Projected nonmonotone search methods for optimization with orthogonality constraints

• Absil PA, Malick J (2012) Projection-like retractions on matrix manifolds. SIAM J Optim 22(1):135–158

  Article  MathSciNet  MATH  Google Scholar 

• Absil PA, Mahony R, Sepulchre R (2007) Optimization algorithms on matrix manifolds. Princeton University Press, Princeton NJ, USA. ISBN:0691132984; 9780691132983

• Barzilai J, Borwein JM (1988) Two-point step size gradient methods. IMA J Numer Anal 8(1):141–148

  Article  MathSciNet  MATH  Google Scholar 

• Boumal N, Mishra B, Absil PA, Sepulchre R (2014) Manopt, a Matlab toolbox for optimization on manifolds. J Mach Learn Res 15(1):1455–1459

  MATH  Google Scholar 

• Dai YH, Fletcher R (2005) Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numerische Mathematik 100(1):21–47

  Article  MathSciNet  MATH  Google Scholar 

• d’Aspremont A, El Ghaoui L, Jordan MI, Lanckriet GR (2007) A direct formulation for sparse PCA using semidefinite programming. SIAM Rev 49(3):434–448

  Article  MathSciNet  MATH  Google Scholar 

• Edelman A, Arias TA, Smith ST (1998) The geometry of algorithms with orthogonality constraints. SIAM J Matrix Anal Appl 20(2):303–353

  Article  MathSciNet  MATH  Google Scholar 

• Eldén L, Park H (1999) A Procrustes problem on the Stiefel manifold. Numerische Mathematik 82(4):599–619

  Article  MathSciNet  MATH  Google Scholar 

• Francisco J, Martini T (2014) Spectral projected gradient method for the procrustes problem. TEMA (São Carlos) 15(1):83–96

  Article  MathSciNet  Google Scholar 

• Francisco J, Bazán FV, Mendonça MW (2017) Non-monotone algorithm for minimization on arbitrary domains with applications to large-scale orthogonal procrustes problem. Appl Numer Math 112:51–64

  Article  MathSciNet  MATH  Google Scholar 

• Gallivan KA, Absil P (2010) Note on the convex hull of the Stiefel manifold. Technical note

• Golub GH, Van Loan CF (1996) Matrix computations. 3rd edn. Johns Hopkins University Press, Baltimore, MD, USA. ISBN:0801854148

• Grubisi I, Pietersz R (2007) Efficient rank reduction of correlation matrices. Linear Algebra Appl 422(2):629–653

  Article  MathSciNet  MATH  Google Scholar 

• Hiriart-Urruty JB (1982) At what points is the projection mapping differentiable? Am Math Mon 89(7):456–458

• Holmes RB (1973) Smoothness of certain metric projections on Hilbert space. Trans Am Math Soc 184:87–100

  Article  MathSciNet  Google Scholar 

• Joho M, Mathis H (2002) 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, pp 273–277

• Journée M, Nesterov Y, Richtárik P, Sepulchre R (2010) Generalized power method for sparse principal component analysis. J Mach Learn Res 11:517–553

  MathSciNet  MATH  Google Scholar 

• Kokiopoulou E, Chen J, Saad Y (2011) Trace optimization and eigenproblems in dimension reduction methods. Numer Linear Algebra Appl 18(3):565–602

  Article  MathSciNet  MATH  Google Scholar 

• Koonin SE, Meredith DC (1990) Computational physics: Fortran version. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA. ISBN:0201127792; 9780201127799

• Liu Y-F, Dai Y-H, Luo ZQ (2011) On the complexity of leakage interference minimization for interference alignment. In: 2011 IEEE 12th international workshop on signal processing advances in wireless communications, pp 471–475. doi:10.1109/SPAWC.2011.5990455

• Manton JH (2002) Optimization algorithms exploiting unitary constraints. IEEE Trans Signal Process 50(3):635–650

  Article  MathSciNet  MATH  Google Scholar 

• Martin RM (2004) Electronic structure: basic theory and practical methods. Cambridge University Press, Cambridge, UK. ISBN:0521782856

• Pietersz R 4, Groenen PJ (2004) Rank reduction of correlation matrices by majorization. Quant Finance 4(6):649–662

• Raydan M (1997) The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J Optim 7(1):26–33

  Article  MathSciNet  MATH  Google Scholar 

• Rebonato R, Jäckel P (1999) The most general methodology to create a valid correlation matrix for risk management and option pricing purposes. J Risk 2(2):17–27

• Saad Y (1992) Numerical methods for large eigenvalue problems. Manchester University Press, Manchester, UK

• Sato H (2014) Riemannian Newton’s method for joint diagonalization on the Stiefel manifold with application to ICA. arXiv:1403.8064 (preprint)

• Schönemann PH (1966) A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1):1–10

• Szabo A, Ostlund NS (2012) Modern quantum chemistry: introduction to advanced electronic structure theory. In: Dover books on chemistry. Dover Publications, Mineola, New York

• Theis FJ, Cason TP, Absil P-A (2009) Soft dimension reduction for ICA by joint diagonalization on the Stiefel manifold. In: Adali T, Jutten C, Romano JMT, Barros AK (eds) Independent Component Analysis and Signal Separation: 8th International Conference, ICA 2009, Paraty, Brazil, March 15–18, 2009. Proceedings, Springer, Berlin, Heidelberg, pp 354–361

• Wen Z, Yin W (2013) A feasible method for optimization with orthogonality constraints. Math Program 142(1–2):397–434

  Article  MathSciNet  MATH  Google Scholar 

• Yang C, Meza JC, Wang LW (2006) A constrained optimization algorithm for total energy minimization in electronic structure calculations. J Comput Phys 217(2):709–721

  Article  MathSciNet  MATH  Google Scholar 

• Yang C, Meza JC, Wang LW (2007) A trust region direct constrained minimization algorithm for the Kohn–Sham equation. SIAM J Sci Comput 29(5):1854–1875

  Article  MathSciNet  MATH  Google Scholar 

• Yang C, Meza JC, Lee B, Wang LW (2009) KSSOLVoa MATLAB toolbox for solving the Kohn–Sham equations. ACM Trans Math Softw (TOMS) 36(2):10

  Article  MATH  Google Scholar 

• Zhang H, Hager WW (2004) A nonmonotone line search technique and its application to unconstrained optimization. SIAM J Optim 14(4):1043–1056

  Article  MathSciNet  MATH  Google Scholar 

• Zhang Z, Du K (2006) Successive projection method for solving the unbalanced procrustes problem. Sci China Ser A 49(7):971–986

  Article  MathSciNet  MATH  Google Scholar 

• Zhao Z, Bai ZJ, Jin XQ (2015) A Riemannian Newton algorithm for nonlinear eigenvalue problems. SIAM J Matrix Anal Appl 36(2):752–774

  Article  MathSciNet  MATH  Google Scholar 

• Zou H, Hastie T, Tibshirani R (2006) Sparse principal component analysis. J Comput Graph Stat 15(2):265–286

  Article  MathSciNet  Google Scholar 

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