IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v84y2022i1d10.1007_s10898-022-01129-z.html
   My bibliography  Save this article

Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization

Author

Listed:
  • Nguyen Xuan Duy Bao

    (University of Science
    Vietnam National University)

  • Phan Quoc Khanh

    (Ton Duc Thang University)

  • Nguyen Minh Tung

    (Banking University of Ho Chi Minh City)

Abstract

We consider higher-order conditions and sensitivity analysis for solutions to equilibrium problems. The conditions for solutions are in terms of quasi-contingent derivatives and involve higher-order complementarity slackness for both the objective and the constraints and under Hölder metric subregularity assumptions. For sensitivity analysis, a formula of this type of derivative of the solution map to a parametric equilibrium problem is established in terms of the same types of derivatives of the data of the problem. Here, the concepts of a quasi-contingent derivative and critical directions are new. We consider open-cone solutions and proper solutions. We also study an important and typical special case: weak solutions of a vector minimization problem with mixed constraints. The results are significantly new and improve recent corresponding results in many aspects.

Suggested Citation

  • Nguyen Xuan Duy Bao & Phan Quoc Khanh & Nguyen Minh Tung, 2022. "Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(1), pages 205-228, September.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:1:d:10.1007_s10898-022-01129-z
    DOI: 10.1007/s10898-022-01129-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01129-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/s10898-022-01129-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. S. J. Li & K. L. Teo & X. Q. Yang, 2008. "Higher-Order Optimality Conditions for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 533-553, June.
    2. Nguyen Hoang Anh & Phan Khanh, 2014. "Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives," Journal of Global Optimization, Springer, vol. 58(4), pages 693-709, April.
    3. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.
    4. H. T. H. Diem & P. Q. Khanh & L. T. Tung, 2014. "On Higher-Order Sensitivity Analysis in Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 463-488, August.
    5. P. Q. Khanh & N. M. Tung, 2015. "Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 68-90, October.
    6. N. L. H. Anh & P. Q. Khanh, 2013. "Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 363-384, August.
    7. E. K. Makarov & N. N. Rachkovski, 1999. "Unified Representation of Proper Efficiency by Means of Dilating Cones," Journal of Optimization Theory and Applications, Springer, vol. 101(1), pages 141-165, April.
    8. Boris Mordukhovich & Wei Ouyang, 2015. "Higher-order metric subregularity and its applications," Journal of Global Optimization, Springer, vol. 63(4), pages 777-795, December.
    9. Phan Khanh & Le Tung, 2013. "First and second-order optimality conditions using approximations for vector equilibrium problems with constraints," Journal of Global Optimization, Springer, vol. 55(4), pages 901-920, April.
    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. H. T. H. Diem & P. Q. Khanh & L. T. Tung, 2014. "On Higher-Order Sensitivity Analysis in Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 463-488, August.
    2. Nguyen Hoang Anh & Phan Khanh, 2014. "Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives," Journal of Global Optimization, Springer, vol. 58(4), pages 693-709, April.
    3. Nguyen Minh Tung, 2020. "New Higher-Order Strong Karush–Kuhn–Tucker Conditions for Proper Solutions in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 448-475, May.
    4. N. L. H. Anh & P. Q. Khanh, 2013. "Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 363-384, August.
    5. M. H. Li & S. J. Li, 2010. "Second-Order Differential and Sensitivity Properties of Weak Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 76-87, January.
    6. Fernando García-Castaño & Miguel Ángel Melguizo-Padial & G. Parzanese, 2023. "Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(3), pages 367-382, June.
    7. Amos Uderzo, 2016. "A Strong Metric Subregularity Analysis of Nonsmooth Mappings Via Steepest Displacement Rate," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 573-599, November.
    8. Nguyen Minh Tung & Nguyen Xuan Duy Bao, 2023. "New Set-Valued Directional Derivatives: Calculus and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 411-437, May.
    9. Phan Quoc Khanh & Nguyen Minh Tung, 2016. "Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 45-69, October.
    10. Nguyen Minh Tung & Nguyen Xuan Duy Bao, 2022. "Higher-order set-valued Hadamard directional derivatives: calculus rules and sensitivity analysis of equilibrium problems and generalized equations," Journal of Global Optimization, Springer, vol. 83(2), pages 377-402, June.
    11. Alexander Y. Kruger, 2016. "Nonlinear Metric Subregularity," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 820-855, December.
    12. Koushik Das & Savin Treanţă & Tareq Saeed, 2022. "Mond-Weir and Wolfe Duality of Set-Valued Fractional Minimax Problems in Terms of Contingent Epi-Derivative of Second-Order," Mathematics, MDPI, vol. 10(6), pages 1-21, March.
    13. Zhenhua Peng & Yihong Xu, 2017. "New Second-Order Tangent Epiderivatives and Applications to Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 128-140, January.
    14. Matthieu Maréchal, 2018. "Metric Subregularity in Generalized Equations," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 541-558, March.
    15. S. Zhu & S. Li & K. Teo, 2014. "Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization," Journal of Global Optimization, Springer, vol. 58(4), pages 673-692, April.
    16. Yi-Hong Xu & Zhen-Hua Peng, 2018. "Second-Order M-Composed Tangent Derivative and Its Applications," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(05), pages 1-20, October.
    17. Huynh Ngai & Nguyen Huu Tron & Michel Théra, 2016. "Directional Hölder Metric Regularity," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 785-819, 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:jglopt:v:84:y:2022:i:1:d:10.1007_s10898-022-01129-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.