IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v203y2024i2d10.1007_s10957-024-02405-6.html
   My bibliography  Save this article

Understanding Badly and Well-Behaved Linear Matrix Inequalities Via Semi-infinite Optimization

Author

Listed:
  • Qinghong Zhang

    (Northern Michigan University)

Abstract

In this paper, we use a linear semi-infinite optimization approach to study badly and well-behaved linear matrix inequalities. We utilize a result on uniform LP duality of linear semi-infinite optimization problems to prove recent results obtained by Pataki. Such an approach not only provides alternative proofs of known results, but also gives new insights about badly and well-behaved linear matrix inequalities in terms of a cone and a linear subspace associated with the corresponding linear semi-infinite systems. Furthermore, when the linear matrix inequality constraint of the primal semidefinite optimization problem is badly behaved, we give a characterization of objective functions for the primal linear semidefinite optimization problem with which strong duality holds.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:2:d:10.1007_s10957-024-02405-6
    DOI: 10.1007/s10957-024-02405-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02405-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-024-02405-6?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. 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)

    Citations

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


    Cited by:

    1. O. I. Kostyukova & T. V. Tchemisova & O. S. Dudina, 2024. "On the Uniform Duality in Copositive Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1940-1966, November.

    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. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    7. 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.
    8. 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.
    9. 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.
    10. Zhi-Quan Luo & Shuzhong Zhang, 1997. "On the extensions of Frank-Wolfe theorem," Tinbergen Institute Discussion Papers 97-122/4, Tinbergen Institute.

    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:203:y:2024:i:2:d:10.1007_s10957-024-02405-6. 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.