IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v145y2010i2d10.1007_s10957-009-9625-1.html
   My bibliography  Save this article

Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

Author

Listed:
  • J. Li

    (China West Normal University)

  • G. Mastroeni

    (University of Pisa)

Abstract

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.

Suggested Citation

  • J. Li & G. Mastroeni, 2010. "Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 355-372, May.
  • Handle: RePEc:spr:joptap:v:145:y:2010:i:2:d:10.1007_s10957-009-9625-1
    DOI: 10.1007/s10957-009-9625-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-009-9625-1
    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/s10957-009-9625-1?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. X.Q. Yang, 2003. "On the Gap Functions of Prevariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 116(2), pages 437-452, February.
    2. N. J. Huang & J. Li & J. C. Yao, 2007. "Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 201-212, May.
    3. X.Q. Yang & J.C. Yao, 2002. "Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 407-417, November.
    4. K. L. Lin & D. P. Yang & J. C. Yao, 1997. "Generalized Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 117-125, January.
    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. Suhel Ahmad Khan & Jia-Wei Chen, 2015. "Gap Functions and Error Bounds for Generalized Mixed Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 767-776, September.
    2. Shengjie Li & Yangdong Xu & Manxue You & Shengkun Zhu, 2018. "Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 609-636, June.
    3. Giovanni P. Crespi & Matteo Rocca & Carola Schrage, 2015. "Variational Inequalities Characterizing Weak Minimality in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 804-824, September.
    4. Xiao-bo Li & Li-wen Zhou & Nan-jing Huang, 2016. "Gap Functions and Global Error Bounds for Generalized Mixed Variational Inequalities on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 830-849, March.
    5. Khan, Suhel Ahmad & Chen, Jia-wei, 2015. "Gap function and global error bounds for generalized mixed quasi variational inequalities," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 71-81.

    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. N. J. Huang & J. Li & J. C. Yao, 2007. "Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 201-212, May.
    2. S. K. Mishra & S. Y. Wang & K. K. Lai, 2008. "Gap Function for Set-Valued Vector Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 77-84, July.
    3. Y. C. Lin, 2009. "On F-Implicit Generalized Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 557-568, September.
    4. Y. Chiang & J. C. Yao, 2004. "Vector Variational Inequalities and the (S)+ Condition," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 271-290, November.
    5. T. Jabarootian & J. Zafarani, 2008. "Generalized Vector Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 15-30, January.
    6. Suhel Ahmad Khan & Jia-Wei Chen, 2015. "Gap Functions and Error Bounds for Generalized Mixed Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 767-776, September.
    7. Ren-you Zhong & Nan-jing Huang, 2011. "Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 317-326, August.
    8. Ren-you Zhong & Zhen Dou & Jiang-hua Fan, 2015. "Degree Theory and Solution Existence of Set-Valued Vector Variational Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 527-549, November.
    9. B. Djafari Rouhani & B. Ahmadi Kakavandi, 2006. "Infinite Time-Dependent Network Equilibria with a Multivalued Cost Function," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 405-415, December.
    10. Bin Chen & Nan-jing Huang, 2013. "Continuity of the solution mapping to parametric generalized vector equilibrium problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1515-1528, August.
    11. T. Antczak, 2007. "Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 71-87, January.
    12. N. Hadjisavvas & S. Schaible, 1998. "From Scalar to Vector Equilibrium Problems in the Quasimonotone Case," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 297-309, February.
    13. G. Mastroeni, 2012. "On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation," Journal of Global Optimization, Springer, vol. 53(2), pages 203-214, June.
    14. Q. H. Ansari & J> C> Yao, 2000. "On Nondifferentiable and Nonconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 475-488, September.
    15. Sonia & Ratna Dev Sarma, 2023. "A topological approach for vector quasi-variational inequalities with set-valued functions," Computational Management Science, Springer, vol. 20(1), pages 1-13, December.
    16. Xing Wang & Nan-Jing Huang, 2013. "Differential Vector Variational Inequalities in Finite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 109-129, July.
    17. Jia-Wei Chen & Zhongping Wan & Yeol Cho, 2013. "Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 33-64, February.
    18. Y. P. Fang & N. J. Huang, 2006. "Feasibility and Solvability for Vector Complementarity Problems1," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 373-390, June.
    19. Xing Wang & Nan-jing Huang, 2014. "A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 633-648, August.
    20. G. Y. Chen & X. Q. Yang, 2002. "Characterizations of Variable Domination Structures via Nonlinear Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 97-110, January.

    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:145:y:2010:i:2:d:10.1007_s10957-009-9625-1. 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.