IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v102y1999i2d10.1023_a1021768121263.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1021768121263
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1023/A:1021768121263?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    4. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jean-Philippe Vial, 1997. "A path-following version of the Todd-Burrell procedure for linear programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 153-167, June.
    2. Gondzio, J. & Sarkissian, R. & Vial, J.-P., 1997. "Using an interior point method for the master problem in a decomposition approach," European Journal of Operational Research, Elsevier, vol. 101(3), pages 577-587, September.
    3. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    4. P. Chardaire & A. Lisser, 2002. "Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems," Operations Research, INFORMS, vol. 50(2), pages 260-276, April.
    5. A. Ouorou & P. Mahey & J.-Ph. Vial, 2000. "A Survey of Algorithms for Convex Multicommodity Flow Problems," Management Science, INFORMS, vol. 46(1), pages 126-147, January.
    6. Alexandre Belloni, 2008. "Norm-Induced Densities and Testing the Boundedness of a Convex Set," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 235-256, February.
    7. I. I. Dikin & C. Roos, 1997. "Convergence of the Dual Variables for the Primal Affine Scaling Method with Unit Steps in the Homogeneous Case," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 305-321, November.
    8. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    9. Li-Zhi Liao, 2014. "A Study of the Dual Affine Scaling Continuous Trajectories for Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 548-568, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:102:y:1999:i:2:d:10.1023_a:1021768121263. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.