A multigrid method for the maximal correlation problem

Xin-Guo Liu; Kun Wang

doi:10.3934/naco.2012.2.785

Article Contents

2012, Volume 2, Issue 4: 785-796. Doi: 10.3934/naco.2012.2.785

This issue Previous Article A note on semicontinuity to a parametric generalized Ky Fan inequality Next Article On product-type generalized block AOR method for augmented linear systems

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

Received: December 2011

Revised: September 2012

Published: November 2012

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 62H20; Secondary: 65F10.

Citation:

Related Papers

Cited by

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: http://www.math.pku.edu.cn:8000/var/preprint/7229.pdf.
[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.