IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v78y2021i3d10.1007_s10589-020-00258-z.html
   My bibliography  Save this article

The Tikhonov regularization for vector equilibrium problems

Author

Listed:
  • Lam Quoc Anh

    (Cantho University)

  • Tran Quoc Duy

    (FPT University)

  • Le Dung Muu

    (Thang Long University)

  • Truong Van Tri

    (University of Ostrava)

Abstract

We consider vector equilibrium problems in real Banach spaces and study their regularized problems. Based on cone continuity and generalized convexity properties of vector-valued mappings, we propose general conditions that guarantee existence of solutions to such problems in cases of monotonicity and nonmonotonicity. First, our study indicates that every Tikhonov trajectory converges to a solution to the original problem. Then, we establish the equivalence between the problem solvability and the boundedness of any Tikhonov trajectory. Finally, the convergence of the Tikhonov trajectory to the least-norm solution of the original problem is discussed.

Suggested Citation

  • Lam Quoc Anh & Tran Quoc Duy & Le Dung Muu & Truong Van Tri, 2021. "The Tikhonov regularization for vector equilibrium problems," Computational Optimization and Applications, Springer, vol. 78(3), pages 769-792, April.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:3:d:10.1007_s10589-020-00258-z
    DOI: 10.1007/s10589-020-00258-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00258-z
    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/s10589-020-00258-z?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. I. V. Konnov, 2019. "Equilibrium formulations of relative optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 137-152, August.
    2. Ouayl Chadli & Qamrul Hasan Ansari & Suliman Al-Homidan, 2017. "Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 726-758, March.
    3. Gábor Kassay & Mihaela Miholca, 2015. "Existence results for vector equilibrium problems given by a sum of two functions," Journal of Global Optimization, Springer, vol. 63(1), pages 195-211, September.
    4. Do Luu, 2016. "Optimality Condition for Local Efficient Solutions of Vector Equilibrium Problems via Convexificators and Applications," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 643-665, November.
    5. Nikolaus Daniels, 2018. "Tikhonov regularization of control-constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 70(1), pages 295-320, May.
    6. Q. Ansari & W. Oettli & D. Schläger, 1997. "A generalization of vectorial equilibria," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 147-152, June.
    7. Yu Deng & Patrick Mehlitz & Uwe Prüfert, 2019. "Optimal control in first-order Sobolev spaces with inequality constraints," Computational Optimization and Applications, Springer, vol. 72(3), pages 797-826, April.
    8. M. Bianchi & N. Hadjisavvas & S. Schaible, 1997. "Vector Equilibrium Problems with Generalized Monotone Bifunctions," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 527-542, March.
    9. Michaël Gaydu, 2012. "Stability properties of the Tikhonov regularization for nonmonotone inclusions," Journal of Global Optimization, Springer, vol. 52(4), pages 843-853, April.
    10. I. Konnov & D. Dyabilkin, 2011. "Nonmonotone equilibrium problems: coercivity conditions and weak regularization," Journal of Global Optimization, Springer, vol. 49(4), pages 575-587, April.
    11. 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.
    12. I. V. Konnov, 2015. "Regularized Penalty Method for General Equilibrium Problems in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 500-513, February.
    13. Dang Hieu, 2017. "New subgradient extragradient methods for common solutions to equilibrium problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 571-594, July.
    14. I. Konnov, 2014. "On penalty methods for non monotone equilibrium problems," Journal of Global Optimization, Springer, vol. 59(1), pages 131-138, May.
    15. Reddy, G.D., 2019. "A class of parameter choice rules for stationary iterated weighted Tikhonov regularization scheme," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 464-476.
    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. Ouayl Chadli & Qamrul Hasan Ansari & Suliman Al-Homidan, 2017. "Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 726-758, March.
    2. Wensheng Jia & Xiaoling Qiu & Dingtao Peng, 2020. "An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality," Mathematics, MDPI, vol. 8(1), pages 1-9, January.
    3. I. V. Konnov, 2019. "Equilibrium formulations of relative optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 137-152, August.
    4. L.J. Lin & Q.H. Ansari & J.Y. Wu, 2003. "Geometric Properties and Coincidence Theorems with Applications to Generalized Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 121-137, April.
    5. Q. H. Ansari & I. V. Konnov & J. C. Yao, 2001. "Existence of a Solution and Variational Principles for Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 481-492, September.
    6. Tran Van Su, 2018. "New optimality conditions for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces," 4OR, Springer, vol. 16(2), pages 173-198, June.
    7. J. Y. Fu, 2006. "Stampacchia Generalized Vector Quasiequilibrium Problems and Vector Saddle Points," Journal of Optimization Theory and Applications, Springer, vol. 128(3), pages 605-619, March.
    8. M. Bianchi & G. Kassay & R. Pini, 2022. "Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization," Journal of Global Optimization, Springer, vol. 82(3), pages 483-498, March.
    9. N. X. Hai & P. Q. Khanh, 2007. "Existence of Solutions to General Quasiequilibrium Problems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 317-327, June.
    10. X. H. Gong, 2008. "Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 35-46, October.
    11. X. H. Gong, 2007. "Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 151-161, May.
    12. I. V. Konnov, 2015. "Regularized Penalty Method for General Equilibrium Problems in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 500-513, February.
    13. 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.
    14. Adela Capătă, 2012. "Optimality Conditions for Extended Ky Fan Inequality with Cone and Affine Constraints and Their Applications," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 661-674, March.
    15. S. H. Hou & H. Yu & G. Y. Chen, 2003. "On Vector Quasi-Equilibrium Problems with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 485-498, December.
    16. Do Luu & Tran Thi Mai, 2018. "Optimality and duality in constrained interval-valued optimization," 4OR, Springer, vol. 16(3), pages 311-337, September.
    17. Yang Yu & Xiaochuan Luo & Huaxi (Yulin) Zhang & Qingxin Zhang, 2019. "The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient," Mathematics, MDPI, vol. 7(5), pages 1-17, April.
    18. 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.
    19. M. Bianchi & R. Pini, 2002. "A Result on Localization of Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 335-343, November.
    20. Yuehu Wang & Baoqing Liu, 2019. "Order-Preservation Properties of Solution Mapping for Parametric Equilibrium Problems and Their Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 881-901, December.

    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:coopap:v:78:y:2021:i:3:d:10.1007_s10589-020-00258-z. 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.