ratio, beta = 0.8 |
Func. | PTBO | PSO | DE | BA | CS | BSO | WWO | WCA | SMS |
---|---|---|---|---|---|---|---|---|---|
F01 | 4.31 × 10−28 | 3.56 × 103 ǂ | 2.65 × 103 ǂ | 1.05 × 10−3 ǂ | 4.46 × 103 ǂ | 3.71 × 10−3 ǂ | 2.62 × 10−28 ξ | 1.58 × 10−28 ξ | 4.63 × 104 ǂ |
5.96 × 10−28 | 2.65 × 103 | 8.98 × 10−1 | 1.29 × 10−4 | 3.10 × 102 | 5.71 × 10−3 | 1.30 × 10−28 | 1.45 × 10−28 | 4.24 × 103 | |
F02 | 8.63 × 105 | 1.12 × 107 ǂ | 1.56 × 107 ξ | 4.82 × 104 ξ | 4.21 × 107 ǂ | 1.17 × 106 ǂ | 1.20 × 106 ǂ | 7.53 × 105 ξ | 4.91 × 108 ǂ |
6.60 × 105 | 1.56 × 107 | 2.05 × 105 | 1.81 × 104 | 4.04 × 106 | 3.35 × 105 | 7.19 × 105 | 7.54 × 105 | 8.49 × 107 | |
F03 | 3.84 × 107 | 4.98 × 1010 ǂ | 4.72 × 1010 ǂ | 8.96 × 108 ǂ | 1.72 × 1010 ǂ | 1.66 × 108 ǂ | 3.98 × 108 ǂ | 2.94 × 109 ǂ | 1.00 × 1010 ǂ |
5.22 × 107 | 4.72 × 1010 | 3.57 × 107 | 3.48 × 108 | 2.11 × 109 | 2.22 × 108 | 6.71 × 108 | 4.30 × 109 | 0.00 | |
F04 | 8.26 × 103 | 5.64 × 103 ξ | 8.41 × 103 ǂ | 2.42 × 104 ǂ | 6.05 × 104 ǂ | 5.44 × 103 ξ | 5.57 × 104 ǂ | 6.55 × 101 ξ | 8.14 × 104 ǂ |
3.71 × 103 | 8.41 × 103 | 1.27 × 103 | 3.91 × 102 | 6.61 × 102 | 2.22 × 103 | 1.05 × 104 | 2.07 × 102 | 9.79 × 103 | |
F05 | 2.64 × 10−15 | 1.58 × 103 ǂ | 1.25 × 103 ǂ | 1.37 × 10−2 ǂ | 1.09 × 103 ǂ | 4.12 × 10−2 ǂ | 5.53 × 10−2 ǂ | 2.95 × 10−12 ǂ | 1.38 × 104 ǂ |
4.44 × 10−15 | 1.25 × 103 | 1.84 × 101 | 2.08 × 10−3 | 6.70 × 101 | 1.74 × 10−2 | 2.90 × 10−1 | 7.89 × 10−12 | 2.66 × 103 | |
ǂ/ξ/~ | 4/1/0 | 4/1/0 | 4/1/0 | 5/0/0 | 4/1/0 | 4/1/0 | 2/3/0 | 5/0/0 |
Function | PTBO | PSO | DE | BA | CS | BSO | WWO | WCA | SMS |
---|---|---|---|---|---|---|---|---|---|
F06 | 2.53 × 101 | 2.51 × 102 ǂ | 3.20 × 101 ǂ | 1.36 ξ | 4.99 × 102 ǂ | 4.30 × 101 ǂ | 5.51 × 101 ǂ | 3.27 × 101 ǂ | 6.18 × 103 ǂ |
2.67 × 101 | 2.11 × 102 | 2.51 × 101 | 6.61 | 5.70 × 101 | 2.84 × 101 | 2.76 × 101 | 2.67 × 101 | 9.16 × 102 | |
F07 | 3.04 × 101 | 1.19 × 102 ǂ | 2.44 × 101 ξ | 2.59 × 108 ǂ | 2.51 × 108 ǂ | 1.59 × 102 ǂ | 8.91 × 101 ǂ | 1.90 × 102 ǂ | 5.50 × 103 ǂ |
1.73 × 101 | 4.67 × 101 | 1.43 × 101 | 4.83 × 107 | 2.44 × 107 | 1.11 × 102 | 3.02 × 101 | 4.00 × 101 | 3.85 × 103 | |
F08 | 2.09 × 101 | 2.09 × 101 ~ | 2.09 × 101 ~ | 2.10 × 101 ~ | 2.11 × 101 ~ | 2.09 × 101 ~ | 2.09 × 101 ~ | 2.09 × 101 ~ | 2.09 × 101 ~ |
6.50 × 10−2 | 5.71 × 10−2 | 4.33 × 10−2 | 6.52 × 10−2 | 6.15 × 10−2 | 7.40 × 10−2 | 5.56 × 10−2 | 5.56 × 10−2 | 5.31 × 10−2 | |
F09 | 1.90 × 101 | 2.18 × 101 ~ | 1.86 × 101 ξ | 5.77 × 101 ǂ | 5.73 × 101 ǂ | 3.42 × 101 ǂ | 2.76 × 101 ǂ | 3.40 × 101 ǂ | 4.00 × 101 ǂ |
5.26 | 2.73 | 4.40 | 3.29 | 1.16 | 3.72 | 4.29 | 3.01 | 1.48 | |
F10 | 3.27 × 10−1 | 7.01 × 102 ǂ | 6.65 × 10−1 ~ | 1.14 ǂ | 9.05 × 102 ǂ | 9.62 × 10−1 ~ | 1.41 × 10−1 ξ | 3.17 × 10−1 ~ | 5.60 × 103 ǂ |
1.77 × 10−1 | 4.13 × 102 | 1.21 | 6.95 × 10−1 | 4.90 × 101 | 2.16 × 10−1 | 7.49 × 10−2 | 2.00 × 10−1 | 6.76 × 102 | |
F11 | 3.92 × 101 | 1.05 × 102 ǂ | 2.27 × 101 ξ | 9.90 × 102 ǂ | 9.90 × 102 ǂ | 6.03 × 102 ǂ | 1.22 × 102 ǂ | 1.05 × 102 ǂ | 7.65 × 102 ǂ |
1.21 × 101 | 5.68 × 101 | 7.87 | 2.71 × 101 | 2.24 × 101 | 1.03 × 102 | 3.80 × 101 | 4.84 × 101 | 6.72 × 101 | |
F12 | 6.32 × 101 | 1.07 × 102 ǂ | 4.14 × 101 ξ | 9.50 × 102 ǂ | 9.52 × 102 ǂ | 5.97 × 102 ǂ | 1.36 × 102 ǂ | 3.38 × 102 ǂ | 7.63 × 102 ǂ |
2.51 × 101 | 3.62 × 101 | 1.21 × 101 | 1.28 × 101 | 1.34 × 101 | 1.04 × 102 | 3.51 × 101 | 9.25 × 101 | 5.78 × 101 | |
F13 | 1.18 × 102 | 2.80 × 102 ǂ | 2.09 × 102 ǂ | 1.12 × 103 ǂ | 1.12 × 103 ǂ | 6.51 × 102 ǂ | 2.07 × 102 ǂ | 3.46 × 102 ǂ | 7.68 × 102 ǂ |
3.07 × 101 | 3.13 × 101 | 3.63 × 101 | 4.65 × 101 | 3.75 × 101 | 7.86 × 101 | 5.72 × 101 | 6.77 × 101 | 5.35 × 101 | |
F14 | 9.47 × 102 | 2.00 × 103 ǂ | 1.46 × 103 ǂ | 5.72 × 103 ǂ | 5.90 × 103 ǂ | 4.42 × 103 ǂ | 3.23 × 103 ǂ | 2.37 × 103 ǂ | 7.72 × 103 ǂ |
2.81 × 102 | 6.45 × 102 | 4.71 × 102 | 4.38 × 102 | 4.38 × 102 | 7.77 × 102 | 6.47 × 102 | 7.83 × 102 | 3.88 × 102 | |
F15 | 4.23 × 103 | 4.02 × 103 ξ | 7.21 × 103 ǂ | 5.02 × 103 ǂ | 4.99 × 103 ǂ | 4.33 × 103 ~ | 5.16 × 103 ǂ | 4.72 × 103 ǂ | 7.59 × 103 ǂ |
7.64 × 102 | 6.52 × 102 | 3.53 × 102 | 2.58 × 102 | 2.57 × 102 | 6.67 × 102 | 1.72 × 103 | 6.74 × 102 | 2.60 × 102 | |
F16 | 3.94 × 10−1 | 1.94 ǂ~ | 2.49 ǂ | 1.98 ǂ | 1.89 ǂ | 3.13 × 10−1 ξ | 2.00 ǂ | 1.63 ǂ | 2.59 ǂ |
1.96 × 10−1 | 4.89 × 10−1 | 3.29 × 10−1 | 9.87 × 10−1 | 8.45 × 10−1 | 1.16 × 10−1 | 8.11 × 10−1 | 4.95 × 10−1 | 3.08 × 10−1 | |
F17 | 6.72 × 101 | 8.45 × 101 ǂ | 6.39 × 101 ξ | 8.29 × 102 ǂ | 8.42 × 102 ǂ | 6.10 × 102 ǂ | 1.43 × 102 ǂ | 2.39 × 102 ǂ | 1.51 × 103 ǂ |
1.63 × 101 | 3.92 × 101 | 9.90 | 1.12 × 101 | 1.41 × 101 | 1.03 × 102 | 3.67 × 101 | 1.09 × 102 | 1.02 × 102 | |
F18 | 6.66 × 101 | 1.26 × 102 ǂ | 2.92 × 102 ǂ | 8.26 × 102 ǂ | 8.33 × 102 ǂ | 5.13 × 102 ǂ | 1.42 × 102 ǂ | 4.66 × 102 ǂ | 1.53 × 103 ǂ |
1.23 × 101 | 4.99 × 101 | 1.86 × 101 | 1.17 × 101 | 1.02 × 101 | 9.43 × 101 | 3.31 × 101 | 1.43 × 102 | 8.20 × 101 | |
F19 | 5.04 × 102 | 2.16 × 103 ǂ | 5.65 × 102 ǂ | 3.37 × 103 ǂ | 1.55 × 103 ǂ | 5.12 × 102 ~ | 5.16 × 102 ~ | 5.14 × 102 ~ | 5.90 × 105 ǂ |
8.36 × 10−1 | 3.38 × 103 | 2.84 | 1.85 × 102 | 7.16 × 101 | 2.41 | 1.71 | 5.60 | 2.78 × 105 | |
F20 | 1.15 × 101 | 1.24 × 101 ǂ | 1.28 × 101 ǂ | 1.50 × 101 ǂ | 1.50 × 101 ǂ | 1.45 × 101 ǂ | 1.29 × 101 ǂ | 1.50 × 101 ǂ | 1.50 × 101 ǂ |
2.12 | 8.98 × 10−1 | 5.20 × 10−1 | 0.00 | 0.00 | 9.18 × 10−2 | 1.00 | 0.00 | 5.71 × 10−2 | |
ǂ/ξ/~ | 12/1/2 | 8/5/2 | 13/1/1 | 14/0/1 | 10/1/4 | 12/1/2 | 10/3/2 | 14/0/1 |
Function | PTBO | PSO | DE | BA | CS | BSO | WWO | WCA | SMS |
---|---|---|---|---|---|---|---|---|---|
F21 | 2.97 × 102 | 4.91 × 102 ǂ | 4.20 × 102 ǂ | 3.01 × 102 ~ | 1.50 × 103 ǂ | 3.48 × 102 ǂ | 3.25 × 102 ~ | 3.51 × 102 ~ | 3.64 × 103 ǂ |
8.01 × 101 | 2.21 × 102 | 8.52 × 101 | 2.68 × 10−2 | 8.81 × 101 | 8.87 × 101 | 8.01 × 101 | 8.51 × 101 | 2.17 × 102 | |
F22 | 8.61 × 102 | 2.17 × 103 ǂ | 1.33 × 103 ǂ | 8.52 × 103 ǂ | 8.52 × 103 ǂ | 5.62 × 103 ǂ | 4.09 × 103 ǂ | 4.01 × 103 ǂ | 8.39 × 103 ǂ |
3.11 × 102 | 6.53 × 102 | 3.94 × 102 | 1.76 × 102 | 1.93 × 102 | 8.61 × 102 | 7.54 × 102 | 1.23 × 103 | 3.46 × 102 | |
F23 | 4.15 × 103 | 4.16 × 103 ~ | 6.61 × 103 ǂ | 7.55 × 103 ǂ | 7.74 × 103 ǂ | 5.67 × 103 ǂ | 5.40 × 103 ~ | 6.64 × 103 ǂ | 8.25 × 103 ǂ |
1.40 × 103 | 8.98 × 102 | 9.13 × 102 | 5.35 × 102 | 3.99 × 102 | 6.67 × 102 | 1.33 × 103 | 6.71 × 102 | 2.72 × 102 | |
F24 | 2.33 × 102 | 2.84 × 102 ǂ | 2.95 × 102 ǂ | 7.03 × 102 ǂ | 7.28 × 102 ǂ | 3.34 × 102 ǂ | 2.63 × 102 ǂ | 3.08 × 102 ǂ | 3.58 × 102 ǂ |
1.04 × 101 | 1.23 × 101 | 1.05 × 101 | 9.88 × 101 | 1.43 × 101 | 1.89 × 101 | 1.44 × 101 | 3.54 × 101 | 1.05 × 101 | |
F25 | 2.74 × 102 | 3.03 × 102 ~ | 2.66 × 102 ξ | 4.48 × 102 ǂ | 4.45 × 102 ǂ | 3.56 × 102 ǂ | 3.04 × 102 ~ | 3.17 × 102 ~ | 3.73 × 102 ǂ |
1.02 × 101 | 9.32 | 7.55 | 2.23 × 101 | 4.24 | 1.76 × 101 | 1.47 × 101 | 1.11 × 101 | 8.47 | |
F26 | 2.51 × 102 | 3.23 × 102 ǂ | 2.70 × 102 ǂ | 5.52 × 102 ǂ | 5.15 × 102 ǂ | 2.38 × 102 ξ | 2.00 × 102 ξ | 3.51 × 102 ~ | 2.62 × 102 ǂ |
6.57 × 101 | 6.95 × 101 | 7.11 × 101 | 3.00 × 101 | 3.47 × 101 | 7.46 × 101 | 3.74 × 10−2 | 7.68 × 101 | 1.61 × 101 | |
F27 | 6.38 × 102 | 9.62 × 102 ǂ | 7.14 × 102 ~ | 3.30 × 103 ǂ | 2.83 × 103 ǂ | 1.36 × 103 ǂ | 9.90 × 102 ǂ | 1.22 × 103 ǂ | 1.50 × 103 ǂ |
1.16 × 102 | 7.94 × 101 | 7.45 × 101 | 2.89 × 102 | 8.94 × 101 | 9.72 × 101 | 1.20 × 102 | 8.93 × 101 | 4.49 × 101 | |
F28 | 2.96 × 102 | 2.11 × 103 ǂ | 3.00 × 102 ~ | 7.53 × 103 ǂ | 8.82 × 103 ǂ | 4.95 × 103 ǂ | 5.36 × 102 ǂ | 2.03 × 103 ǂ | 5.34 × 103 ǂ |
2.83 × 101 | 5.50 × 102 | 1.09 × 10−1 | 4.46 × 102 | 2.61 × 102 | 7.15 × 102 | 5.90 × 102 | 1.69 × 103 | 4.17 × 102 | |
ǂ/ξ/~ | 6/0/2 | 5/1/2 | 7/0/1 | 8/0/0 | 7/0/1 | 4/1/3 | 5/0/3 | 8/0/0 |
PTBO vs. | PSO | DE | BA | CS | BSO | WWO | WCA | SMS |
---|---|---|---|---|---|---|---|---|
F01–05 | 4/1/0 | 4/1/0 | 4/1/0 | 5/0/0 | 4/1/0 | 4/1/0 | 2/3/0 | 5/0/0 |
F06–20 | 12/1/2 | 8/5/2 | 13/1/1 | 14/0/1 | 10/1/4 | 12/1/2 | 10/3/2 | 14/0/1 |
F21–28 | 6/0/2 | 5/1/2 | 7/0/1 | 8/0/0 | 7/0/1 | 4/1/3 | 5/0/3 | 8/0/0 |
ǂ/ξ/~ | 22/2/4 | 17/7/4 | 24/4/2 | 27/0/1 | 21/2/5 | 20/3/5 | 17/6/5 | 27/0/1 |
PTBO vs. | R+ | R− | p-Value |
---|---|---|---|
PSO | 305 | 73 | 2.69 × 10−4 |
DE | 215 | 163 | 3.16 × 10−1 |
BA | 325 | 53 | 1.19 × 10−4 |
CS | 368 | 10 | 3.79 × 10−6 |
BSO | 340 | 38 | 1.57 × 10−4 |
WWO | 359 | 19 | 2.52 × 10−5 |
WCA | 320 | 58 | 1.20 × 10−3 |
SMS | 360 | 18 | 2.96 × 10−5 |
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Cao, Z.; Wang, L. An Optimization Algorithm Inspired by the Phase Transition Phenomenon for Global Optimization Problems with Continuous Variables. Algorithms 2017, 10, 119. https://doi.org/10.3390/a10040119
Cao Z, Wang L. An Optimization Algorithm Inspired by the Phase Transition Phenomenon for Global Optimization Problems with Continuous Variables. Algorithms. 2017; 10(4):119. https://doi.org/10.3390/a10040119
Chicago/Turabian StyleCao, Zijian, and Lei Wang. 2017. "An Optimization Algorithm Inspired by the Phase Transition Phenomenon for Global Optimization Problems with Continuous Variables" Algorithms 10, no. 4: 119. https://doi.org/10.3390/a10040119