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

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