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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

Chengjin Li

College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350007, P. R. China

Received  March 2020 Revised  September 2020 Published  September 2020

Fund Project: This work was supported by the National Natural Science Foundations of China (11301080, 11526053), Science Foundation of Fujian Province of China (2016J05003), the Foundation of the Education Department of Fujian Province of China (JA15106), and the Project of Nonlinear analysis and its applications (IRTL1206)

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.

Keywords: Projection, iterative algorithm, positive semidefinite, least squares problem.
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 & Optimization, 2020, 10 (4) : 511-520. doi: 10.3934/naco.2020048
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. G. Woodgate, Efficient stiffness matrix estimation for elastic structures, Computers and Structures, 69 (1998), 79-84.   Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. G. Woodgate, Efficient stiffness matrix estimation for elastic structures, Computers and Structures, 69 (1998), 79-84.   Google Scholar

Table .  Comparing of Algorithm 2.4 and Algorithm 3.2
Parameters Spending time Parameters Spending time
$ (m,n,p,(r)) $ $ type $ $ tim1 $ $ tim2 $ $ (m,n,p,(r)) $ $ type $ $ tim1 $ $ tim2 $
(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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Parameters Spending time Parameters Spending time
$ (m,n,p,(r)) $ $ type $ $ tim1 $ $ tim2 $ $ (m,n,p,(r)) $ $ type $ $ tim1 $ $ tim2 $
(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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