• 论文 •

### 一类线性约束矩阵不等式及其最小二乘问题

1. 湖南大学数学与计量经济学院, 长沙 410082
• 收稿日期:2015-07-15 出版日期:2016-04-15 发布日期:2016-05-13
• 基金资助:

国家自然科学基金(11201136)资助项目.

Zhou Xi, Lei Yuan, Qiao Wenlong. A CLASS OF LINEAR CONSTRAINED MATRIX INEQUALITY AND ITS LEAST SQUARES PROBLEM[J]. Mathematica Numerica Sinica, 2016, 38(2): 171-186.

### A CLASS OF LINEAR CONSTRAINED MATRIX INEQUALITY AND ITS LEAST SQUARES PROBLEM

Zhou Xi, Lei Yuan, Qiao Wenlong

1. College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
• Received:2015-07-15 Online:2016-04-15 Published:2016-05-13

A class of linear constrained matrix inequality and its least squares problem, which is equivalent to the corresponding matrix inequality smallest nonnegative deviation problem, is considered in this paper. Some related literatures have proposed an iteration method to solve the smallest nonnegative deviation problem, however, a great deal of computation for this algorithm is required for large scale problems because a constrained least squares subproblem should be solved exactly at each iteration. Based on the existing algorithm, a modified iteration method is proposed to improve the computational efficiency. In this iteration process, the whole required computation has been reduced by implementing the matrix form LSQR method to solve a constrained least squares subproblem over a Krylov subspace of low dimension in finite steps. Furthermore, the convergence of the modified algorithm is analyzed by using the projection theorem and related matrix analysis methods. Finally, several numerical experiments are presented to verify the theoretical results and the effectiveness of the iteration method.

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 [1] Jbilou K, Messaoudi A, Sadok H. Global FOM and GMRES algorithms for matrix equations[J]. Applied Numerical Mathematics, 1999, 31(1):49-63.[2] Friswell M, Mottershead J E. Finite element model updating in structural dynamics[M]. Springer Science & Business Media, 1995.[3] Zhang J H. Modeling of dynamical systems[M]. Beijing:National Defence and Industry Press, 2000.[4] Chen T W, Francis B, Hagiwara T. Optimal sampled-data control systems[J]. Proceedings of the IEEE, 1998, 86(4):741-741.[5] Penrose R. A generalized inverse for matrices[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1955, 51(3):406-413.[6] Dai H, Lancaster P. Linear matrix equations from an inverse problem of vibration theory[J]. Linear Algebra and Its Applications, 1996, 246:31-47.[7] 廖安平, 白中治. 矩阵方程ATXA=D的双对称最小二乘解[J]. 计算数学, 2002, 24(1):9-20.[8] 王明辉, 魏木生, 姜同松. 子矩阵约束下矩阵方程AXB=E的极小范数最小二乘对称解[J]. 计算数学, 2007, 29(2):147-154.[9] 袁仕芳, 廖安平, 雷渊. 矩阵方程AXB+CYD=E的对称极小范数最小二乘解[J]. 计算数学, 2007, 29(2):203-216.[10] 周海林. 矩阵方程AXB+CYD=E的M对称解的迭代算法[J]. 计算数学, 2015, 37(2):186-198.[11] 李姣芬, 彭振赟, 彭靖静. 矩阵不等式约束下矩阵方程AX=B的双对称解[J]. 计算数学, 2013, 37(2):137-150.[12] Andersson L E, Elfving T. A constrained procrustes problem[J]. SIAM Journal on Matrix Analysis and Applications, 1997, 18(1):124-139.[13] Flanders H, Wimmer H K. On the matrix equations AX-XB=C and AX-YB=C[J]. SIAM Journal on Applied Mathematics, 1977, 32(4):707-710.[14] Xu G P, Wei M S, Zheng D S. On solutions of matrix equation AXB+CYD=F[J]. Linear Algebra and Its Applications, 1998, 279(1):93-109.[15] Paige C C, Saunders M A. Towards a generalized singular value decomposition[J]. SIAM Journal on Numerical Analysis, 1981, 18(3):398-405.[16] Peng Z Y, Hu X Y. The reflexive and anti-reflexive solutions of the matrix equation AX=B[J]. Linear Algebra and Its Applications, 2003, 375:147-155.[17] Bouyouli R, Jbilou K, Sadaka R, Sadok H. Convergence properties of some block Krylov subspace methods for multiple linear systems[J]. Journal of Computational and Appled Mathematies, 2006, 196(2):498-511.[18] Darnell D, Morgan R B, Wileox W. Deflated GMRES for systems with multiple shifts and multiple right-hand sides[J]. Linear Algebra and its Applieations, 2008, 429(10):2415-2434.[19] Salkuyeh D K. CG-type algorithms to solve symmetric matrix equations[J]. Applied Mathematics and Computation, 2006, 172(2):985-999.[20] Guo K H, Hu X Y, Zhang L. A new iteration method for the matrix equation AX=B[J]. Applied Mathematics and Computation, 2007, 187(2):1434-1441.[21] Li F L, Gong L S, Hu X Y, Zhang L. Successive projection iterative method for solving matrix AX=B[J]. Journal of Computational and Appled Mathematic, 2010, 234(8):2405-2410.[22] Herman G T. A relaxation method for reconstructing objects from noisy X-rays[J]. Mathematical Programming, 1975, 8(1):1-19.[23] Herman G T. Image Reconstruction from Projections:The Fundamentals of Computerized Tomog-raphy[M]. Academic Press, New York, 1982.[24] Bramley R, Winnicka B. Solving linear inequalities in a least squares sense[J]. SIAM Journal on Scientific Computing, 1996, 17(1):275-286.[25] Pinar M C. Newton's method for linear inequality systems[J]. European Journal of Operational Reserach, 1998, 107(3):710-719.[26] 王日爽. 泛函分析与最优化理论[M]. 北京:北京航天航空大学出版社, 2003.[27] Moreau J J. Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[J]. Comptes Rendus de l'Académie des sciences, Paris, 1962, 255:238-240.[28] Wierzbicki A P, Kurcyusz S. Projection on a cone, penalty functionals and duality theory for problems with inequaltity constraints in Hilbert space[J]. SIAM Journal on Control and Optimization, 1977, 15(1):25-56.
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