IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v58y2014i1p109-135.html
   My bibliography  Save this article

An entire space polynomial-time algorithm for linear programming

Author

Listed:
  • Da Tian

Abstract

We propose an entire space polynomial-time algorithm for linear programming. First, we give a class of penalty functions on entire space for linear programming by which the dual of a linear program of standard form can be converted into an unconstrained optimization problem. The relevant properties on the unconstrained optimization problem such as the duality, the boundedness of the solution and the path-following lemma, etc, are proved. Second, a self-concordant function on entire space which can be used as penalty for linear programming is constructed. For this specific function, more results are obtained. In particular, we show that, by taking a parameter large enough, the optimal solution for the unconstrained optimization problem is located in the increasing interval of the self-concordant function, which ensures the feasibility of solutions. Then by means of the self-concordant penalty function on entire space, a path-following algorithm on entire space for linear programming is presented. The number of Newton steps of the algorithm is no more than $$O(nL\log (nL/ {\varepsilon }))$$ , and moreover, in short step, it is no more than $$O(\sqrt{n}\log (nL/{\varepsilon }))$$ . Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Da Tian, 2014. "An entire space polynomial-time algorithm for linear programming," Journal of Global Optimization, Springer, vol. 58(1), pages 109-135, January.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:1:p:109-135
    DOI: 10.1007/s10898-013-0048-z
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-013-0048-z
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10898-013-0048-z?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. O. Shevchenko, 2009. "Recursive Construction of Optimal Self-Concordant Barriers for Homogeneous Cones," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 339-354, February.
    2. Frenk, J.B.G. & Gromicho, J.A.S. & Zhang, S., 1994. "A deep cut ellipsoid algorithm for convex programming," Econometric Institute Research Papers 11633, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. R. Polyak & I. Griva, 2004. "Primal-Dual Nonlinear Rescaling Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 111-156, July.
    4. E. A. Papa Quiroz & P. R. Oliveira, 2007. "New Self-Concordant Barrier for the Hypercube," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 475-490, December.
    5. J. Burke & S. Xu, 2002. "Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 53-76, January.
    6. NESTEROV, Yu., 2006. "Constructing self-concordant barriers for convex cones," LIDAM Discussion Papers CORE 2006030, 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. Da Tian, 2015. "An exterior point polynomial-time algorithm for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 61(1), pages 51-78, May.

    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. Da Tian, 2015. "An exterior point polynomial-time algorithm for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 61(1), pages 51-78, May.
    2. Komunjer, Ivana, 2005. "Quasi-maximum likelihood estimation for conditional quantiles," Journal of Econometrics, Elsevier, vol. 128(1), pages 137-164, September.
    3. Carrizosa, E. & Frenk, J.B.G., 1996. "Dominating Sets for Convex Functions with some Applications," Econometric Institute Research Papers EI 9657-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Pinheiro, Ricardo B.N.M. & Lage, Guilherme G. & da Costa, Geraldo R.M., 2019. "A primal-dual integrated nonlinear rescaling approach applied to the optimal reactive dispatch problem," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1137-1153.
    5. Mehdi Karimi & Levent Tunçel, 2020. "Primal–Dual Interior-Point Methods for Domain-Driven Formulations," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 591-621, May.
    6. Jean-Pierre Dussault & Mathieu Frappier & Jean Charles Gilbert, 2019. "A lower bound on the iterative complexity of the Harker and Pang globalization technique of the Newton-min algorithm for solving the linear complementarity problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(4), pages 359-380, December.
    7. Efsun Kürüm & Kasirga Yildirak & Gerhard-Wilhelm Weber, 2012. "A classification problem of credit risk rating investigated and solved by optimisation of the ROC curve," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(3), pages 529-557, September.
    8. Chris Coey & Lea Kapelevich & Juan Pablo Vielma, 2022. "Solving Natural Conic Formulations with Hypatia.jl," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2686-2699, September.
    9. Liwei Zhang & Jian Gu & Xiantao Xiao, 2011. "A class of nonlinear Lagrangians for nonconvex second order cone programming," Computational Optimization and Applications, Springer, vol. 49(1), pages 61-99, May.
    10. Quoc Tran-Dinh & Anastasios Kyrillidis & Volkan Cevher, 2018. "A Single-Phase, Proximal Path-Following Framework," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1326-1347, November.
    11. U. Faigle & M. Hunting & W. Kern & R. Prakash & K. Supowit, 1997. "Simplices by point-sliding and the Yamnitsky-Levin algorithm," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(1), pages 131-142, February.
    12. M. Gonçalves & J. Melo & L. Prudente, 2015. "Augmented Lagrangian methods for nonlinear programming with possible infeasibility," Journal of Global Optimization, Springer, vol. 63(2), pages 297-318, October.
    13. E. Carrizosa & J. B. G. Frenk, 1998. "Dominating Sets for Convex Functions with Some Applications," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 281-295, February.
    14. Marielena Fonseca Tófoli & Edilaine Martins Soler & Antonio Roberto Balbo & Edméa Cássia Baptista & Leonardo Nepomuceno, 2020. "Interior/exterior-point methods with inertia correction strategy for solving optimal reactive power flow problems with discrete variables," Annals of Operations Research, Springer, vol. 286(1), pages 243-263, March.
    15. NESTEROV, Yu., 2006. "Nonsymmetric potential-reduction methods for general cones," LIDAM Discussion Papers CORE 2006034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. G. C. Bento & O. P. Ferreira & P. R. Oliveira, 2012. "Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 88-107, July.

    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:jglopt:v:58:y:2014:i:1:p:109-135. 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.