A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds
Abstract
:1. Introduction
- The iteration operation on the Riemannian manifold can reduce the modeling error and improve the expression ability of nonlinear problems;
- It can make full use of the intrinsic geometric structure of the objective function and embed the constraints into the search space to solve the unconstrained optimization problems based on the constrained search space, such as the low-rank matrix constraint problems and the positive semidefinite programming problems [11];
- Solving the target directly on the Riemannian manifold can reduce the damage to the original structural information caused by the vectorization of high-dimensional data.
2. Preliminaries
2.1. The Riemannian Manifold
2.2. Basic Operators on Manifold
2.3. Test Problems
2.3.1. Semidefinite Programming Problem
2.3.2. Secant-Based Data Dimensionality Reduction Problem
2.3.3. Robust Principal Component Analysis
2.4. The Quantum-Behaved Particle Swarm Optimization Algorithm
3. The Proposed RQPSO Algorithm
3.1. The QPSO Algorithm on Riemannian Manifold
- 1.
- Randomly select a point from the population as the mean best position ;
- 2.
- For particle , randomly select a point along the geodesic line between and . Generate a tangent from to , and then randomly select a point on the tangent and retract it to on the manifold;
- 3.
- Generate a tangent in the same direction as that from to at the point produced in (2). Then, retract to on the manifold at point and obtain a new population.
Algorithm 1 Quantum Particle Swarm Optimization on Riemannian Manifold (RQPSO) |
|
3.2. Quantum Behavior of the RQPSO Algorithm
4. Experimental Studies
4.1. Parameters Analysis
4.1.1. The Effect of on the Performance of RQPSO
4.1.2. The Effect of on the Performance of RQPSO
4.1.3. The Effect of Parameters on the Performance of RQPSO
4.2. Comparison Experiments
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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0.01 | 0.05 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1 | Average | ||
---|---|---|---|---|---|---|---|---|---|---|
0.01 | 8.19 × 102 | 7.94 × 102 | 7.73 × 102 | 8.89 × 102 | 8.90 × 102 | 8.14 × 102 | 7.59 × 102 | 8.60 × 102 | 8.25 × 102 | |
0.05 | 7.90 × 102 | 8.16 × 102 | 8.54 × 102 | 8.50 × 102 | 8.74 × 102 | 9.46 × 102 | 8.38 × 102 | 8.38 × 102 | 8.51 × 102 | |
0.1 | 7.86 × 102 | 8.87 × 102 | 8.59 × 102 | 8.77 × 102 | 8.41 × 102 | 8.06 × 102 | 8.33 × 102 | 7.55 × 102 | 8.30 × 102 | |
0.3 | 8.94 × 102 | 8.32 × 102 | 8.12 × 102 | 8.04 × 102 | 8.16 × 102 | 7.89 × 102 | 8.48 × 102 | 8.85 × 102 | 8.35 × 102 | |
0.5 | 8.32 × 102 | 9.04 × 102 | 8.16 × 102 | 7.42 × 102 | 7.99 × 102 | 7.98 × 102 | 8.37 × 102 | 7.67 × 102 | 8.12 × 102 | |
0.7 | 9.15 × 102 | 8.22 × 102 | 8.46 × 102 | 8.38 × 102 | 8.56 × 102 | 7.47 × 102 | 8.74 × 102 | 8.82 × 102 | 8.48 × 102 | |
0.9 | 9.33 × 102 | 9.30 × 102 | 8.65 × 102 | 8.22 × 102 | 9.00 × 102 | 8.46 × 102 | 8.96 × 102 | 8.35 × 102 | 8.79 × 102 | |
1 | 9.78 × 102 | 9.26 × 102 | 8.13 × 102 | 8.44 × 102 | 8.30 × 102 | 9.07 × 102 | 9.75 × 102 | 8.98 × 102 | 8.97 × 102 | |
Average | 8.68 × 102 | 8.64 × 102 | 8.30 × 102 | 8.33 × 102 | 8.51 × 102 | 8.32 × 102 | 8.58 × 102 | 8.40 × 102 |
RQPSO | DE | PSO | SD | ||
---|---|---|---|---|---|
50 | 3 | −6.1752 × 10−2(4.9 × 10−3) | −2.6914 × 10−2(3.2 × 10−3) | −2.6504 × 10−2(3.0 × 10−3) | −6.1607 × 10−2(4.7 × 10−3) |
50 | 5 | −1.0293 × 10−1(5.9 × 10−3) | −4.0609 × 10−2(5.1 × 10−3) | −3.5358 × 10−2(6.4 × 10−3) | −1.0289 × 10−1(5.8 × 10−3) |
50 | 7 | −1.4504 × 10−1(9.9 × 10−3) | −5.0532 × 10−2(8.0 × 10−3) | −4.3870 × 10−2(5.4 × 10−3) | −1.4231 × 10−1(1.2 × 10−2) |
50 | 9 | −1.9149 × 10−1(1.1 × 10−2) | −5.8637 × 10−2(1.4 × 10−2) | −5.2653 × 10−2(1.0 × 10−2) | −1.9150 × 10−1(1.1 × 10−2) |
100 | 3 | −4.5067 × 10−2(1.3 × 10−3) | −1.5092 × 10−2(2.3 × 10−3) | −1.2780 × 10−2(1.4 × 10−3) | −4.4553 × 10−2(1.0 × 10−3) |
100 | 5 | −7.2857 × 10−2(3.0 × 10−3) | −1.9977 × 10−2(3.8 × 10−3) | −1.8332 × 10−2(2.0 × 10−3) | −7.2205 × 10−2(3.2 × 10−3) |
100 | 7 | −1.0462 × 10−1(3.9 × 10−3) | −2.9372 × 10−2(3.2 × 10−3) | −2.5277 × 10−2(8.0 × 10−4) | −1.0509 × 10−1(3.9 × 10−3) |
100 | 9 | −1.3619 × 10−1(6.0 × 10−4) | −3.1572 × 10−2(4.5 × 10−3) | −2.8639 × 10−2(9.0 × 10−4) | −1.3589 × 10−1(1.1 × 10−3) |
150 | 3 | −3.6335 × 10−2(1.3 × 10−3) | −8.6850 × 10−3(1.0 × 10−3) | −8.8120 × 10−3(9.0 × 10−4) | −3.5789 × 10−2(1.3 × 10−3) |
150 | 5 | −6.0190 × 10−2(2.6 × 10−3) | −1.2079 × 10−2(1.6 × 10−3) | −1.2042 × 10−2(1.4 × 10−3) | −5.9307 × 10−2(2.8 × 10−3) |
150 | 7 | −8.6039 × 10−2(7.0 × 10−4) | −1.6084 × 10−2(5.3 × 10−3) | −1.6833 × 10−2(2.2 × 10−3) | −8.4907 × 10−2(2.0 × 10−3) |
150 | 9 | −1.1066 × 10−1(2.8 × 10−3) | −2.1269 × 10−2(3.2 × 10−3) | −1.9045 × 10−2(3.0 × 10−4) | −1.0970 × 10−1(2.6 × 10−3) |
200 | 3 | −3.1323 × 10−2(1.0 × 10−3) | −6.9490 × 10−3(1.7 × 10−3) | −7.1720 × 10−3(3.0 × 10−4) | −3.0667 × 10−2(9.0 × 10−4) |
200 | 5 | −5.3004 × 10−2(2.0 × 10−3) | −9.8660 × 10−3(2.2 × 10−3) | −9.9820 × 10−3(1.4 × 10−3) | −5.1632 × 10−2(3.4 × 10−3) |
200 | 7 | −7.2677 × 10−2(1.9 × 10−3) | −1.1306 × 10−2(2.7 × 10−3) | −1.2830 × 10−2(1.1 × 10−3) | −7.2031 × 10−2(1.9 × 10−3) |
200 | 9 | −9.3857 × 10−2(3.2 × 10−3) | −1.2133 × 10−2(3.7 × 10−3) | −1.5398 × 10−2(1.4 × 10−3) | −9.2505 × 10−2(4.1 × 10−3) |
250 | 3 | −2.8527 × 10−2(5.0 × 10−4) | −5.6330 × 10−3(1.6 × 10−3) | −5.8580 × 10−3(9.0 × 10−4) | −2.7597 × 10−2(1.1 × 10−3) |
250 | 5 | −4.7342 × 10−2(1.0 × 10−3) | −7.0910 × 10−3(1.5 × 10−3) | −8.5570 × 10−3(1.1 × 10−3) | −4.6141 × 10−2(1.2 × 10−3) |
250 | 7 | −6.5982 × 10−2(4.0 × 10−3) | −1.0577 × 10−2(9.0 × 10−4) | −1.0990 × 10−2(2.0 × 10−4) | −6.5406 × 10−2(4.3 × 10−3) |
250 | 9 | −8.4324 × 10−2(4.0 × 10−4) | −1.1266 × 10−2(3.4 × 10−3) | −1.3213 × 10−2(1.6 × 10−3) | −8.5312 × 10−2(4.0 × 10−4) |
RQPSO | DE | PSO | SD | ||
---|---|---|---|---|---|
50 | 3 | −5.1412 × 10−1(3.6 × 10−2) | −3.6914 × 10−1(3.9 × 10−2) | −2.2311 × 10−1(3.7 × 10−2) | −1.5535 × 10−1(4.0 × 10−2) |
50 | 5 | −7.2470 × 10−1(4.3 × 10−2) | −5.4939 × 10−1(4.4 × 10−2) | −3.4967 × 10−1(3.9 × 10−2) | −2.9375 × 10−1(1.3 × 10−1) |
50 | 7 | −8.4877 × 10−1(2.4 × 10−2) | −6.4097 × 10−1(4.4 × 10−2) | −4.2784 × 10−1(3.7 × 10−2) | −3.7782 × 10−1(1.4 × 10−1) |
50 | 9 | −9.5784 × 10−1(2.5 × 10−2) | −6.4802 × 10−1(2.5 × 10−2) | −4.7087 × 10−1(4.2 × 10−2) | −4.5747 × 10−1(2.7 × 10−2) |
50 | 11 | −9.7983 × 10−1(4.8 × 10−3) | −7.6750 × 10−1(3.8 × 10−2) | −5.2336 × 10−1(2.5 × 10−2) | −5.4464 × 10−1(5.9 × 10−2) |
100 | 3 | −4.5088 × 10−1(6.3 × 10−2) | −2.7348 × 10−1(2.8 × 10−2) | −1.6399 × 10−1(2.9 × 10−2) | −1.2822 × 10−1(6.4 × 10−2) |
100 | 5 | −7.0184 × 10−1(5.8 × 10−2) | −3.9794 × 10−1(3.5 × 10−2) | −2.3570 × 10−1(1.6 × 10−2) | −2.1470 × 10−1(5.6 × 10−2) |
100 | 7 | −8.4863 × 10−1(3.8 × 10−2) | −4.8878 × 10−1(5.2 × 10−2) | −2.8928 × 10−1(4.1 × 10−2) | −3.2030 × 10−1(1.2 × 10−1) |
100 | 9 | −9.3729 × 10−1(9.1 × 10−3) | −5.6107 × 10−1(5.5 × 10−2) | −3.3634 × 10−1(2.7 × 10−2) | −4.0881 × 10−1(6.0 × 10−2) |
100 | 11 | −9.6894 × 10−1(2.1 × 10−2) | −5.7446 × 10−1(2.5 × 10−2) | −3.6837 × 10−1(3.9 × 10−2) | −3.9402 × 10−1(3.8 × 10−2) |
150 | 3 | −4.5232 × 10−1(4.0 × 10−2) | −2.1901 × 10−1(1.5 × 10−2) | −1.3250 × 10−1(3.0 × 10−2) | −1.0771 × 10−1(4.3 × 10−2) |
150 | 5 | −7.1913 × 10−1(3.6 × 10−2) | −3.5151 × 10−1(2.1 × 10−2) | −1.8971 × 10−1(2.0 × 10−2) | −2.2270 × 10−1(4.9 × 10−2) |
150 | 7 | −8.4141 × 10−1(5.1 × 10−2) | −4.4002 × 10−1(1.7 × 10−2) | −2.4282 × 10−1(3.1 × 10−2) | −2.6939 × 10−1(1.5 × 10−1) |
150 | 9 | −9.3148 × 10−1(2.7 × 10−2) | −4.6331 × 10−1(2.5 × 10−2) | −2.6749 × 10−1(1.9 × 10−2) | −3.507 × 10−1(6.2 × 10−2) |
150 | 11 | −9.5823 × 10−1(1.9 × 10−2) | −5.4493 × 10−1(2.9 × 10−2) | −3.0970 × 10−1(2.1 × 10−2) | −3.9021 × 10−1(1.0 × 10−1) |
200 | 3 | −4.5178 × 10−1(4.9 × 10−2) | −1.8594 × 10−1(1.7 × 10−2) | −1.1556 × 10−1(1.9 × 10−2) | - 9.438 × 10−2(3.1 × 10−2) |
200 | 5 | −7.0589 × 10−1(4.6 × 10−2) | −3.0637 × 10−1(1.3 × 10−2) | −1.6907 × 10−1(1.2 × 10−2) | −1.9180 × 10−1(6.6 × 10−2) |
200 | 7 | −8.4805 × 10−1(4.5 × 10−2) | −3.8202 × 10−1(2.7 × 10−2) | −2.1149 × 10−1(1.3 × 10−2) | −2.5795 × 10−1(1.1 × 10−1) |
200 | 9 | −9.1661 × 10−1(4.3 × 10−2) | −4.2616 × 10−1(2.2 × 10−2) | −2.3520 × 10−1(2.9 × 10−2) | −3.1237 × 10−1(8.2 × 10−2) |
200 | 11 | −9.5415 × 10−1(2.1 × 10−2) | −4.3796 × 10−1(2.0 × 10−2) | −2.6385 × 10−1(2.2 × 10−2) | −3.1408 × 10−1(7.2 × 10−2) |
250 | 3 | −4.4076 × 10−1(5.2 × 10−2) | −1.6892 × 10−1(1.2 × 10−2) | −1.0321 × 10−1(2.2 × 10−2) | −1.1178 × 10−1(5.2 × 10−2) |
250 | 5 | −7.1519 × 10−1(6.4 × 10−2) | −2.6108 × 10−1(2.1 × 10−2) | −1.4771 × 10−1(1.7 × 10−2) | −1.6973 × 10−1(8.9 × 10−2) |
250 | 7 | −8.2571 × 10−1(4.9 × 10−2) | −3.4541 × 10−1(3.7 × 10−2) | −1.8112 × 10−1(1.2 × 10−2) | −2.3421 × 10−1(9.1 × 10−2) |
250 | 9 | −9.1166 × 10−1(5.3 × 10−2) | −3.7130 × 10−1(2.6 × 10−2) | −2.1360 × 10−1(2.2 × 10−2) | −2.8784 × 10−1(7.0 × 10−2) |
250 | 11 | −9.4470 × 10−1(3.3 × 10−2) | −3.9195 × 10−1(1.6 × 10−2) | −2.3519 × 10−1(2.0 × 10−2) | −3.1495 × 10−1(5.8 × 10−2) |
300 | 3 | −4.3421 × 10−1(4.8 × 10−2) | −1.5870 × 10−1(1.1 × 10−2) | −8.8721 × 10−2(1.9 × 10−2) | −1.0580 × 10−1(6.7 × 10−2) |
300 | 5 | −7.1296 × 10−1(3.0 × 10−2) | −2.5536 × 10−1(1.0 × 10−2) | −1.3469 × 10−1(2.6 × 10−2) | −2.0357 × 10−1(1.1 × 10−1) |
300 | 7 | −8.4278 × 10−1(4.1 × 10−2) | −3.1615 × 10−1(1.5 × 10−2) | −1.7673 × 10−1(2.4 × 10−2) | −2.4069 × 10−1(1.2 × 10−1) |
300 | 9 | −9.1008 × 10−1(2.5 × 10−2) | −3.6700 × 10−1(2.1 × 10−2) | −1.8787 × 10−1(2.5 × 10−2) | −2.2254 × 10−1(6.7 × 10−2) |
300 | 11 | −9.2532 × 10−1(1.6 × 10−2) | −3.6645 × 10−1(1.4 × 10−2) | −2.1657 × 10−1(1.6 × 10−2) | −2.9021 × 10−1(7.4 × 10−2) |
RQPSO | DE | PSO | SD | |||
---|---|---|---|---|---|---|
0.1 | 50 | 3 | 7.5654 × 102(5.4 × 101) | 8.7453 × 102(9.1 × 101) | 9.6618 × 102(1.0 × 102) | 7.6972 × 102(5.1 × 101) |
0.1 | 50 | 5 | 5.8195 × 102(5.3 × 101) | 7.1959 × 102(6.7 × 101) | 9.2457 × 102(5.4 × 101) | 5.9085 × 102(3.4 × 101) |
0.1 | 50 | 7 | 5.2952 × 102(6.0 × 101) | 6.8135 × 102(8.3 × 101) | 9.5001 × 102(9.1 × 101) | 5.0008 × 102(4.6 × 101) |
0.1 | 50 | 9 | 4.1113 × 102(2.1 × 101) | 5.4901 × 102(6.1 × 101) | 8.2810 × 102(5.6 × 101) | 3.8311 × 102(3.6 × 101) |
0.1 | 50 | 11 | 3.3373 × 102(3.1 × 101) | 4.9010 × 102(7.4 × 101) | 8.0673 × 102(7.6 × 101) | 3.2929 × 102(8.3 × 101) |
0.1 | 100 | 3 | 1.3832 × 103(1.3 × 102) | 1.56508 × 103(1.3 × 102) | 1.6717 × 103(1.2 × 102) | 1.3888 × 102(9.8 × 101) |
0.1 | 100 | 5 | 1.1628 × 103(8.3 × 101) | 1.4193 × 103(1.2 × 102) | 1.7113 × 103(1.4 × 102) | 1.1765 × 103(5.6 × 101) |
0.1 | 100 | 7 | 9.1449 × 102(1.5 × 102) | 1.2066 × 103(1.6 × 102) | 1.5274 × 103(1.9 × 102) | 9.1183 × 102(1.4 × 102) |
0.1 | 100 | 9 | 8.3650 × 102(1.8 × 102) | 1.1813 × 103(1.2 × 102) | 1.5791 × 103(1.4 × 102) | 8.3439 × 102(1.1 × 102) |
0.1 | 100 | 11 | 7.2025 × 102(6.1 × 101) | 1.0461 × 103(6.2 × 10¹) | 1.5885 × 103(1.3 × 102) | 6.9774 × 102(3.5 × 101) |
0.1 | 150 | 3 | 1.9575 × 103(9.6 × 101) | 2.2522 × 103(8.3 × 101) | 2.3563 × 103(6.9 × 101) | 1.9948 × 103(4.8 × 101) |
0.1 | 150 | 5 | 1.6554 × 103(2.1 × 102) | 2.0202 × 103(1.8 × 102) | 2.2774 × 103(1.7 × 102) | 1.6435 × 102(2.3 × 102) |
0.1 | 150 | 7 | 1.4248 × 103(9.2 × 101) | 1.8835 × 103(2.0 × 102) | 2.2789 × 103(2.3 × 102) | 1.4109 × 103(1.4 × 102) |
0.1 | 150 | 9 | 1.1981 × 103(1.1 × 102) | 1.7073 × 103(1.3 × 102) | 2.1912 × 103(2.0 × 102) | 1.2012 × 103(8.1 × 101) |
0.1 | 150 | 11 | 1.0890 × 102(1.4 × 102) | 1.6137 × 103(1.5 × 102) | 2.1808 × 103(1.9 × 102) | 1.1446 × 103(1.5 × 102) |
0.1 | 200 | 3 | 2.4460 × 103(3.2 × 102) | 2.7945 × 103(4.1 × 102) | 2.9424 × 103(3.7 × 102) | 2.4734 × 103(3.3 × 102) |
0.1 | 200 | 5 | 2.1402 × 103(6.3 × 101) | 2.6536 × 103(1.8 × 102) | 2.9612 × 103(1.4 × 102) | 2.1360 × 103(6.9 × 102) |
0.1 | 200 | 7 | 1.8318 × 103(9.5 × 101) | 2.4882 × 103(1.2 × 102) | 2.8832 × 103(2.5 × 102) | 1.8842 × 103(1.3 × 102) |
0.1 | 200 | 9 | 1.6339 × 103(1.8 × 102) | 2.3872 × 103(1.8 × 102) | 2.9148 × 103(1.8 × 102) | 1.6754 × 103(9.6 × 101) |
0.1 | 200 | 11 | 1.4853 × 103(1.3 × 102) | 2.3016 × 103(4.9 × 101) | 2.8639 × 103(5.4 × 101) | 1.6038 × 103(1.5 × 102) |
0.01 | 50 | 3 | 7.1973 × 101(2.6 × 101) | 1.5195 × 102(1.1 × 101) | 2.0471 × 102(2.8 × 101) | 7.4976 × 101(3.1 × 101) |
0.01 | 50 | 5 | 5.1583 × 101(3.2 × 101) | 1.0843 × 102(2.5 × 101) | 1.7699 × 102(1.5 × 101) | 6.2757 × 101(3.0 × 101) |
0.01 | 50 | 7 | 4.7165 × 101(2.6 × 101) | 8.3367 × 101(2.4 × 101) | 1.7559 × 102(1.7 × 101) | 5.9280 × 101(2.8 × 101) |
0.01 | 50 | 9 | 2.9900 × 101(1.2 × 101) | 5.5275 × 101(1.1 × 101) | 1.3470 × 102(4.2 × 101) | 3.7299 × 101(1.3 × 101) |
0.01 | 50 | 11 | 3.2033 × 101(1.3 × 101) | 4.9114 × 101(1.5 × 101) | 1.0878 × 102(1.7 × 101) | 4.0297 × 101(1.4 × 101) |
0.01 | 100 | 3 | 1.2717 × 102(2.6 × 101) | 2.6042 × 102(3.9 × 101) | 2.9370 × 102(2.3 × 101) | 1.3677 × 101(1.6 × 101) |
0.01 | 100 | 5 | 9.7236 × 101(3.5 × 101) | 2.0219 × 102(7.3 × 101) | 2.7024 × 102(7.7 × 101) | 1.1542 × 102(2.5 × 101) |
0.01 | 100 | 7 | 7.3738 × 101(2.4 × 101) | 1.5125 × 102(1.9 × 101) | 2.3166 × 102(6.2 × 101) | 9.2338 × 101(1.2 × 101) |
0.01 | 100 | 9 | 6.9656 × 101(2.7 × 101) | 1.5284 × 102(5.0 × 101) | 2.8383 × 102(7.4 × 101) | 8.5461 × 101(2.1 × 101) |
0.01 | 100 | 11 | 5.6931 × 101(2.5 × 101) | 1.1947 × 102(2.1 × 101) | 2.1254 × 102(4.6 × 101) | 9.1455 × 101(3.0 × 101) |
0.01 | 150 | 3 | 1.7892 × 102(6.9 × 101) | 3.5798 × 102(8.2 × 101) | 4.0475 × 102(1.1 × 102) | 1.9433 × 102(8.0 × 101) |
0.01 | 150 | 5 | 1.4577 × 102(2.6 × 101) | 2.8052 × 102(5.1 × 101) | 3.4794 × 102(5,0 × 101) | 1.6277 × 102(2.9 × 101) |
0.01 | 150 | 7 | 1.5671 × 102(4.2 × 101) | 2.8247 × 102(3.7 × 101) | 3.6967 × 102(4.2 × 101) | 1.7908 × 102(4.9 × 101) |
0.01 | 150 | 9 | 1.1167 × 102(1.9 × 101) | 2.2275 × 102(3.4 × 101) | 3.1155 × 102(4.2 × 101) | 1.4133 × 102(1.4 × 101) |
0.01 | 150 | 11 | 1.0527 × 102(1.4 × 101) | 1.9597 × 102(2.3 × 101) | 2.9205 × 102(5.1 × 101) | 1.4177 × 102(2.1 × 101) |
0.01 | 200 | 3 | 2.0611 × 102(3.8 × 101) | 4.2597 × 102(1.7 × 102) | 4.4865 × 102(1.4 × 102) | 2.1420 × 102(4.6 × 101) |
0.01 | 200 | 5 | 1.5747 × 102(6.1 × 101) | 3.2010 × 102(2.7 × 101) | 3.5707 × 102(2.4 × 101) | 1.7202 × 102(6.8 × 101) |
0.01 | 200 | 7 | 1.5383 × 102(6.8 × 101) | 3.0628 × 102(8.2 × 101) | 3.9345 × 102(1.3 × 102) | 1.7395 × 102(6.3 × 101) |
0.01 | 200 | 9 | 1.6872 × 102(3.1 × 101) | 3.4866 × 102(8.1 × 101) | 4.3993 × 102(5.9 × 101) | 2.1596 × 102(3.5 × 101) |
0.01 | 200 | 11 | 1.6727 × 102(3.2 × 101) | 2.8346 × 102(4.4 × 101) | 3.5677 × 102(1.7 × 101) | 2.0155 × 102(6.7 × 101) |
0.001 | 50 | 3 | 2.5855 × 101(9.3 × 100) | 1.2590 × 102(3.6 × 101) | 1.5201 × 102(3.4 × 101) | 3.4484 × 101(1.7 × 101) |
0.001 | 50 | 5 | 8.3620 × 100(7.0 × 100) | 6.1819 × 101(2.1 × 101) | 1.2659 × 102(2.2 × 101) | 9.3607 × 100(1.1 × 101) |
0.001 | 50 | 7 | 6.7883 × 100(5.5 × 100) | 3.8484 × 101(6.1 × 100) | 1.0937 × 102(1.1 × 101) | 6.4283 × 100(1.1 × 101) |
0.001 | 50 | 9 | 6.4813 × 100(4.4 × 100) | 2.5437 × 101(8.9 × 100) | 9.5061 × 101(3.0 × 101) | 6.5565 × 100(1.0 × 101) |
0.001 | 50 | 11 | 3.6024 × 100(1.4 × 100) | 1.1593 × 101(2.4 × 100) | 7.9012 × 101(4.3 × 101) | 2.1094 × 100(7.7 × 10−1) |
0.001 | 100 | 3 | 2.8134 × 101(1.3 × 101) | 1.9378 × 102(6.1 × 101) | 2.3001 × 102(4.5 × 101) | 2.9517 × 101(1.4 × 101) |
0.001 | 100 | 5 | 1.5818 × 101(5.3 × 100) | 1.8256 × 102(4.2 × 101) | 2.4844 × 102(5.2 × 101) | 2.1447 × 101(8.4 × 100) |
0.001 | 100 | 7 | 1.1198 × 101(5.6 × 100) | 1.1245 × 102(3.5 × 101) | 1.7682 × 102(2.7 × 101) | 1.2736 × 101(9.5 × 100) |
0.001 | 100 | 9 | 1.3117 × 10¹(5.5 × 100) | 9.3403 × 101(4.7 × 101) | 1.8227 × 102(8.1 × 101) | 1.8515 × 101(1.4 × 101) |
0.001 | 100 | 11 | 7.6223 × 100(4.7 × 100) | 6.2281 × 101(1.4 × 101) | 1.4409 × 102(3.1 × 101) | 1.0546 × 101(1.0 × 101) |
0.001 | 150 | 3 | 3.7829 × 101(1.0 × 101) | 2.2759 × 102(1.1 × 102) | 2.5950 × 102(1.1 × 102) | 4.2692 × 101(1.4 × 101) |
0.001 | 150 | 5 | 2.7834 × 101(1.5 × 101) | 2.1556 × 102(6.3 × 101) | 2.7175 × 102(6.7 × 101) | 4.0423 × 101(2.5 × 101) |
0.001 | 150 | 7 | 1.4826 × 101(1.2 × 101) | 1.3634 × 102(3.1 × 101) | 1.9849 × 102(5.9 × 101) | 1.9988 × 101(1.5 × 101) |
0.001 | 150 | 9 | 1.6358 × 101(6.0 × 100) | 1.5928 × 102(3.5 × 101) | 2.4219 × 102(8.5 × 101) | 3.1013 × 101(1.6 × 101) |
0.001 | 150 | 11 | 1.8158 × 101(5.1 × 100) | 1.2162 × 102(2.6 × 101) | 1.9174 × 102(3.0 × 101) | 2.4575 × 101(8.0 × 100) |
0.001 | 200 | 3 | 3.2492 × 101(8.1 × 100) | 2.6359 × 102(1.0 × 102) | 2.8572 × 102(9.5 × 101) | 3.8216 × 101(1.6 × 101) |
0.001 | 200 | 5 | 2.6752 × 101(9.8 × 100) | 2.1531 × 102(9.5 × 101) | 2.5783 × 102(1.1 × 102) | 3.6128 × 101(1.7 × 101) |
0.001 | 200 | 7 | 2.6361 × 101(1.9 × 101) | 1.8300 × 102(1.4 × 102) | 2.2845 × 102(1.6 × 102) | 3.4054 × 101(2.6 × 101) |
0.001 | 200 | 9 | 1.8149 × 101(8.3 × 100) | 1.6877 × 102(6.9 × 101) | 2.1808 × 102(1.0 × 102) | 2.7946 × 101(1.1 × 101) |
0.001 | 200 | 11 | 2.2633 × 101(5.2 × 100) | 1.5670 × 102(3.5 × 101) | 2.0801 × 102(4.1 × 101) | 3.0098 × 101(3.9 × 100) |
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Halimu, Y.; Zhou, C.; You, Q.; Sun, J. A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds. Mathematics 2022, 10, 4168. https://doi.org/10.3390/math10224168
Halimu Y, Zhou C, You Q, Sun J. A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds. Mathematics. 2022; 10(22):4168. https://doi.org/10.3390/math10224168
Chicago/Turabian StyleHalimu, Yeerjiang, Chao Zhou, Qi You, and Jun Sun. 2022. "A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds" Mathematics 10, no. 22: 4168. https://doi.org/10.3390/math10224168
APA StyleHalimu, Y., Zhou, C., You, Q., & Sun, J. (2022). A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds. Mathematics, 10(22), 4168. https://doi.org/10.3390/math10224168