列正交约束下广义Sylvester方程极小化问题的有效算法

Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (2): 479-495.

Previous Articles Next Articles

An Effective Algorithm for Generalized Sylvester Equation Minimization Problem Under Columnwise Orthogonal Constraints

Yueyuan Liu,Kai Wang,Shujuan Qin,Jiaofen Li*()

^{School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guangxi Guilin 541004}

Received:2020-03-26 Online:2021-04-26 Published:2021-04-29
Contact: Jiaofen Li E-mail:lixiaogui1290@163.com
Supported by:
the NSFC(11761024);the NSFC(11561015);the NSFC(11961012);the GUET Excellent Graduate Thesis Program(2019YJSPY03);the GUET Graduate Innovation Project(2020YJSCX02);the NSF of Guangxi Province(2017GXNSFBA198082)

Abstract

Abstract:

This paper presents an efficient algorithm for solving the generalized Sylvester equation minimization problem under columnwise orthogonal constraints. Based on some geometric properties of the Stiefel manifold and the MPRP conjugate gradient method in Euclidean space, a Riemannian MPRP conjugate gradient algorithm with Armijo-type line search is proposed to solve the presented problem, and its global convergence is also established. An attractive property of the proposed method is that the direction generated by the method is always a descent direction for the objective function. Some numerical tests are given to show the efficiency of the proposed method. Comparisons with some existing methods are also given.

Key words: Generalized Sylvester equation, Minimization problem, Columnwise orthogonal constraint, Riemannian conjugate gradient method

CLC Number:

O151.1

Yueyuan Liu,Kai Wang,Shujuan Qin,Jiaofen Li. An Effective Algorithm for Generalized Sylvester Equation Minimization Problem Under Columnwise Orthogonal Constraints[J].Acta mathematica scientia,Series A, 2021, 41(2): 479-495.

Trendmd

References

1	Dehghan M , Hajarian M . An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Applied Mathematical Modelling, 2010, 34 (3): 639- 654
2	Hajarian M . Developing BiCOR and CORS methods for coupled Sylvester-transpose and periodic Sylvester matrix equations. Applied Mathematical Modelling, 2015, 39 (19): 6073- 6084
3	Hajarian M . Extending the CGLS algorithm for least squares solutions of the generalized Sylvester-transpose matrix equations. Journal of the Franklin Institute, 2016, 353 (5): 1168- 1185
4	Li S K , Huang T Z . LSQR iterative method for generalized coupled Sylvester matrix equations. Applied Mathematical Modelling, 2012, 36 (8): 3545- 3554
5	Ding F , Chen T . On iterative solutions of general coupled matrix equations. SIAM Journal on Control and Optimization, 2006, 44 (6): 2269- 2284
6	Bouhamidi A , Jbilou K , Raydan M . Convex constrained optimization for large-scale generalized Sylvester equations. Computational Optimization and Applications, 2011, 48 (2): 233- 253
7	Bouhamidi A , Enkhbat R , Jbilou K . Conditional gradient Tikhonov method for a convex optimization problem in image restoration. Journal of Computational and Applied Mathematics, 2014, 255, 580- 592
8	Li J F , Li W , Huang R . An efficient method for solving a matrix least squares problem over a matrix inequality constraint. Computational Optimization and Applications, 2016, 63 (2): 393- 423
9	Escalante R , Raydan M . Dykstra's algorithm for a constrained least-squares matrix problem. Numerical Linear Algebra with Applications, 1996, 3 (6): 459- 471
10	Escalante R , Raydan M . Dykstra's algorithm for constrained least-squares rectangular matrix problems. Computers & Mathematics with Applications, 1998, 35 (6): 73- 79
11	Monsalve M , Moreno J , Escalante R , Raydan M . Selective alternating projections to find the nearest SDD+ matrix. Applied Mathematics and Computation, 2003, 145 (2/3): 205- 220
12	Li J F , Hu X Y , Zhang L . Dykstra's algorithm for constrained least-squares doubly symmetric matrix problems. Theoretical Computer Science, 2010, 411 (31-33): 2818- 2826
13	Li J F , Li W , Vong S W , et al. A Riemannian optimization approach for solving the generalized eigenvalue problem for nonsquare matrix pencils. Journal of Scientific Computing, 2020, 82 (3): 1- 43
14	Meng C , Hu X , Zhang L . The skew-symmetric orthogonal solutions of the matrix equation AX=B. Linear Algebra and its Applications, 2005, 402, 303- 318
15	Kiers H A . Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems. Computational statistics & data analysis, 2002, 41 (1): 157- 170
16	Kanamori T , Takeda A . Numerical study of learning algorithms on Stiefel manifold. Computational Management Science, 2014, 11 (4): 319- 340
17	Lai R , Osher S . A splitting method for orthogonality constrained problems. Journal of Scientific Computing, 2014, 58 (2): 431- 449
18	Chen W , Ji H , You Y . An augmented Lagrangian method for 1-regularized optimization problems with orthogonality constraints. SIAM Journal on Scientific Computing, 2016, 38 (4): B570- B592
19	Sato H , Iwai T . A Riemannian optimization approach to the matrix singular value decomposition. SIAM Journal on Optimization, 2013, 23 (1): 188- 212
20	Zhu X . A Riemannian conjugate gradient method for optimization on the Stiefel manifold. Computational Optimization and Applications, 2017, 67 (1): 73- 110
21	Vandereycken B . Low-rank matrix completion by Riemannian optimization. SIAM Journal on Optimization, 2013, 23 (2): 1214- 1236
22	Zhao Z , Jin X Q , Bai Z J . A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM Journal on Numerical Analysis, 2016, 54 (4): 2015- 2035
23	Yao T T , Bai Z J , Zhao Z , Ching W K . A Riemannian Fletcher-Reeves conjugate gradient method for doubly stochastic inverse eigenvalue problems. SIAM Journal on Matrix Analysis and Applications, 2016, 37 (1): 215- 234
24	Yao T T , Bai Z J , Zhao Z . A Riemannian variant of the Fletcher-Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata. Numerical Linear Algebra with Applications, 2019, 26 (2): e2221
25	Zhang L , Zhou W , Li D H . A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence. IMA Journal of Numerical Analysis, 2006, 26 (4): 629- 640
26	Absil P A , Mahony R , Sepulchre R . Optimization Aalgorithms on Matrix Manifolds. Princeton: Princeton University Press, 2009
27	Wen Z , Yin W . A feasible method for optimization with orthogonality constraints. Mathematical Programming, 2013, 142 (1/2): 397- 434
28	Oviedo H , Lara H , Dalmau O . A non-monotone linear search algorithm with mixed direction on Stiefel manifold. Optimization Methods and Software, 2019, 34 (2): 437- 457

[1]	Li Jiaofen, Kong Lvyuan, Song Jiashuo, Wen Yaqiong. A Riemannian Optimization Approach for a Class of Matrix Trace Function Extremum Problem in Feature Extraction [J]. Acta mathematica scientia,Series A, 2024, 44(4): 1012-1036.
[2]	ZHU Li, XIA Fu-Quan . Levitin-Polyak Well-posedness of Generalized Mixed Variational Inequalities [J]. Acta mathematica scientia,Series A, 2012, 32(4): 633-643.
[3]	YUAN Shi-Fang, LIAO An-Ping, LEI Yuan. Minimization Problem for Hermitian Matrices over the Quaternion Field [J]. Acta mathematica scientia,Series A, 2009, 29(5): 1298-1306.
[4]	Peng Jianwen;Zhu Daoli. Strictly B -Preinvex Functions [J]. Acta mathematica scientia,Series A, 2006, 26(2): 200-206.

An Effective Algorithm for Generalized Sylvester Equation Minimization Problem Under Columnwise Orthogonal Constraints

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 4

Metrics

Comments

Recommended 0