IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v203y2024i3d10.1007_s10957-024-02546-8.html
   My bibliography  Save this article

Formulas for Calculating Generalized Differentials with Respect to a Set and Their Applications

Author

Listed:
  • Vo Duc Thinh

    (Zhejiang Normal University
    Dong Thap University)

  • Xiaolong Qin

    (Zhejiang Normal University)

  • Jen-Chih Yao

    (China Medical University Hospital
    Azerbaijan State University of Economics)

Abstract

This paper provides formulas for calculating Fréchet and limiting normal cones with respect to a set of sets and the limiting coderivative with respect to a set of set-valued mappings. These calculations are obtained under some qualification constraints and are expressed in the similar forms of the ones of Fréchet and limiting normal cones and the limiting coderivative. By using these new formulas, we state explicit necessary optimality conditions with respect to a set for optimization problems with equilibrium constraints under certain qualification conditions. Some examples are also presented.

Suggested Citation

  • Vo Duc Thinh & Xiaolong Qin & Jen-Chih Yao, 2024. "Formulas for Calculating Generalized Differentials with Respect to a Set and Their Applications," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2784-2817, December.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:3:d:10.1007_s10957-024-02546-8
    DOI: 10.1007/s10957-024-02546-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02546-8
    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/s10957-024-02546-8?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. Thai Doan Chuong & Do Sang Kim, 2016. "Hölder-Like Property and Metric Regularity of a Positive-Order for Implicit Multifunctions," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 596-611, May.
    2. N. D. Yen, 1995. "Lipschitz Continuity of Solutions of Variational Inequalities with a Parametric Polyhedral Constraint," Mathematics of Operations Research, INFORMS, vol. 20(3), pages 695-708, August.
    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. V. D. Thinh & T. D. Chuong & N. L. H. Anh, 2023. "Second order analysis for robust inclusion systems and applications," Journal of Global Optimization, Springer, vol. 85(1), pages 81-110, January.
    2. Wang, Jian & He, Xiaozheng & Peeta, Srinivas & Wang, Wei, 2022. "Globally convergent line search algorithm with Euler-based step size-determination method for continuous network design problem," Transportation Research Part B: Methodological, Elsevier, vol. 163(C), pages 119-144.
    3. Michael Patriksson & R. Tyrrell Rockafellar, 2003. "Sensitivity Analysis of Aggregated Variational Inequality Problems, with Application to Traffic Equilibria," Transportation Science, INFORMS, vol. 37(1), pages 56-68, February.
    4. Le Tuan & Gue Lee & Pham Sach, 2010. "Upper semicontinuity result for the solution mapping of a mixed parametric generalized vector quasiequilibrium problem with moving cones," Journal of Global Optimization, Springer, vol. 47(4), pages 639-660, August.
    5. Tong, Jun & Hu, Jiaqiao & Hu, Jianqiang, 2017. "Computing equilibrium prices for a capital asset pricing model with heterogeneous beliefs and margin-requirement constraints," European Journal of Operational Research, Elsevier, vol. 256(1), pages 24-34.
    6. X. P. Ding & C. L. Luo, 1999. "On Parametric Generalized Quasi-Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 100(1), pages 195-205, January.
    7. Patriksson, Michael, 2008. "On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs," Transportation Research Part B: Methodological, Elsevier, vol. 42(10), pages 843-860, December.
    8. Kien, B.T. & Wong, M.-M. & Wong, N.C. & Yao, J.C., 2009. "Degree theory for generalized variational inequalities and applications," European Journal of Operational Research, Elsevier, vol. 192(3), pages 730-736, February.
    9. Michael Patriksson, 2004. "Sensitivity Analysis of Traffic Equilibria," Transportation Science, INFORMS, vol. 38(3), pages 258-281, August.
    10. M. A. Noor, 1997. "Sensitivity Analysis for Quasi-Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 399-407, November.
    11. Thai Doan Chuong & Do Sang Kim, 2018. "Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications," Annals of Operations Research, Springer, vol. 267(1), pages 81-99, August.
    12. Jafari, Ehsan & Boyles, Stephen D., 2016. "Improved bush-based methods for network contraction," Transportation Research Part B: Methodological, Elsevier, vol. 83(C), pages 298-313.
    13. D. Aussel & J. Cotrina, 2013. "Stability of Quasimonotone Variational Inequality Under Sign-Continuity," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 653-667, September.
    14. Daniela Osorio-González & José Guadalupe Flores-Muñiz & Nataliya Kalashnykova & Viacheslav Kalashnikov, 2024. "Consistent Conjectural Variations Equilibrium for a Bilevel Human Migration Model," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2354-2369, December.
    15. Lucian Coroianu, 2016. "Best Lipschitz Constants of Solutions of Quadratic Programs," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 853-875, September.
    16. Lu, Shu & (Marco) Nie, Yu, 2010. "Stability of user-equilibrium route flow solutions for the traffic assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 44(4), pages 609-617, May.
    17. P. Q. Khanh & L. M. Luu, 2007. "Lower Semicontinuity and Upper Semicontinuity of the Solution Sets and Approximate Solution Sets of Parametric Multivalued Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 329-339, June.
    18. Shu Lu, 2008. "Sensitivity of Static Traffic User Equilibria with Perturbations in Arc Cost Function and Travel Demand," Transportation Science, INFORMS, vol. 42(1), pages 105-123, February.
    19. Michael Patriksson & R. Tyrrell Rockafellar, 2002. "A Mathematical Model and Descent Algorithm for Bilevel Traffic Management," Transportation Science, INFORMS, vol. 36(3), pages 271-291, August.
    20. Shu Lu & Stephen M. Robinson, 2008. "Variational Inequalities over Perturbed Polyhedral Convex Sets," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 689-711, August.

    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:3:d:10.1007_s10957-024-02546-8. 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.