• Previous Article
    Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty
  • NACO Home
  • This Issue
  • Next Article
    Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization
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

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.

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
References:
[1]

B. AkessonJ. JorgensenN. 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. KrislockJ. LangJ. VarahD. 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. LiS. 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. LiD. 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. RinnanM. AnderssonC. 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. TohM. 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. 

show all references

References:
[1]

B. AkessonJ. JorgensenN. 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. KrislockJ. LangJ. VarahD. 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. LiS. 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. LiD. 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. RinnanM. AnderssonC. 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. TohM. 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. 

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
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
[1]

Xiao-Wen Chang, David Titley-Peloquin. An improved algorithm for generalized least squares estimation. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 451-461. doi: 10.3934/naco.2020044

[2]

Zhuoyi Xu, Yong Xia, Deren Han. On box-constrained total least squares problem. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 439-449. doi: 10.3934/naco.2020043

[3]

Anwa Zhou, Jinyan Fan. A semidefinite relaxation algorithm for checking completely positive separable matrices. Journal of Industrial and Management Optimization, 2019, 15 (2) : 893-908. doi: 10.3934/jimo.2018076

[4]

Zhou Sheng, Gonglin Yuan, Zengru Cui, Xiabin Duan, Xiaoliang Wang. An adaptive trust region algorithm for large-residual nonsmooth least squares problems. Journal of Industrial and Management Optimization, 2018, 14 (2) : 707-718. doi: 10.3934/jimo.2017070

[5]

Yazheng Dang, Marcus Ang, Jie Sun. An inertial triple-projection algorithm for solving the split feasibility problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022019

[6]

Ouafa Belguidoum, Hassina Grar. An improved projection algorithm for variational inequality problem with multivalued mapping. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022002

[7]

Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial and Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057

[8]

Chenchen Wu, Dachuan Xu, Xin-Yuan Zhao. An improved approximation algorithm for the $2$-catalog segmentation problem using semidefinite programming relaxation. Journal of Industrial and Management Optimization, 2012, 8 (1) : 117-126. doi: 10.3934/jimo.2012.8.117

[9]

Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082

[10]

Hassan Mohammad, Mohammed Yusuf Waziri, Sandra Augusta Santos. A brief survey of methods for solving nonlinear least-squares problems. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 1-13. doi: 10.3934/naco.2019001

[11]

Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15

[12]

Mila Nikolova. Analytical bounds on the minimizers of (nonconvex) regularized least-squares. Inverse Problems and Imaging, 2008, 2 (1) : 133-149. doi: 10.3934/ipi.2008.2.133

[13]

Yanyan Hu, Fubao Xi, Min Zhu. Least squares estimation for distribution-dependent stochastic differential delay equations. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1505-1536. doi: 10.3934/cpaa.2022027

[14]

Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863

[15]

Ya-Zheng Dang, Zhong-Hui Xue, Yan Gao, Jun-Xiang Li. Fast self-adaptive regularization iterative algorithm for solving split feasibility problem. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1555-1569. doi: 10.3934/jimo.2019017

[16]

Tianyu Yang, Yang Yang. A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem. Inverse Problems and Imaging, 2022, 16 (1) : 1-18. doi: 10.3934/ipi.2021038

[17]

Ashif Mustafa, Manideepa Saha. A generalized projection iterative method for solving non-singular linear systems. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022009

[18]

Qingzhi Yang. The revisit of a projection algorithm with variable steps for variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (2) : 211-217. doi: 10.3934/jimo.2005.1.211

[19]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015

[20]

Simai He, Min Li, Shuzhong Zhang, Zhi-Quan Luo. A nonconvergent example for the iterative water-filling algorithm. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 147-150. doi: 10.3934/naco.2011.1.147

 Impact Factor: 

Metrics

  • PDF downloads (160)
  • HTML views (198)
  • Cited by (0)

Other articles
by authors

[Back to Top]