IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v92y1997i3d10.1023_a1022699205336.html
   My bibliography  Save this article

Generalized Quasi-Variational Inequalities Without Continuities

Author

Listed:
  • P. Cubiotti

Abstract

Given a nonempty set $$X \subseteq \mathbb{R}^n $$ and two multifunctions $$\beta :X \to 2^X ,\phi :X \to 2^{\mathbb{R}^n } $$ , we consider the following generalized quasi-variational inequality problem associated with X, β φ: Find $$(\bar x,\bar z) \in X \times \mathbb{R}^n $$ such that $$\bar x \in \beta (\bar x),\bar z \in \phi (\bar x){\text{, and sup}}_{y \in \beta (\bar x)} \left\langle {\bar z,\bar x - y} \right\rangle \leqslant 0$$ . We prove several existence results in which the multifunction φ is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case β(x(≡X.

Suggested Citation

  • P. Cubiotti, 1997. "Generalized Quasi-Variational Inequalities Without Continuities," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 477-495, March.
    • Handle: RePEc:spr:joptap:v:92:y:1997:i:3:d:10.1023_a:1022699205336
    DOI: 10.1023/A:1022699205336
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022699205336
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022699205336?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. Romesh Saigal, 1976. "Extension of the Generalized Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 260-266, August.
    2. D. Chan & J. S. Pang, 1982. "The Generalized Quasi-Variational Inequality Problem," Mathematics of Operations Research, INFORMS, vol. 7(2), pages 211-222, May.
    3. Jen-Chih Yao, 1995. "Generalized-Quasi-Variational Inequality Problems with Discontinuous Mappings," Mathematics of Operations Research, INFORMS, vol. 20(2), pages 465-478, May.
    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. M. Balaj & L. J. Lin, 2013. "Existence Criteria for the Solutions of Two Types of Variational Relation Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 232-246, February.
    2. M. Fakhar & J. Zafarani, 2008. "Generalized Symmetric Vector Quasiequilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 136(3), pages 397-409, March.
    3. P. Cubiotti, 2003. "Existence Theorem for the Discontinuous Generalized Quasivariational Inequality Problem," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 623-633, December.
    4. R. P. Agarwal & M. Balaj & D. O’Regan, 2012. "A Unifying Approach to Variational Relation Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 417-429, November.
    5. B. T. Kien & N. C. Wong & J. C. Yao, 2007. "On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 515-530, December.
    6. P. H. Sach, 2008. "On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 337-350, 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. Paolo Cubiotti & Jen-Chih Yao, 1997. "Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 213-228, June.
    2. Flam, Sjur & Ruszczynski, A., 2006. "Computing Normalized Equilibria in Convex-Concave Games," Working Papers 2006:9, Lund University, Department of Economics.
    3. Monica Milasi, 2014. "Existence theorem for a class of generalized quasi-variational inequalities," Journal of Global Optimization, Springer, vol. 60(4), pages 679-688, December.
    4. X. P. Ding, 1997. "Generalized Variational-Like Inequalities with Nonmonotone Set-Valued Mappings," Journal of Optimization Theory and Applications, Springer, vol. 95(3), pages 601-613, December.
    5. Tian, Guoqiang, 1991. "Generalized quasi-variational-like inequality problem," MPRA Paper 41219, University Library of Munich, Germany, revised 26 May 1992.
    6. Jacqueline Boucher & Yves Smeers, 2001. "Alternative Models of Restructured Electricity Systems, Part 1: No Market Power," Operations Research, INFORMS, vol. 49(6), pages 821-838, December.
    7. N. Minh & N. Tan, 2006. "On the Existence of Solutions of Quasi-equilibrium Problems with Constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 17-31, August.
    8. O. Chaldi & Z. Chbani & H. Riahi, 2000. "Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 105(2), pages 299-323, May.
    9. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    10. Massimo Pappalardo & Giandomenico Mastroeni & Mauro Passacantando, 2016. "Merit functions: a bridge between optimization and equilibria," Annals of Operations Research, Springer, vol. 240(1), pages 271-299, May.
    11. S. Park, 1997. "Generalized Equilibrium Problems and Generalized Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 409-417, November.
    12. M. Fakhar & J. Zafarani, 2008. "Generalized Symmetric Vector Quasiequilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 136(3), pages 397-409, March.
    13. Ankur A. Kulkarni & Uday V. Shanbhag, 2012. "Revisiting Generalized Nash Games and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 175-186, July.
    14. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    15. S. Kum, 1998. "An Example of Generalized Quasi-Variational Inequality," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 253-258, October.
    16. Zhou, Jing & Lam, William H.K. & Heydecker, Benjamin G., 2005. "The generalized Nash equilibrium model for oligopolistic transit market with elastic demand," Transportation Research Part B: Methodological, Elsevier, vol. 39(6), pages 519-544, July.
    17. Laura Scrimali, 2012. "Infinite Dimensional Duality Theory Applied to Investment Strategies in Environmental Policy," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 258-277, July.
    18. L. F. Bueno & G. Haeser & F. Lara & F. N. Rojas, 2020. "An Augmented Lagrangian method for quasi-equilibrium problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 737-766, July.
    19. SCRIMALI, Laura, 2006. "A quasi-variational inequality approach to the financial equilibrium problem," LIDAM Discussion Papers CORE 2006108, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    20. Q. H. Ansari & T. C. Lai & J. C. Yao, 1999. "On the Equivalence of Extended Generalized Complementarity and Generalized Least-Element Problems," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 277-288, 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:92:y:1997:i:3:d:10.1023_a:1022699205336. 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.