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Volumn 22, Issue 1, 2012, Pages 135-158

Projection-like retractions on matrix manifolds

Author keywords

Equality constrained optimization; Feasible optimization method; Fixed rank matrices; Matrix manifold; Projection; Retraction; Spectral manifold; Stiefel manifold

Indexed keywords

EQUALITY-CONSTRAINED OPTIMIZATION; OPTIMIZATION METHOD; PROJECTION; RETRACTION; SPECTRAL MANIFOLD; STIEFEL MANIFOLD;

EID: 84861639358     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/100802529     Document Type: Article
Times cited : (358)

References (36)
  • 3
    • 84861587966 scopus 로고    scopus 로고
    • Optimization on manifolds: Methods and applications
    • M. Diehl, F. Glineur, E. Jarlebring, and W. Michiels, eds., Springer-Verlag, Berlin
    • P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization on manifolds: Methods and applications, in Recent Trends in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring, and W. Michiels, eds., Springer-Verlag, Berlin, 2010, pp. 125-144.
    • (2010) Recent Trends in Optimization and its Applications in Engineering , pp. 125-144
    • Absil, P.-A.1    Mahony, R.2    Sepulchre, R.3
  • 4
    • 0345447647 scopus 로고    scopus 로고
    • Newton's method on riemannian manifolds and a geometric model for the human spine
    • R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens, and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal., 22 (2002), pp. 359-390.
    • (2002) IMA J. Numer. Anal. , vol.22 , pp. 359-390
    • Adler, R.L.1    Dedieu, J.-P.2    Margulies, J.Y.3    Martens, M.4    Shub, M.5
  • 7
    • 33750892341 scopus 로고    scopus 로고
    • Geometrical interpretation of proximal-type algorithms in structured nonsmooth optimization
    • A. Daniilidis, W. Hare, and J. Malick, Geometrical interpretation of proximal-type algorithms in structured nonsmooth optimization, Optimization, 55 (2006), pp. 481-503.
    • (2006) Optimization , vol.55 , pp. 481-503
    • Daniilidis, A.1    Hare, W.2    Malick, J.3
  • 9
    • 0000802374 scopus 로고
    • The approximation of one matrix by another of lower rank
    • C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), pp. 211-218.
    • (1936) Psychometrika , vol.1 , pp. 211-218
    • Eckart, C.1    Young, G.2
  • 10
    • 0032216898 scopus 로고    scopus 로고
    • The geometry of algorithms with orthogonality constraints
    • 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.
    • (1998) SIAM J. Matrix Anal. Appl. , vol.20 , pp. 303-353
    • Edelman, A.1    Arias, T.A.2    Smith, S.T.3
  • 11
    • 0020145445 scopus 로고
    • Minimizing a differentiable function over a differential manifold
    • D. Gabay, Minimizing a differentiable function over a differential manifold, J. Optim. Theory Appl., 37 (1982), pp. 177-219.
    • (1982) J. Optim. Theory Appl. , vol.37 , pp. 177-219
    • Gabay, D.1
  • 13
    • 0004235829 scopus 로고
    • Optimization and dynamical systems
    • Springer-Verlag London, London
    • U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Commun. Control Engrg. Ser., Springer-Verlag London, London, 1994.
    • (1994) Commun. Control Engrg. Ser.
    • Helmke, U.1    Moore, J.B.2
  • 14
    • 0004206422 scopus 로고
    • Category theory: An introduction
    • Allyn and Bacon, Boston, MA
    • H. Herrlich and G. E. Strecker, Category Theory: An Introduction, Ser. Advanced Math., Allyn and Bacon, Boston, MA, 1973.
    • (1973) Ser. Advanced Math.
    • Herrlich, H.1    Strecker, G.E.2
  • 16
    • 0038886281 scopus 로고
    • At what point is the projection mapping differentiable
    • J.-B. Hiriart-Urruty, At what point is the projection mapping differentiable, AmericanMath. Monthly, 1983, pp. 456-458.
    • (1983) AmericanMath. Monthly , pp. 456-458
    • Hiriart-Urruty, J.-B.1
  • 19
    • 84966219708 scopus 로고
    • Smoothness of certain metric projection on hilbert space
    • R. B. Holmes, Smoothness of certain metric projection on Hilbert space, Trans. Amer. Math. Soc., 184 (1973), pp. 87-100.
    • (1973) Trans. Amer. Math. Soc. , vol.184 , pp. 87-100
    • Holmes, R.B.1
  • 20
    • 0004151494 scopus 로고
    • Cambridge University Press, Cambridge, UK
    • R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1989.
    • (1989) Matrix Analysis
    • Horn, R.A.1    Johnson, C.R.2
  • 22
    • 0030304775 scopus 로고    scopus 로고
    • Convex analysis on the hermitian matrices
    • A. S. Lewis, Convex analysis on the Hermitian matrices, SIAM J. Optim., 6 (1996), pp. 164-177.
    • (1996) SIAM J. Optim. , vol.6 , pp. 164-177
    • Lewis, A.S.1
  • 23
    • 48549094732 scopus 로고    scopus 로고
    • Alternating projections on manifolds
    • A. Lewis and J. Malick, Alternating projections on manifolds, Math. Oper. Res., 33 (2007).
    • (2007) Math. Oper. Res. , vol.33
    • Lewis, A.1    Malick, J.2
  • 24
    • 33748333622 scopus 로고
    • The gradient projection method along geodesics
    • D. G. Luenberger, The gradient projection method along geodesics, Management Sci., 18 (1972), pp. 620-631.
    • (1972) Management Sci. , vol.18 , pp. 620-631
    • Luenberger, D.G.1
  • 26
    • 27244447941 scopus 로고    scopus 로고
    • Newton methods for convex minimization: Connection among u-lagrangian, riemannian newton and sqp methods
    • J. Malick and S. A. Miller, Newton methods for convex minimization: Connection among U-Lagrangian, Riemannian Newton and SQP methods, Math. Program., 104 (2005).
    • (2005) Math. Program. , vol.104
    • Malick, J.1    Miller, S.A.2
  • 27
    • 0036503069 scopus 로고    scopus 로고
    • Optimization algorithms exploiting unitary constraints
    • J. H. Manton, Optimization algorithms exploiting unitary constraints, IEEE Trans. Signal Process., 50 (2002), pp. 635-650.
    • (2002) IEEE Trans. Signal Process. , vol.50 , pp. 635-650
    • Manton, J.H.1
  • 28
    • 0033469859 scopus 로고    scopus 로고
    • The u-lagrangian of the maximum eigenvalue function
    • F. Oustry, The U-Lagrangian of the maximum eigenvalue function, SIAM J. Optim., 9 (1999), pp. 526-549.
    • (1999) SIAM J. Optim. , vol.9 , pp. 526-549
    • Oustry, F.1
  • 29
    • 0002928322 scopus 로고    scopus 로고
    • A second-order bundle method to minimize the maximum eigenvalue function
    • F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function, Math. Program., 89 (2000), pp. 1-34.
    • (2000) Math. Program. , vol.89 , pp. 1-34
    • Oustry, F.1
  • 30
    • 0009317622 scopus 로고
    • Some remarks on dynamical systems and numerical analysis
    • L. Lara-Carrero and J. Lewowicz, eds., Equinoccio, U. Simón Bolívar, Caracas, Venezuela
    • M. Shub, Some remarks on dynamical systems and numerical analysis, in Proceedings of the 7th ELAM, L. Lara-Carrero and J. Lewowicz, eds., Equinoccio, U. Simón Bolívar, Caracas, Venezuela, 1986, pp. 69-92.
    • (1986) Proceedings of the 7th ELAM , pp. 69-92
    • Shub, M.1
  • 32
    • 0000188055 scopus 로고
    • Optimization techniques on riemannian manifolds
    • A. Bloch, ed., Fields Inst. Commun. American Mathematical Society, Providence, RI
    • S. T. Smith, Optimization techniques on Riemannian manifolds, in Hamiltonian and Gradient Flows, Algorithms and Control, A. Bloch, ed., Fields Inst. Commun. 3, American Mathematical Society, Providence, RI, 1994, pp. 113-136.
    • (1994) Hamiltonian and Gradient Flows, Algorithms and Control , vol.3 , pp. 113-136
    • Smith, S.T.1
  • 33
    • 0003274450 scopus 로고
    • Convex functions and optimization methods on riemannian manifolds
    • Kluwer Academic, Dordrecht, The Netherlands
    • C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Math. Appl. 297, Kluwer Academic, Dordrecht, The Netherlands, 1994.
    • (1994) Math. Appl. , vol.297
    • Udrişte, C.1
  • 36
    • 0004145937 scopus 로고
    • Oxford Science Publications, New York
    • R. Webster, Convexity, Oxford Science Publications, New York, 1994.
    • (1994) Convexity
    • Webster, R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.