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On Well Definedness of the Central Path

Author

Listed:
  • L. M. Graña Drummond

    (COPPE-UFRJ)

  • B. F. Svaiter

    (Jardim Botânico)

Abstract

We study the well definedness of the central path for a linearly constrained convex programming problem with smooth objective function. We prove that, under standard assumptions, existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given. We show that, under an additional assumption on the objective function, the central path converges to the analytic center of the optimal set.

Suggested Citation

  • L. M. Graña Drummond & B. F. Svaiter, 1999. "On Well Definedness of the Central Path," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 223-237, August.
    • Handle: RePEc:spr:joptap:v:102:y:1999:i:2:d:10.1023_a:1021768121263
    DOI: 10.1023/A:1021768121263
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    References listed on IDEAS

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    1. de GHELLINCK, Guy & VIAL, Jean-Philippe, 1986. "A polynomial Newton method for linear programming," LIDAM Reprints CORE 724, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Nimrod Megiddo & Michael Shub, 1989. "Boundary Behavior of Interior Point Algorithms in Linear Programming," Mathematics of Operations Research, INFORMS, vol. 14(1), pages 97-146, February.
    3. de GHELLI NCK, G. & VIAL, J.-Ph., 1986. "A polynomial Newton method for linear programming," LIDAM Discussion Papers CORE 1986014, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. M. Paul Laiu & André L. Tits, 2019. "A constraint-reduced MPC algorithm for convex quadratic programming, with a modified active set identification scheme," Computational Optimization and Applications, Springer, vol. 72(3), pages 727-768, April.
    2. María J. Cánovas & Marco A. López & Juan Parra & F. Javier Toledo, 2006. "Lipschitz Continuity of the Optimal Value via Bounds on the Optimal Set in Linear Semi-Infinite Optimization," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 478-489, August.
    3. Pedro Borges & Claudia Sagastizábal & Mikhail Solodov, 2021. "Decomposition Algorithms for Some Deterministic and Two-Stage Stochastic Single-Leader Multi-Follower Games," Computational Optimization and Applications, Springer, vol. 78(3), pages 675-704, April.
    4. Roger Behling & Clovis Gonzaga & Gabriel Haeser, 2014. "Primal-Dual Relationship Between Levenberg–Marquardt and Central Trajectories for Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 705-717, September.

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