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

A multigrid method for the maximal correlation problem

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).
Citation: Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control and 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.

[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.

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[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.

[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

[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

[15]

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

[16]

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

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[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.

[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

[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

[15]

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

[16]

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

[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.

[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.

[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.

[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.

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