A multigrid method for the maximal correlation problem

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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

Abstract
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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. 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. doi: 10.1023/A:1009739827008.
[3]	G. H. Golub and C. F. Van Loan, "Matrix Computations,", Third Edition, (1996).
[4]	S. M. Grzegórski, On the convergence of the method of alternating projections for multivariate symmetric eigenvalue problem,, Numan 2010, (2010), 15.
[5]	W. Hackbusch, "Multi-grid Method and Applications,", Springer-Verlag, (1985).
[6]	M. Hanafi and J. M. F. Ten Berge, Global optimality of the successive Maxbet algorithm,, Psychometrika, 68 (2003), 97. doi: 10.1007/BF02296655.
[7]	P. Horst, Relations among m sets of measures,, Psychometrika, 26 (1961), 129. 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).
[9]	J. R. Kettenring, Canonical analysis of several sets of variables,, Biometrika, 58 (1971), 433. 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.
[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. 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, (2008).
[14]	L.-H. Xu, "Numerical Methods for the Multivariate Eigenvalue Problem,", M.S. Thesis, (2008), 10423.
[15]	S.-F. Xu, "Matrix Computations: Theory and Methods,", Peking University Press, (1995).
[16]	Y. Y. Ye, Approximating quadratic programming with bound and quadratic constraints,, Math. Program., 84 (1999), 219. doi: 10.1007/s10107980012a.
[17]	L.-H. Zhang and M. T. Chu, Computing absolute maximum correlation,, IMA J. Numer. Anal., 32 (2012), 163. 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. 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. 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. 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. 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. doi: 10.1023/A:1009739827008.
[3]	G. H. Golub and C. F. Van Loan, "Matrix Computations,", Third Edition, (1996).
[4]	S. M. Grzegórski, On the convergence of the method of alternating projections for multivariate symmetric eigenvalue problem,, Numan 2010, (2010), 15.
[5]	W. Hackbusch, "Multi-grid Method and Applications,", Springer-Verlag, (1985).
[6]	M. Hanafi and J. M. F. Ten Berge, Global optimality of the successive Maxbet algorithm,, Psychometrika, 68 (2003), 97. doi: 10.1007/BF02296655.
[7]	P. Horst, Relations among m sets of measures,, Psychometrika, 26 (1961), 129. 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).
[9]	J. R. Kettenring, Canonical analysis of several sets of variables,, Biometrika, 58 (1971), 433. 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.
[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. 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, (2008).
[14]	L.-H. Xu, "Numerical Methods for the Multivariate Eigenvalue Problem,", M.S. Thesis, (2008), 10423.
[15]	S.-F. Xu, "Matrix Computations: Theory and Methods,", Peking University Press, (1995).
[16]	Y. Y. Ye, Approximating quadratic programming with bound and quadratic constraints,, Math. Program., 84 (1999), 219. doi: 10.1007/s10107980012a.
[17]	L.-H. Zhang and M. T. Chu, Computing absolute maximum correlation,, IMA J. Numer. Anal., 32 (2012), 163. 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. 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. 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. doi: 10.3934/naco.2012.2.129.

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