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Central Paths in Semidefinite Programming, Generalized Proximal-Point Method and Cauchy Trajectories in Riemannian Manifolds

Author

Listed:
  • J. X. Cruz Neto

    (Universidade Federal do Piauí)

  • O. P. Ferreira

    (Universidade Federal de Goiás)

  • P. R. Oliveira

    (Universidade Federal do Rio de Janeiro)

  • R. C. M. Silva

    (Universidade Federal de Amazonas)

Abstract

The relationships among the central path in the context of semidefinite programming, generalized proximal-point method and Cauchy trajectory in a Riemannian manifolds is studied in this paper. First, it is proved that the central path associated to a general function is well defined. The convergence and characterization of its limit point is established for functions satisfying a certain continuity property. Also, the generalized proximal-point method is considered and it is proved that the correspondingly generated sequence is contained in the central path. As a consequence, both converge to the same point. Finally, it is proved that the central path coincides with the Cauchy trajectory in a Riemannian manifold.

Suggested Citation

  • J. X. Cruz Neto & O. P. Ferreira & P. R. Oliveira & R. C. M. Silva, 2008. "Central Paths in Semidefinite Programming, Generalized Proximal-Point Method and Cauchy Trajectories in Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 227-242, November.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:2:d:10.1007_s10957-008-9422-2
    DOI: 10.1007/s10957-008-9422-2
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    References listed on IDEAS

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    1. Halicka, M. & de Klerk, E. & Roos, C., 2005. "Limiting behavior of the central path in semidefinite optimization," Other publications TiSEM 82985463-0467-4c61-8be1-1, Tilburg University, School of Economics and Management.
    2. NESTEROV , Yu. & TODD, Mike, 2002. "On the Riemannian geometry defined by self-concordant barriers and interior-point methods," LIDAM Reprints CORE 1595, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. A. N. Iusem, 1998. "On Some Properties of Generalized Proximal Point Methods for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 337-362, February.
    4. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
    5. Halická, M. & de Klerk, E. & Roos, C., 2002. "On the convergence of the central path in semidefinite optimization," Other publications TiSEM 9ca12b89-1208-46aa-8d70-4, Tilburg University, School of Economics and Management.
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    Cited by:

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