IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v77y2013i3p305-321.html
   My bibliography  Save this article

Certification aspects of the fast gradient method for solving the dual of parametric convex programs

Author

Listed:
  • Stefan Richter
  • Colin Jones
  • Manfred Morari

Abstract

This paper examines the computational complexity certification of the fast gradient method for the solution of the dual of a parametric convex program. To this end, a lower iteration bound is derived such that for all parameters from a compact set a solution with a specified level of suboptimality will be obtained. For its practical importance, the derivation of the smallest lower iteration bound is considered. In order to determine it, we investigate both the computation of the worst case minimal Euclidean distance between an initial iterate and a Lagrange multiplier and the issue of finding the largest step size for the fast gradient method. In addition, we argue that optimal preconditioning of the dual problem cannot be proven to decrease the smallest lower iteration bound. The findings of this paper are of importance in embedded optimization, for instance, in model predictive control. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Stefan Richter & Colin Jones & Manfred Morari, 2013. "Certification aspects of the fast gradient method for solving the dual of parametric convex programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 305-321, June.
    • Handle: RePEc:spr:mathme:v:77:y:2013:i:3:p:305-321
    DOI: 10.1007/s00186-012-0420-7
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-012-0420-7
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-012-0420-7?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. DEVOLDER, Olivier, 2011. "Stochastic first order methods in smooth convex optimization," LIDAM Discussion Papers CORE 2011070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2012. "Double smoothing technique for large-scale linearly constrained convex optimization," LIDAM Reprints CORE 2423, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. repec:cor:louvrp:-2423 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    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. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Egor Gladin & Alexander Gasnikov & Pavel Dvurechensky, 2025. "Accuracy Certificates for Convex Minimization with Inexact Oracle," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-23, January.
    3. Pavel Dvurechensky & Yurii Nesterov & Vladimir Spokoiny, 2015. "Primal-Dual Methods for Solving Infinite-Dimensional Games," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 23-51, July.
    4. J. O. Royset & E. Y. Pee, 2012. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 855-882, December.
    5. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2013. "Intermediate gradient methods for smooth convex problems with inexact oracle," LIDAM Discussion Papers CORE 2013017, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Masoud Ahookhosh & Arnold Neumaier, 2018. "Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 )," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 110-145, April.
    7. Renato D. C. Monteiro & Camilo Ortiz & Benar F. Svaiter, 2016. "An adaptive accelerated first-order method for convex optimization," Computational Optimization and Applications, Springer, vol. 64(1), pages 31-73, May.
    8. Radu Boţ & Christopher Hendrich, 2015. "A variable smoothing algorithm for solving convex optimization problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 124-150, April.
    9. Taylor, A. & Hendrickx, J. & Glineur, F., 2015. "Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods," LIDAM Discussion Papers CORE 2015013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Vishwajit Hegde & Arvind S. Menon & L. A. Prashanth & Krishna Jagannathan, 2021. "Online Estimation and Optimization of Utility-Based Shortfall Risk," Papers 2111.08805, arXiv.org, revised Nov 2023.
    11. Radu Boţ & Christopher Hendrich, 2013. "A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems," Computational Optimization and Applications, Springer, vol. 54(2), pages 239-262, March.
    12. Olga Yufereva & Michael Persiianov & Pavel Dvurechensky & Alexander Gasnikov & Dmitry Kovalev, 2024. "Decentralized convex optimization on time-varying networks with application to Wasserstein barycenters," Computational Management Science, Springer, vol. 21(1), pages 1-31, June.
    13. Quoc Tran-Dinh, 2017. "Adaptive smoothing algorithms for nonsmooth composite convex minimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 425-451, April.
    14. Jueyou Li & Guo Chen & Zhaoyang Dong & Zhiyou Wu, 2016. "A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints," Computational Optimization and Applications, Springer, vol. 64(3), pages 671-697, July.
    15. Lorenzo Rosasco & Silvia Villa & Bang Công Vũ, 2016. "Stochastic Forward–Backward Splitting for Monotone Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 388-406, May.
    16. Dvurechensky, Pavel & Gorbunov, Eduard & Gasnikov, Alexander, 2021. "An accelerated directional derivative method for smooth stochastic convex optimization," European Journal of Operational Research, Elsevier, vol. 290(2), pages 601-621.

    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:mathme:v:77:y:2013:i:3:p:305-321. 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.