A non-monotone linear search algorithm with mixed direction on Stiefel manifold

References

• P.A. Absil and J. Malick, Projection-like retractions on matrix manifolds, SIAM J. Optim. 22 (2012), pp. 135–158. doi: 10.1137/100802529
   Web of Science ®Google Scholar
• P.A. Absil, R. Mahony, and R. Sepulchre, Optimization algorithms on matrix manifolds, Princeton University Press, Princeton, NJ, 2009.
   Google Scholar
• J. Barzilai and J.M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal. 8 (1988), pp. 141–148. doi: 10.1093/imanum/8.1.141
   Web of Science ®Google Scholar
• N. Boumal, B. Mishra, P.A. Absil, and R. Sepulchre, Manopt, a Matlab toolbox for optimization on manifolds, J. Mach. Learn. Res. 15 (2014), pp. 1455–1459.
   Web of Science ®Google Scholar
• O.S.D. Cedeño and H.F.O. Leon, Projected nonmonotone search methods for optimization with orthogonality constraints, Comput. Appl. Math. (2017), pp. 1–27. Available at https://doi.org/10.1007/s40314-017-0501-6.
   Google Scholar
• Y.H. Dai and R. Fletcher, Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming, Numer. Math. 100 (2005), pp. 21–47. doi: 10.1007/s00211-004-0569-y
   Web of Science ®Google Scholar
• A. d'Aspremont, L. El Ghaoui, M.I. Jordan, and G.R.G. Lanckriet, A direct formulation for sparse PCA using semidefinite programming, SIAM Rev. 49 (2007), pp. 434–448. doi: 10.1137/050645506
   Web of Science ®Google Scholar
• A. Edelman, T.A. Arias, and S.T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20 (1998), pp. 303–353. doi: 10.1137/S0895479895290954
   Web of Science ®Google Scholar
• L. Eldén and H. Park, A procrustes problem on the Stiefel manifold, Numer. Math. 82 (1999), pp. 599–619. doi: 10.1007/s002110050432
   Web of Science ®Google Scholar
• J. Francisco, F.V. Bazán, and M.W. Mendonça, Non-monotone algorithm for minimization on arbitrary domains with applications to large-scale orthogonal procrustes problem, Appl. Numer. Math. 112 (2017), pp. 51–64. doi: 10.1016/j.apnum.2016.09.018
   Web of Science ®Google Scholar
• J. De Bem Francisco and T. Martini, Spectral projected gradient method for the procrustes problem, TEMA (São Carlos) 15 (2014), pp. 83–96. doi: 10.5540/tema.2014.015.01.0083
   Google Scholar
• G.H. Golub and C.F. Van Loan, Matrix computations, 1996.
   Google Scholar
• I. Grubisi and R. Pietersz, Efficient rank reduction of correlation matrices, Linear Algebr. Appl. 422 (2007), pp. 629–653. doi: 10.1016/j.laa.2006.11.024
   Web of Science ®Google Scholar
• B. Iannazzo and M. Porcelli, The Riemannian Barzilai–Borwein method with nonmonotone line search and the matrix geometric mean computation, IMA J. Numer. Anal. (2017), p. drx015.
   Web of Science ®Google Scholar
• M. Joho and H. Mathis, Joint diagonalization of correlation matrices by using gradient methods with application to blind signal separation, in Sensor Array and Multichannel Signal Processing Workshop Proceedings, 2002, IEEE, Rosslyn, VA, 2002, pp. 273–277.
   Google Scholar
• M. Journée, Y. Nesterov, P. Richtárik, and R. Sepulchre, Generalized power method for sparse principal component analysis, J. Mach. Learn. Res. 11 (2010), pp. 517–553.
   Web of Science ®Google Scholar
• E. Kokiopoulou, J. Chen, and Y. Saad, Trace optimization and eigenproblems in dimension reduction methods, Numer. Linear Algebra Appl. 18 (2011), pp. 565–602. doi: 10.1002/nla.743
   Web of Science ®Google Scholar
• R. Lai and S. Osher, A splitting method for orthogonality constrained problems, J. Sci. Comput. 58 (2014), pp. 431–449. doi: 10.1007/s10915-013-9740-x
   Web of Science ®Google Scholar
• Y.F. Liu, Y.H. Dai, and Z.Q. Luo, On the complexity of leakage interference minimization for interference alignment, in 2011 IEEE 12th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), June, IEEE, 2011, pp. 471–475.
   Google Scholar
• J.H. Manton, Optimization algorithms exploiting unitary constraints, IEEE Trans. Signal Process. 50 (2002), pp. 635–650. doi: 10.1109/78.984753
   Web of Science ®Google Scholar
• R.M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, Cambridge, 2004.
   Google Scholar
• J. Nocedal and S.J. Wright, Numerical Optimization, 2nd ed., Springer, New York, 2006.
   Google Scholar
• R. Pietersz 4 and P.J.F. Groenen, Rank reduction of correlation matrices by majorization, Quant. Finan. 4 (2004), pp. 649–662. doi: 10.1080/14697680400016182
   Web of Science ®Google Scholar
• M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim. 7 (1997), pp. 26–33. doi: 10.1137/S1052623494266365
   Web of Science ®Google Scholar
• R. Rebonato and P. Jäckel, The most general methodology to create a valid correlation matrix for risk management and option pricing purposes, J. Risk 2 (1999), pp. 17–27. doi: 10.21314/JOR.2000.023
   Google Scholar
• Y. Saad, Numerical Methods for Large Eigenvalue Problems, Vol. 158, SIAM, Manchester, 1992.
   Google Scholar
• P.H. Schönemann, A generalized solution of the orthogonal procrustes problem, Psychometrika 31 (1966), pp. 1–10. doi: 10.1007/BF02289451
   Web of Science ®Google Scholar
• A. Szabo and N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, 1st ed., Dover Publications, Inc., Mineola, 1996.
   Google Scholar
• F.J. Theis, T.P. Cason, and P.A. Absil, Soft dimension reduction for ICA by joint diagonalization on the Stiefel manifold, in Independent Component Analysis and Signal Separation, T. Adali, C. Jutten, J.M.T. Romano and A.K. Barros, eds., Springer, Berlin, 2009, pp. 354–361.
   Google Scholar
• Z. Wen and W. Yin, A feasible method for optimization with orthogonality constraints, Math. Program. 142 (2013), pp. 397–434. doi: 10.1007/s10107-012-0584-1
   Web of Science ®Google Scholar
• C. Yang, J.C. Meza, and L.W. Wang, A constrained optimization algorithm for total energy minimization in electronic structure calculations, J. Comput. Phys. 217 (2006), pp. 709–721. doi: 10.1016/j.jcp.2006.01.030
   Web of Science ®Google Scholar
• C. Yang, J.C. Meza, and L.W. Wang, A trust region direct constrained minimization algorithm for the Kohn–Sham equation, SIAM J. Sci. Comput. 29 (2007), pp. 1854–1875. doi: 10.1137/060661442
   Web of Science ®Google Scholar
• C. Yang, J.C. Meza, B. Lee, and L.W. Wang, KSSOLVoa MATLAB toolbox for solving the Kohn–Sham equations, ACM Trans. Math. Softw. 36 (2009), pp. 1–34. doi: 10.1145/1499096.1499099
   Web of Science ®Google Scholar
• H. Zhang and W.W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), pp. 1043–1056. doi: 10.1137/S1052623403428208
   Web of Science ®Google Scholar
• Z. Zhao, Z.J. Bai, and X.Q. Jin, A Riemannian Newton algorithm for nonlinear eigenvalue problems, SIAM J. Matrix Anal. Appl. 36 (2015), pp. 752–774. doi: 10.1137/140967994
   Web of Science ®Google Scholar
• H. Zou, T. Hastie, and R. Tibshirani, Sparse principal component analysis, J. Comput. Graph. Stat. 15 (2006), pp. 265–286. doi: 10.1198/106186006X113430
   Web of Science ®Google Scholar