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Comparing of Algorithm 2.4 and Algorithm 3.2
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December
2020, 10(4): 511-520.
doi: 10.3934/naco.2020048
Parameter-related projection-based iterative algorithm for a kind of generalized positive semidefinite least squares problem
College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350007, P. R. China |
A projection-based iterative algorithm, which is related to a single parameter (or the multiple parameters), is proposed to solve the generalized positive semidefinite least squares problem introduced in this paper. The single parameter (or the multiple parameters) projection-based iterative algorithms converges to the optimal solution under certain condition, and the corresponding numerical results are shown too.
Mathematics Subject Classification:
Primary: 65K05; Secondary: 90C30.
Citation:
Chengjin Li. Parameter-related projection-based iterative algorithm for a kind of generalized positive semidefinite least squares problem. Numerical Algebra, Control and Optimization,
2020, 10
(4)
: 511-520.
doi: 10.3934/naco.2020048
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References:
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A generalized autocovariance least-squares method for Kalman filter tuning, Journal of Proccess Control, 18 (2008), 769-779.
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Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation, SIAM Journal on Control and Optimization, 26 (1988), 537-556.
doi: 10.1137/0326032. |
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Z. X. Chan and D. F. Sun,
Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming, SIAM Journal on Optimization, 19 (2008), 370-396.
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H. Dai and P. Lancaster,
Linear matrix equations from an inverse problem of vibration theory, Linear Algebra and Its Applications, 246 (1996), 31-47.
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N. Gillis and P. Sharma,
A semi-analytical approach for the positive semidefinite procrustes problem, Linear Algebra and Its Applications, 540 (2018), 112-137.
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N. Krislock, J. Lang, J. Varah, D. K. Pai and H. P. Seidel,
Local compliance estimation via positive semidefinite constrained least squares, IEEE transactions on Robotics, 20 (2004), 1007-1011.
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C. J. Li, S. G. Zhang and H. H. Wu,
The proximal point iterative algorithm for the generalized semidefinite least squares problem, ACTA Mathematicae Applicatae Sinica, (in Chinese), 42 (2019), 371-384.
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X. Li, D. F. Sun and K. C. Toh,
A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Mathematical Programming, 155 (2016), 333-373.
doi: 10.1007/s10107-014-0850-5. |
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A. Liao and Z. Bai,
Least-squares solution of AXB = D over symmetric positive semidefinite matrices X, Journal of Computational Mathematics, 21 (2003), 175-182.
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Y. Nesterov, Introductory Lectures On Convex Optimization: A Basic Course, Springer Science and Business Media, 2004.
doi: 10.1007/978-1-4419-8853-9. |
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H. D. Qi,
Conditional quadratic semidefinite programming: examples and methods, Journal of Operations Research Society of China, 2 (2014), 143-170.
doi: 10.1007/s40305-014-0048-9. |
| [12] |
H. D. Qi,
A convex matrix optimization for the additive constant problem in multidimensional scaling with application to locally linear embedding, SIAM Journal on Optimization, 26 (2016), 2564-2590.
doi: 10.1137/15M1043133. |
| [13] |
H. D. Qi and D. Sun,
A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 360-385.
doi: 10.1137/050624509. |
| [14] |
A. Rinnan, M. Andersson, C. Ridder and B. Engelsen,
Recursive weighted partial least squares(rPLS): An efficient varible selection method using PLS, Journal of Chemometrics, 28 (2014), 439-447.
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M. L. Stein,
Spatial variation of total column ozone on a global scale, The Annals of Applied Statistics, 1 (2007), 191-210.
doi: 10.1214/07-AOAS106. |
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K. C. Toh, M. J. Todd and R. H. Tutuncu,
SDPT3 - a Matlab software package for semidefinite programming, Optimization Methods and Software, 11 (1999), 545-581.
doi: 10.1080/10556789908805762. |
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K. G. Woodgate,
Efficient stiffness matrix estimation for elastic structures, Computers and Structures, 69 (1998), 79-84.
|
show all references
References:
| [1] |
B. Akesson, J. Jorgensen, N. Poulsen and S. Jorgensen,
A generalized autocovariance least-squares method for Kalman filter tuning, Journal of Proccess Control, 18 (2008), 769-779.
|
| [2] |
J. C. Allwright,
Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation, SIAM Journal on Control and Optimization, 26 (1988), 537-556.
doi: 10.1137/0326032. |
| [3] |
Z. X. Chan and D. F. Sun,
Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming, SIAM Journal on Optimization, 19 (2008), 370-396.
doi: 10.1137/070681235. |
| [4] |
H. Dai and P. Lancaster,
Linear matrix equations from an inverse problem of vibration theory, Linear Algebra and Its Applications, 246 (1996), 31-47.
doi: 10.1016/0024-3795(94)00311-4. |
| [5] |
N. Gillis and P. Sharma,
A semi-analytical approach for the positive semidefinite procrustes problem, Linear Algebra and Its Applications, 540 (2018), 112-137.
doi: 10.1016/j.laa.2017.11.023. |
| [6] |
N. Krislock, J. Lang, J. Varah, D. K. Pai and H. P. Seidel,
Local compliance estimation via positive semidefinite constrained least squares, IEEE transactions on Robotics, 20 (2004), 1007-1011.
|
| [7] |
C. J. Li, S. G. Zhang and H. H. Wu,
The proximal point iterative algorithm for the generalized semidefinite least squares problem, ACTA Mathematicae Applicatae Sinica, (in Chinese), 42 (2019), 371-384.
|
| [8] |
X. Li, D. F. Sun and K. C. Toh,
A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Mathematical Programming, 155 (2016), 333-373.
doi: 10.1007/s10107-014-0850-5. |
| [9] |
A. Liao and Z. Bai,
Least-squares solution of AXB = D over symmetric positive semidefinite matrices X, Journal of Computational Mathematics, 21 (2003), 175-182.
|
| [10] |
Y. Nesterov, Introductory Lectures On Convex Optimization: A Basic Course, Springer Science and Business Media, 2004.
doi: 10.1007/978-1-4419-8853-9. |
| [11] |
H. D. Qi,
Conditional quadratic semidefinite programming: examples and methods, Journal of Operations Research Society of China, 2 (2014), 143-170.
doi: 10.1007/s40305-014-0048-9. |
| [12] |
H. D. Qi,
A convex matrix optimization for the additive constant problem in multidimensional scaling with application to locally linear embedding, SIAM Journal on Optimization, 26 (2016), 2564-2590.
doi: 10.1137/15M1043133. |
| [13] |
H. D. Qi and D. Sun,
A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 360-385.
doi: 10.1137/050624509. |
| [14] |
A. Rinnan, M. Andersson, C. Ridder and B. Engelsen,
Recursive weighted partial least squares(rPLS): An efficient varible selection method using PLS, Journal of Chemometrics, 28 (2014), 439-447.
|
| [15] |
M. L. Stein,
Spatial variation of total column ozone on a global scale, The Annals of Applied Statistics, 1 (2007), 191-210.
doi: 10.1214/07-AOAS106. |
| [16] |
K. C. Toh, M. J. Todd and R. H. Tutuncu,
SDPT3 - a Matlab software package for semidefinite programming, Optimization Methods and Software, 11 (1999), 545-581.
doi: 10.1080/10556789908805762. |
| [17] |
K. G. Woodgate,
Efficient stiffness matrix estimation for elastic structures, Computers and Structures, 69 (1998), 79-84.
|
| Parameters | Spending time | Parameters | Spending time | ||||
| (71, 65, 67) | T1 | 62.1 | 15.6 | (59, 8, 30) | T1 | 0.017 | 0.016 |
| (53, 39, 48) | 2.13 | 1.95 | (103, 61, 85) | 1.82 | 1.09 | ||
| (101, 33, 61) | 0.25 | 0.19 | (102, 78, 90) | 8.95 | 5.61 | ||
| (102, 19, 95) | 0.040 | 0.039 | (93, 20, 92) | 0.041 | 0.038 | ||
| (66, 48, 53) | 10.1 | 3.28 | (58, 18, 45) | 0.05 | 0.05 | ||
| (62, 4, 38) | T2 | 0.009 | 0.008 | (49, 41, 42) | T2 | 115.6 | 8.67 |
| (71, 11, 24) | 0.04 | 0.02 | (94, 44, 92) | 0.94 | 0.28 | ||
| (92, 43, 64) | 1.99 | 0.45 | (98, 19, 93) | 0.05 | 0.03 | ||
| (93, 7, 48) | 0.014 | 0.012 | (73, 4, 30) | 0.008 | 0.007 | ||
| (86, 64, 72) | 32.9 | 3.99 | (78, 39, 62) | 1.54 | 0.34 | ||
| (94, 16, 85, (6)) | T3 | 0.03 | 0.02 | (91, 43, 83, (21)) | T3 | 0.90 | 0.20 |
| (60, 36, 53, (25)) | 1.18 | 0.28 | (54, 11, 24, (9)) | 0.04 | 0.02 | ||
| (31, 8, 10, (7)) | 0.12 | 0.03 | (86, 9, 83, (1)) | 0.019 | 0.015 | ||
| (78, 55, 69, (7)) | 20.7 | 0.77 | (97, 17, 33, (15)) | 0.08 | 0.04 | ||
| (98, 43, 51, (3)) | 10.3 | 0.44 | (42, 10, 36, (9)) | 0.023 | 0.019 | ||
| Parameters | Spending time | Parameters | Spending time | ||||
| (71, 65, 67) | T1 | 62.1 | 15.6 | (59, 8, 30) | T1 | 0.017 | 0.016 |
| (53, 39, 48) | 2.13 | 1.95 | (103, 61, 85) | 1.82 | 1.09 | ||
| (101, 33, 61) | 0.25 | 0.19 | (102, 78, 90) | 8.95 | 5.61 | ||
| (102, 19, 95) | 0.040 | 0.039 | (93, 20, 92) | 0.041 | 0.038 | ||
| (66, 48, 53) | 10.1 | 3.28 | (58, 18, 45) | 0.05 | 0.05 | ||
| (62, 4, 38) | T2 | 0.009 | 0.008 | (49, 41, 42) | T2 | 115.6 | 8.67 |
| (71, 11, 24) | 0.04 | 0.02 | (94, 44, 92) | 0.94 | 0.28 | ||
| (92, 43, 64) | 1.99 | 0.45 | (98, 19, 93) | 0.05 | 0.03 | ||
| (93, 7, 48) | 0.014 | 0.012 | (73, 4, 30) | 0.008 | 0.007 | ||
| (86, 64, 72) | 32.9 | 3.99 | (78, 39, 62) | 1.54 | 0.34 | ||
| (94, 16, 85, (6)) | T3 | 0.03 | 0.02 | (91, 43, 83, (21)) | T3 | 0.90 | 0.20 |
| (60, 36, 53, (25)) | 1.18 | 0.28 | (54, 11, 24, (9)) | 0.04 | 0.02 | ||
| (31, 8, 10, (7)) | 0.12 | 0.03 | (86, 9, 83, (1)) | 0.019 | 0.015 | ||
| (78, 55, 69, (7)) | 20.7 | 0.77 | (97, 17, 33, (15)) | 0.08 | 0.04 | ||
| (98, 43, 51, (3)) | 10.3 | 0.44 | (42, 10, 36, (9)) | 0.023 | 0.019 | ||
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