IDEAS home Printed from https://ideas.repec.org/p/ems/eureir/1544.html
   My bibliography  Save this paper

Global error bounds for convex conic problems

Author

Listed:
  • Zhang, S.

Abstract

In this paper Lipschitzian type error bounds are derived for general convex conic problems under various regularity conditions. Specifically, it is shown that if the recession directions satisfy Slater's condition then a global Lipschitzian type error bound holds. Alternatively, if the feasible region is bounded, then the ordinary Slater condition guarantees a global Lipschitzian type error bound. These can be considered as generalizations of previously known results for inequality systems. Moreover, some of the results are also generalized to the intersection of multiple cones. Under Slater's condition alone, a global Lipschitzian type error bound may not hold. However, it is shown that such an error bound holds for a specific region. For linear systems we show that the constant involved in Hoffman's error bound can be estimated by the so-called condition number for linear programming.

Suggested Citation

  • Zhang, S., 1998. "Global error bounds for convex conic problems," Econometric Institute Research Papers EI 9830, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:1544
    as

    Download full text from publisher

    File URL: https://repub.eur.nl/pub/1544/1544_ps.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michael J. Todd, 1990. "A Dantzig-Wolfe-Like Variant of Karmarkar's Interior-Point Linear Programming Algorithm," Operations Research, INFORMS, vol. 38(6), pages 1006-1018, December.
    2. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    3. A.G. Holder & J.F. Sturm & S. Zhang, 1998. "Analytic Central Path, Sensitivity Analysis and Parametric Linear Programming," Tinbergen Institute Discussion Papers 98-003/4, Tinbergen Institute.
    4. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1997. "Duality Results for Conic Convex Programming," Econometric Institute Research Papers EI 9719/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    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. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    2. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    3. Holder, A.G. & Sturm, J.F. & Zhang, S., 1998. "Analytic central path, sensitivity analysis and parametric linear programming," Econometric Institute Research Papers EI 9801, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1998. "Conic convex programming and self-dual embedding," Econometric Institute Research Papers EI 9815, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. Kenneth O. Kortanek & Guolin Yu & Qinghong Zhang, 2021. "Strong duality for standard convex programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 413-436, December.
    6. Yao, D.D. & Zhang, S. & Zhou, X.Y., 1999. "LQ control without Ricatti equations: deterministic systems," Econometric Institute Research Papers EI 9913-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Dennis Cheung & Felipe Cucker & Javier Peña, 2003. "Unifying Condition Numbers for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 609-624, November.
    8. E. A. Yıldırım, 2003. "An Interior-Point Perspective on Sensitivity Analysis in Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 649-676, November.
    9. Qinghong Zhang, 2024. "Understanding Badly and Well-Behaved Linear Matrix Inequalities Via Semi-infinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1820-1846, November.
    10. Yoshiyuki Sekiguchi & Hayato Waki, 2021. "Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 52-72, January.
    11. Hayato Waki & Masakazu Muramatsu, 2013. "Facial Reduction Algorithms for Conic Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 188-215, July.
    12. O. I. Kostyukova & T. V. Tchemisova, 2022. "On strong duality in linear copositive programming," Journal of Global Optimization, Springer, vol. 83(3), pages 457-480, July.
    13. A.G. Holder & J.F. Sturm & S. Zhang, 1998. "Analytic Central Path, Sensitivity Analysis and Parametric Linear Programming," Tinbergen Institute Discussion Papers 98-003/4, Tinbergen Institute.
    14. Brinkhuis, J. & Zhang, S., 2003. "A D-induced duality and its applications," Econometric Institute Research Papers EI 2003-42, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    15. Zhi-Quan Luo & Shuzhong Zhang, 1997. "On the extensions of Frank-Wolfe theorem," Tinbergen Institute Discussion Papers 97-122/4, Tinbergen Institute.
    16. 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:ems:eureir:1544. 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: RePub (email available below). General contact details of provider: https://edirc.repec.org/data/feeurnl.html .

    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.