Quantum Dynamical Interpretation of the Mean Strategy
Abstract
:1. Introduction
- (1)
- Utilizing a novel theoretical method, quantum dynamics, to study and analyze the mean strategy of the optimization algorithm, offering a fresh perspective for optimization algorithm research.
- (2)
- Employing the double well function and the CEC2013 test suite, conducting controlled experiments, performing comprehensive analyses of the mean strategy, and demonstrating its effectiveness.
- (3)
- Analyzing the convergence process of the optimization algorithm using the wave function.
2. Related Work
3. Quantum Dynamic Frame for Swarm Intelligence Algorithms
3.1. Quantum Dynamic Equation of Optimization Problem
3.2. Diffusion Model of QDF
3.3. Reaction–Diffusion Model of QDF
3.4. Quantum Harmonic Oscillator Model of QDF
3.5. Basic Iteration Process of Optimization Algorithm
Algorithm 1 Pseudocode of BIP with mean strategy |
|
3.6. Ground State Convergence Theory
4. Theory of Mean Strategy
4.1. Analysis of the Characteristics of the Quantum Harmonic Oscillator Model
4.2. The Physical Implications of the Mean Strategy
4.3. Mean Strategy of SI Algorithm
Algorithm 2 Pseudocode of BIP with mean strategy |
Input: particles number k, left boundary , right boundary , domain , scale Output:
|
5. Experiment
5.1. Test Functions
5.1.1. Double Well Function
5.1.2. CEC2013 Test Suite
5.2. Parameter Settings
5.3. Convergence Analysis
5.4. Exploration and Exploitation
5.5. Solution Accuracy Discussion of Mean Strategy
5.6. Stability Discussion of the Mean Strategy
5.7. Effectiveness Discussion of the Mean Strategy
5.8. Solving Speed Discussion of Mean Strategy
6. Conclusions and Future Works
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
SI | Swarm Intelligence |
PSO | Particle Swarm Optimization |
QPSO | Quantum-behaved Particle swarm Optimization |
WQPSO | Weighted Quantum-behaved Particle swarm Optimization |
MPSO | Median-oriented Particle Swarm Optimization |
MGWO | Mean Gray Wolf Optimization |
QMBA | Quantum-behaved Bat Algorithm with Mean Best Position Directed |
EHO | Elephant Herding Optimization |
CS | Cuckoo Search |
GQBA | Gaussian Quantum Bat Algorithm |
SHADE | Success-History Based Parameter Adaptation Differential Evolution |
CSHADE | Center-based mutation for Success-History based parameter Adaptation |
Differential Evolution | |
MQHOA | Multi-scale quantum harmonic optimization algorithm |
QDF | Quantum Dynamic Frame |
QDE | Quantum Dynamic Equation |
BIP | Basic Iteration Process |
DWF | Double Well Function |
MQHOA | Multi-scale Quantum Harmonic Optimization Algorithm |
TELA | Two Energy Level Approximation |
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SI Algorithm | Quantum Mechanics |
---|---|
Objective Function | Potential Field |
Solution to Optimization Problem | Wave function |
Optimal Solution | Ground State Wave Function |
Local Optimal Solution | Excited State Wave Function |
Dynamic Equation of SI Algorithm | Schrödinger Equation |
Iteration Process | Evolution of Wave Function |
Probability Distribution of the Solution | Modulus Square of the Wave Function |
Reception for Worse Solution | Two Energy Level Approximation |
Population Sampling | Kinetic Energy |
Categories | No. | Functions |
---|---|---|
Unimodal Functions | 1 | Sphere Function |
2 | Rotated High Conditioned Elliptic Function | |
3 | Rotated Bent Cigar Function | |
4 | Rotated Discus Function | |
5 | Different Powers Function | |
Basic Multimodal Functions | 6 | Rotated Rosenbrocks Function |
7 | Rotated Schaffers F7 Function | |
8 | Rotated Ackleys Function | |
9 | Rotated Weierstrass Function | |
10 | Rotated Griewanks Function | |
11 | Rastrigins Function | |
12 | Rotated Rastrigins Function | |
13 | Non-Continuous Rotated Rastrigins Function | |
14 | Schwefel’s Function | |
15 | Rotated Schwefel’s Function | |
16 | Rotated Katsuura Function | |
17 | Lunacek Bi_Rastrigin Function | |
18 | Rotated Lunacek Bi_Rastrigin Function | |
19 | Expanded Griewanks plus Rosenbrocks Function | |
20 | Expanded Scaffers F6 Function | |
Composition Functions | 21 | Composition Function 1 (Rotated) |
22 | Composition Function 2 (Unrotated) | |
23 | Composition Function 3 (Rotated) | |
24 | Composition Function 4 (Rotated) | |
25 | Composition Function 5 (Rotated) | |
26 | Composition Function 6 (Rotated) | |
27 | Composition Function 7 (Rotated) | |
28 | Composition Function 8 (Rotated) |
Parameter | k | Repeat Times | Max Iterations | |
---|---|---|---|---|
setting | 30 | 2 | 51 | 300,000 |
BIP | BIP-Mean | MQHOA-wmn | |||||||
---|---|---|---|---|---|---|---|---|---|
MEAN-ERR | STD-ERR | MIN-ERR | MEAN-ERR | STD-ERR | MIN-ERR | MEAN-ERR | STD-ERR | MIN-ERR | |
1.02 × 102 | 1.09 × 102 | 1.24 × 101 | 3.34 × 10−13 | 1.15 × 10−13 | 2.27 × 10−13 | 2.19 × 101 | 1.03 × 101 | 6.64 × 100 | |
2.34 × 108 | 5.19 × 107 | 1.02 × 108 | 1.43 × 106 | 2.45 × 105 | 1.02 × 106 | 3.06 × 107 | 4.20 × 106 | 2.41 × 107 | |
3.57 × 1010 | 2.32 × 1010 | 1.25 × 1010 | 1.54 × 107 | 1.66 × 107 | 8.51 × 105 | 1.22 × 109 | 3.84 × 108 | 5.98 × 108 | |
5.44 × 104 | 7.80 × 103 | 3.61 × 104 | 3.11 × 104 | 3.86 × 103 | 2.25 × 104 | 1.37 × 104 | 2.19 × 103 | 9.78 × 103 | |
3.87 × 103 | 7.35 × 102 | 1.51 × 103 | 2.02 × 10−3 | 3.04 × 10−4 | 1.42 × 10−3 | 1.75 × 102 | 9.78 × 101 | 1.21 × 102 | |
5.23 × 102 | 6.53 × 101 | 3.25 × 102 | 2.33 × 101 | 2.14 × 101 | 4.41 × 10−1 | 9.90 × 101 | 1.81 × 101 | 5.57 × 101 | |
1.62 × 102 | 4.90 × 101 | 1.15 × 102 | 1.41 × 101 | 9.58 × 100 | 2.49 × 100 | 4.57 × 101 | 8.71 × 100 | 3.15 × 101 | |
2.10 × 101 | 4.54 × 10−2 | 2.08 × 101 | 2.10 × 101 | 4.77 × 10−2 | 2.08 × 101 | 2.09 × 101 | 5.13 × 10−2 | 2.08 × 101 | |
3.92 × 101 | 1.01 × 100 | 3.65 × 101 | 3.96 × 101 | 1.09 × 100 | 3.53 × 101 | 3.94 × 101 | 1.44 × 100 | 3.63 × 101 | |
9.29 × 102 | 9.79 × 101 | 7.07 × 102 | 2.45 × 10−1 | 1.89 × 10−1 | 6.16 × 10−2 | 5.83 × 101 | 6.90 × 100 | 4.24 × 101 | |
3.09 × 102 | 2.03 × 101 | 2.67 × 102 | 1.66 × 102 | 5.32 × 101 | 6.96 × 100 | 2.14 × 102 | 1.36 × 101 | 1.83 × 102 | |
3.15 × 102 | 1.31 × 101 | 2.86 × 102 | 1.56 × 102 | 5.47 × 101 | 2.19 × 101 | 2.12 × 102 | 1.28 × 101 | 1.82 × 102 | |
3.12 × 102 | 1.61 × 101 | 2.66 × 102 | 1.84 × 102 | 1.95 × 101 | 1.07 × 102 | 2.16 × 102 | 1.40 × 101 | 1.73 × 102 | |
7.40 × 103 | 2.46 × 102 | 6.90 × 103 | 7.29 × 103 | 3.02 × 102 | 6.56 × 103 | 7.37 × 103 | 2.20 × 102 | 6.89 × 103 | |
7.38 × 103 | 2.89 × 102 | 6.51 × 103 | 7.35 × 103 | 2.82 × 102 | 6.63 × 103 | 7.27 × 103 | 3.12 × 102 | 6.31 × 103 | |
2.51 × 100 | 2.68 × 10−1 | 1.91 × 100 | 2.53 × 100 | 2.92 × 10−1 | 1.77 × 100 | 2.40 × 100 | 2.85 × 10−1 | 1.85 × 100 | |
3.48 × 102 | 1.21 × 101 | 3.19 × 102 | 2.10 × 102 | 1.24 × 101 | 1.79 × 102 | 2.39 × 102 | 1.21 × 101 | 1.99 × 102 | |
3.49 × | 2.61 × | 3.12 × | 2.12 × | 1.18 × | 1.88 × | 2.41 × | 9.65 × | 2.25 × | |
3.08 × | 2.43 × | 3.75 × | 1.52 × | 1.40 × | 1.13 × | 2.02 × | 1.29 × | 1.76 × | |
1.45 × | 2.25 × | 1.39 × | 1.19 × | 2.69 × | 1.12 × | 1.26 × | 4.78 × | 1.17 × | |
1.25 × | 9.05 × | 1.04 × | 3.46 × | 9.09 × | 2.00 × | 4.12 × | 9.46 × | 3.39 × | |
7.74 × | 3.07 × | 6.85 × | 7.65 × | 3.39 × | 6.80 × | 7.67 × | 3.44 × | 6.74 × | |
7.91 × | 2.80 × | 7.10 × | 7.76 × | 3.24 × | 7.09 × | 7.82 × | 3.16 × | 7.10 × | |
3.15 × | 1.13 × | 2.96 × | 2.30 × | 6.72 × | 2.16 × | 2.45 × | 2.51 × | 2.37 × | |
3.47 × | 5.10 × | 3.31 × | 2.89 × | 2.81 × | 2.51 × | 2.92 × | 4.08 × | 2.35 × | |
2.24 × | 5.86 × | 2.11 × | 2.10 × | 3.08 × | 2.00 × | 2.22 × | 5.02 × | 2.01 × | |
1.40 × | 2.94 × | 1.32 × | 6.58 × | 7.29 × | 5.12 × | 7.92 × | 3.58 × | 6.96 × | |
2.19 × | 2.10 × | 1.00 × | 3.00 × | 1.29 × | 3.00 × | 4.28 × | 3.64 × | 3.75 × |
p-Value | Crouping | BIP-Mean vs. BIP | MQHOA-wmn vs. BIP | BIP-Mean vs. MQHOA-wmn | |
---|---|---|---|---|---|
9.49 × | 9.7555 × | 1.46343 × | |||
3.30 × | 3.01986 × | 3.01986 × | |||
3.30 × | 3.01986 × | 3.01986 × | |||
4.70 × | 3.01986 × | 3.01986 × | |||
3.30 × | 3.01986 × | 3.01986 × | |||
3.30 × | 3.01986 × | 1.20567 × | |||
3.30 × | 3.01986 × | 1.77691 × | |||
7.13 × | 0.122352926 | 0.923442132 | |||
1.75 × | 0.482516904 | 0.325526587 | |||
3.30 × | 3.01986 × | 3.01608 × | |||
3.30 × | 3.01986 × | 7.08811 × | |||
3.30 × | 3.01986 × | 2.0338 × | |||
3.30 × | 3.01986 × | 1.84999 × | |||
1.41 × | 0.911708975 | 0.363222313 | |||
5.12 × | 0.067868861 | 0.982307053 | |||
6.54 × | 0.200948897 | 0.059427915 | |||
3.30 × | 3.01986 × | 4.1825 × | |||
3.30 × | 3.01986 × | 4.97517 × | |||
3.30 × | 3.01986 × | 3.01986 × | |||
3.30 × | 3.01986 × | 3.96477 × | |||
4.09 × | 3.01986 × | 9.45134 × | |||
1.43 × | 0.311187643 | 0.853381737 | |||
1.63 × | 0.728265296 | 0.233988916 | |||
3.30 × | 3.01986 × | 4.07716 × | |||
3.30 × | 1.95678 × | 0.970516051 | |||
1.63 × | 1.10772 × | 0.371077032 | |||
3.30 × | 3.01986 × | 5.96731 × | |||
1.39 × | 3.01986 × | 1.21178 × |
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Wang, F.; Wang, P.; Jiao, Y. Quantum Dynamical Interpretation of the Mean Strategy. Entropy 2024, 26, 719. https://doi.org/10.3390/e26090719
Wang F, Wang P, Jiao Y. Quantum Dynamical Interpretation of the Mean Strategy. Entropy. 2024; 26(9):719. https://doi.org/10.3390/e26090719
Chicago/Turabian StyleWang, Fang, Peng Wang, and Yuwei Jiao. 2024. "Quantum Dynamical Interpretation of the Mean Strategy" Entropy 26, no. 9: 719. https://doi.org/10.3390/e26090719