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Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation

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Abstract

The necessary condition established in Part I of this paper for the global maximizers of the maximization problem

$\mathop {\max }\limits_V \left\{ {\frac{{tr(V^ \top AV)}} {{tr(V^ \top BV)}} + tr(V^ \top CV)} \right\} $

over the Stiefel manifold \(\{ V \in \mathbb{R}^{m \times \ell } |V^ \top V = I_\ell \} (\ell < m)\), naturally leads to a self-consistent-field (SCF) iteration for computing a maximizer. In this part, we analyze the global and local convergence of the SCF iteration, and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration. This is one of the advantages of the SCF iteration over optimization-based methods. Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.

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References

  1. Absil P A, Mahony R, Sepulchre R. Optimization Algorithms On Matrix Manifolds. Princeton, NJ: Princeton University Press, 2008

    Book  MATH  Google Scholar 

  2. Bai Z, Demmel J, Dongarra J, et al. eds. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Philadelphia: SIAM, 2000

    Google Scholar 

  3. Davis C, Kahan W. The rotation of eigenvectors by a perturbation, III. SIAM J Numer Anal, 1970, 7: 1–46

    Article  MATH  MathSciNet  Google Scholar 

  4. Demmel J. Applied Numerical Linear Algebra. Philadelphia: SIAM, 1997

    Book  MATH  Google Scholar 

  5. Edelman A, Arias T A, Smith S T. The geometry of algorithms with orthogonality constraints. SIAM J Matrix Anal Appl, 1999, 20: 03–353

    Article  MathSciNet  Google Scholar 

  6. Golub G H, Van Loan C F. Matrix Computations 3rd ed. Baltimore, Maryland: Johns Hopkins University Press, 1996

    MATH  Google Scholar 

  7. Horn R A, Johnson C R. Topics in Matrix Analysis. Cambridge: Cambridge University Press, 1991

    Book  MATH  Google Scholar 

  8. Knyazev A V. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J Sci Comput, 2001, 23: 517–541

    Article  MATH  MathSciNet  Google Scholar 

  9. Knyazev A V, Argentati M E. Rayleigh-Ritz majorization error bounds with applications to FEM. SIAM J Matrix Anal Appl, 2010, 31: 1521–1537

    Article  MATH  MathSciNet  Google Scholar 

  10. Liu X, Wang X, Wen Z, et al. On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory. ArXiv:1302.6022, 2013

    Google Scholar 

  11. Martin R M. Electronic Structure: Basic Theory and Practical Methods. Cambridge, UK: Cambridge University Press, 2004

    Book  Google Scholar 

  12. Milnor J W, Stasheff J D. Characteristic Classes. Princetou/Tokyo: Princeton University Press & University of Tokyo Press, 1974

    MATH  Google Scholar 

  13. Ngo T, Bellalij M, Saad Y. The trace ratio optimization problem for dimensionality reduction. SIAM J Matrix Anal Appl, 2010, 31: 2950–2971

    Article  MATH  MathSciNet  Google Scholar 

  14. Nocedal J, Wright S. Numerical Optimization, 2nd ed. New York: Springer, 2006

    MATH  Google Scholar 

  15. Parlett B N. The Symmetric Eigenvalue Problem. Philadelphia: SIAM, 1998

    Book  MATH  Google Scholar 

  16. Saad Y. Numerical Methods for Large Eigenvalue Problems. Manchester, UK: Manchester University Press, 1992

    MATH  Google Scholar 

  17. Saad Y, Chelikowsky J R, Shontz S M. Numerical methods for electronic structure calculations of materials. SIAM Rev, 2010, 52: 3–54

    Article  MATH  MathSciNet  Google Scholar 

  18. Saunders V R, Hillier I H. A “level-shifting” method for converging closed shell Hartree-Fock wave functions. Internat J Quantum Chem, 1973, 7: 699–705

    Article  Google Scholar 

  19. Sleijpen G L G, van der Vorst H A. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J Matrix Anal Appl, 1996, 17: 401–425

    Article  MATH  MathSciNet  Google Scholar 

  20. Stewart G W, Sun J G. Matrix Perturbation Theory. Boston: Academic Press, 1990

    MATH  Google Scholar 

  21. Szabo A, Ostlund N S. Modern Quantum Chemistry: An Introduction To Advanced Electronic Structure Theory. New York: Dover, 1996

    Google Scholar 

  22. Thøgersen L, Olsen J, Yeager D, et al. The trust-region self-consistent field method: Towards a black-box optimization in Hartree-Fock and Kohn-Sham theories. J Chem Phys, 2004, 121: 16–27

    Article  Google Scholar 

  23. Wang H, Yan S, Xu D, et al. Trace ratio vs. ratio trace for dimensionality reduction. In: IEEE Conference on Computer Vision and Pattern Recognition, 2007, CVPR’07, 1–8

    Google Scholar 

  24. Wedin P Å. On angles between subspaces. In: Kågström B, Ruhe A, eds. Matrix Pencils. New York: Springer, 1983, 263–285

    Google Scholar 

  25. Wen Z, Milzarek A, Ulbrich M, et al. Adaptive regularized self-consistent field iteration with exact Hessian for electronic structure calculation. SIAM J Sci Comput, 2013, 35: A1299–A1324

    Article  MATH  MathSciNet  Google Scholar 

  26. Wen Z, Yin W. A feasible method for optimization with orthogonality constraints. Math Programming, 2013, 142: 397–434

    Article  MATH  MathSciNet  Google Scholar 

  27. Yang C, Gao W, Meza J C. On the convergence of the self-consistent field iteration for a class of nonlinear eigenvalue problems. SIAM J Matrix Anal Appl, 2009, 30: 1773–1788

    Article  MATH  MathSciNet  Google Scholar 

  28. Yang C, Meza J C, Lee B, et al. KSSOLV — a MATLAB toolbox for solving the Kohn-Sham equations. ACM Trans Math Softw, 2009, 36: 1–35

    Article  MathSciNet  Google Scholar 

  29. Yang C, Meza J C, Wang L W. A trust region direct constrained minimization algorithm for the Kohn-Sham equation. SIAM J Sci Comput, 2007, 29: 1854–1875

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhang L H. On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere. Comput Opt Appl, 2013, 54: 111–139

    Article  MATH  Google Scholar 

  31. Zhang L H, Li R C. Maximization of the sum of the trace ratio on the Stiefel manifold. Technical Report 2013-04. Department of Mathematics, University of Texas at Arlington, May 2013, http://www.uta.edu/math/preprint/

    Google Scholar 

  32. Zhang L H, Li R C. Maximization of the sum of the trace ratio on the Stiefel manifold, I: Theory. Sci China Math, 2014, doi: 10.1007/s11425-014-4824-0, in press

    Google Scholar 

  33. Zhang L H, Liao L Z, Ng M K. Fast algorithms for the generalized Foley-Sammon discriminant analysis. SIAM J Matrix Anal Appl, 2010, 31: 1584–1605

    Article  MATH  MathSciNet  Google Scholar 

  34. Zhang L H, Liao L Z, Ng M K. Superlinear convergence of a general algorithm for the generalized Foley-Sammon discriminant analysis. J Optim Theory Appl, 2013, 157: 853–865

    Article  MATH  MathSciNet  Google Scholar 

  35. Zhang L H, Yang W, Liao L Z. A note on the trace quotient problem. J Optim Lett, 2013, doi: 10.1007/s11590-013-0680-z, in press

    Google Scholar 

  36. Zhang X, Zhu J, Wen Z, Zhou A. Gradient type optimization methods for electronic structure calculations. ArXiv:1308.2864, 2013

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    LeiHong Zhang

  2. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019-0408, USA

    RenCang Li

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  1. LeiHong Zhang
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  2. RenCang Li
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Correspondence to RenCang Li.

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Zhang, L., Li, R. Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation. Sci. China Math. 58, 1549–1566 (2015). https://doi.org/10.1007/s11425-014-4825-z

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