A Riemannian Dennis-Moré Condition


In this paper, we generalize from Euclidean spaces to Riemannian manifolds an important result in optimization that guarantees Riemannian quasi-Newton algorithms converge superlinearly.



This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.

This work was performed in part while the first author was a Visiting Professor at the Institut de mathématiques pures et appliquées (MAPA) at Université catholique de Louvain.


  1. 1.
    Absil, P.A., Baker, C.G., Gallivan, K.A.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7(3), 303–330 (2007) doi:10.1007/s10208-005-0179-9 MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, New Jersey (2008) MATHGoogle Scholar
  3. 3.
    Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22(3), 359–390 (2002) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Baker, C.G.: Riemannian manifold trust-region methods with applications to eigenproblems. Ph.D. thesis, School of Computational Science, Florida State University (2008) Google Scholar
  5. 5.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Springer, New Jersey (1983) MATHGoogle Scholar
  6. 6.
    Dreisigmeyer, D.W.: Direct search algorithms over Riemannian manifolds (2006). Optimization Online 2007-08-1742 Google Scholar
  7. 7.
    Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constrains. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gabay, D.: Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl. 37(2), 177–219 (1982) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Helmke, U., Moore, J.: Optimization and Dynamical Systems. Springer, Berlin (1994) Google Scholar
  10. 10.
    Qi, C.: Numerical optimization on Riemannian manifolds. Ph.D. thesis, Florida State University, Tallahassee, FL, USA (2011) Google Scholar
  11. 11.
    Smith, S.T.: Optimization techniques on Riemannian manifolds. In: Bloch, A. (ed.) Hamiltonian and Gradient Flows, Algorithms and Control. Fields Inst. Commun., vol. 3, pp. 113–136. Amer. Math. Soc., Providence (1994) Google Scholar
  12. 12.
    Yang, Y.: Globally convergent optimization algorithms on Riemannian manifolds: Uniform framework for unconstrained and constrained optimization. J. Optim. Theory Appl. 132(2), 245–265 (2007). doi:10.1007/s10957-006-9081-0 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of Mathematical Engineering, ICTEAM InstituteUniversité catholique de LouvainLouvain-la-NeuveBelgium

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