American Institute of Mathematical Sciences

Advanced
2012, 2(4): 785-796. doi: 10.3934/naco.2012.2.785

A multigrid method for the maximal correlation problem

Xin-Guo Liu 1, and  Kun Wang 1,

1. 

School of Mathematical Science, Ocean University of China, Qiaodao 266100, China, China

Received  December 2011 Revised  September 2012 Published  November 2012

In this note, the continuity results of weak vector solutions and global vector solutions to a parametric generalized Ky Fan inequality are established by using a new scalarization method. Our results improve the corresponding ones of Li and Fang (J. Optim. Theory Appl. 147: 507-515, 2010).
Keywords: alternating projection method, starting point., correlation, Multivariate statistics, maximal correlation problem, multivariate eigenvalue problem, Horst method, global solution, Gauss-Seidel method.
Mathematics Subject Classification: Primary: 62H20; Secondary: 65F1.
Citation: Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 785-796. doi: 10.3934/naco.2012.2.785
References:
[1]

M. T. Chu and J. L. Watterson, On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method, SIAM J. Sci. Comput., 14 (1993), 1089-1106. doi: 10.1137/0914066.  Google Scholar

[2]

M. Y. Fu, Z. Q. Luo and Y. Y. Ye, Approximation algorithms for quadratic programming, J. Comb. Optim., 2 (1998), 29-50. doi: 10.1023/A:1009739827008.  Google Scholar

[3]

G. H. Golub and C. F. Van Loan, "Matrix Computations," Third Edition, The Johns Hopkins University Press, Baltimore, 1996.  Google Scholar

[4]

S. M. Grzegórski, On the convergence of the method of alternating projections for multivariate symmetric eigenvalue problem, Numan 2010, Conference in Numerical Analysis, Chania, Greece, Sept 15-18, 2010. Google Scholar

[5]

W. Hackbusch, "Multi-grid Method and Applications," Springer-Verlag, New York, 1985. Google Scholar

[6]

M. Hanafi and J. M. F. Ten Berge, Global optimality of the successive Maxbet algorithm, Psychometrika, 68 (2003), 97-103. doi: 10.1007/BF02296655.  Google Scholar

[7]

P. Horst, Relations among m sets of measures, Psychometrika, 26 (1961), 129-149. doi: 10.1007/BF02289710.  Google Scholar

[8]

D.-K. Hu, The convergence property of a algorithm about generalized eigenvalue and eigenvector of positive definite matrix, China-Japan Symposium on Statistics, 1984, Beijing, China. Google Scholar

[9]

J. R. Kettenring, Canonical analysis of several sets of variables, Biometrika, 58 (1971), 433-451. doi: 10.1093/biomet/58.3.433.  Google Scholar

[10]

Z.-Y. Liu, J. Qian and S.-F. Xu, On the double eigenvalue problem,, preprint. Available online: , ().   Google Scholar

[11]

J.-G. Sun, An algorithm for the solution of multiparameter eigenvalue problem, Math. Numer. Sinica(Chinese), 8 (1986), 137-149.  Google Scholar

[12]

J. M. F. Ten Berge, Generalized approaches to the MAXBET problem and the MAXDIFF problem, with applications to canonical correlations, Psychometrika, 53 (1988), 487-494. doi: 10.1007/BF02294402.  Google Scholar

[13]

T. L. Van Noorden and J. Barkmeijer, The multivariate eigenvalue problem: A new application, theory and a subspace accelerated power method, Universiteit Utrecht, preprint, 2008. Available online: http://www.math.uu.nl/publications/preprints/1308.ps Google Scholar

[14]

L.-H. Xu, "Numerical Methods for the Multivariate Eigenvalue Problem," M.S. Thesis, Department of Mathematics, Ocean University of China, 2008, ( in Chinese). Available from: http://cdmd.cnki.com.cn/Article/CDMD-10423-2008175406.htm Google Scholar

[15]

S.-F. Xu, "Matrix Computations: Theory and Methods," Peking University Press, Bejing, 1995 (Chinese). Google Scholar

[16]

Y. Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program., 84 (1999), 219-226. doi: 10.1007/s10107980012a.  Google Scholar

[17]

L.-H. Zhang and M. T. Chu, Computing absolute maximum correlation, IMA J. Numer. Anal., 32 (2012), 163-184. doi: 10.1093/imanum/drq029.  Google Scholar

[18]

L.-H. Zhang and L.-Z. Liao, An alternating variable method for the maximal correlation problem, J. Global Optim., 54 (2012), 199-218. doi: 10.1007/s10898-011-9758-2.  Google Scholar

[19]

L.-H. Zhang, L.-Z. Liao and L.-M. Sun, Towards the global solution of the maximal correlation problem, J. Global Optim., 49 (2011), 91-107. doi: 10.1007/s10898-010-9536-6.  Google Scholar

[20]

L. Zhang, Y. Xu and Z. Jin, An efficient algorithm for convex quadratic semi-definite optimization, Numer. Algebra Control Optim., 2 (2012), 129-144. doi: 10.3934/naco.2012.2.129.  Google Scholar

show all references

References:
[1]

M. T. Chu and J. L. Watterson, On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method, SIAM J. Sci. Comput., 14 (1993), 1089-1106. doi: 10.1137/0914066.  Google Scholar

[2]

M. Y. Fu, Z. Q. Luo and Y. Y. Ye, Approximation algorithms for quadratic programming, J. Comb. Optim., 2 (1998), 29-50. doi: 10.1023/A:1009739827008.  Google Scholar

[3]

G. H. Golub and C. F. Van Loan, "Matrix Computations," Third Edition, The Johns Hopkins University Press, Baltimore, 1996.  Google Scholar

[4]

S. M. Grzegórski, On the convergence of the method of alternating projections for multivariate symmetric eigenvalue problem, Numan 2010, Conference in Numerical Analysis, Chania, Greece, Sept 15-18, 2010. Google Scholar

[5]

W. Hackbusch, "Multi-grid Method and Applications," Springer-Verlag, New York, 1985. Google Scholar

[6]

M. Hanafi and J. M. F. Ten Berge, Global optimality of the successive Maxbet algorithm, Psychometrika, 68 (2003), 97-103. doi: 10.1007/BF02296655.  Google Scholar

[7]

P. Horst, Relations among m sets of measures, Psychometrika, 26 (1961), 129-149. doi: 10.1007/BF02289710.  Google Scholar

[8]

D.-K. Hu, The convergence property of a algorithm about generalized eigenvalue and eigenvector of positive definite matrix, China-Japan Symposium on Statistics, 1984, Beijing, China. Google Scholar

[9]

J. R. Kettenring, Canonical analysis of several sets of variables, Biometrika, 58 (1971), 433-451. doi: 10.1093/biomet/58.3.433.  Google Scholar

[10]

Z.-Y. Liu, J. Qian and S.-F. Xu, On the double eigenvalue problem,, preprint. Available online: , ().   Google Scholar

[11]

J.-G. Sun, An algorithm for the solution of multiparameter eigenvalue problem, Math. Numer. Sinica(Chinese), 8 (1986), 137-149.  Google Scholar

[12]

J. M. F. Ten Berge, Generalized approaches to the MAXBET problem and the MAXDIFF problem, with applications to canonical correlations, Psychometrika, 53 (1988), 487-494. doi: 10.1007/BF02294402.  Google Scholar

[13]

T. L. Van Noorden and J. Barkmeijer, The multivariate eigenvalue problem: A new application, theory and a subspace accelerated power method, Universiteit Utrecht, preprint, 2008. Available online: http://www.math.uu.nl/publications/preprints/1308.ps Google Scholar

[14]

L.-H. Xu, "Numerical Methods for the Multivariate Eigenvalue Problem," M.S. Thesis, Department of Mathematics, Ocean University of China, 2008, ( in Chinese). Available from: http://cdmd.cnki.com.cn/Article/CDMD-10423-2008175406.htm Google Scholar

[15]

S.-F. Xu, "Matrix Computations: Theory and Methods," Peking University Press, Bejing, 1995 (Chinese). Google Scholar

[16]

Y. Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program., 84 (1999), 219-226. doi: 10.1007/s10107980012a.  Google Scholar

[17]

L.-H. Zhang and M. T. Chu, Computing absolute maximum correlation, IMA J. Numer. Anal., 32 (2012), 163-184. doi: 10.1093/imanum/drq029.  Google Scholar

[18]

L.-H. Zhang and L.-Z. Liao, An alternating variable method for the maximal correlation problem, J. Global Optim., 54 (2012), 199-218. doi: 10.1007/s10898-011-9758-2.  Google Scholar

[19]

L.-H. Zhang, L.-Z. Liao and L.-M. Sun, Towards the global solution of the maximal correlation problem, J. Global Optim., 49 (2011), 91-107. doi: 10.1007/s10898-010-9536-6.  Google Scholar

[20]

L. Zhang, Y. Xu and Z. Jin, An efficient algorithm for convex quadratic semi-definite optimization, Numer. Algebra Control Optim., 2 (2012), 129-144. doi: 10.3934/naco.2012.2.129.  Google Scholar

[1]

Xin Yang, Nan Wang, Lingling Xu. A parallel Gauss-Seidel method for convex problems with separable structure. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 557-570. doi: 10.3934/naco.2020051
[2]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117
[3]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189
[4]

Jinkui Liu, Shengjie Li. Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2017, 13 (1) : 283-295. doi: 10.3934/jimo.2016017
[5]

Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure & Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359
[6]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
[7]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135
[8]

Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029
[9]

Zhongyi Huang. Tailored finite point method for the interface problem. Networks & Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91
[10]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042
[11]

Jia Cai, Junyi Huo. Sparse generalized canonical correlation analysis via linearized Bregman method. Communications on Pure & Applied Analysis, 2020, 19 (8) : 3933-3945. doi: 10.3934/cpaa.2020173
[12]

Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 529-537. doi: 10.3934/naco.2011.1.529
[13]

Xing Li, Chungen Shen, Lei-Hong Zhang. A projected preconditioned conjugate gradient method for the linear response eigenvalue problem. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 389-412. doi: 10.3934/naco.2018025
[14]

Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034
[15]

Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317
[16]

Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487
[17]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial & Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255
[18]

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 & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057
[19]

Wen-ling Zhao, Dao-jin Song. A global error bound via the SQP method for constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 775-781. doi: 10.3934/jimo.2007.3.775
[20]

Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043

 Impact Factor: 

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences