IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v181y2019i3d10.1007_s10957-019-01484-0.html
   My bibliography  Save this article

Second-Order Strong Karush/Kuhn–Tucker Conditions for Proper Efficiencies in Multiobjective Optimization

Author

Listed:
  • Min Feng

    (Chongqing University)

  • Shengjie Li

    (Chongqing University)

Abstract

In this paper, strong Karush/Kuhn–Tucker conditions are studied for smooth multiobjective optimization with inequality constraints. We introduce a new second-order regularity condition of Abadie type in terms of the second-order directional derivatives and then obtain a second-order strong Karush/Kuhn–Tucker necessary condition at a Borwein-properly efficient solution. Simultaneously, we also use an example to show that, if the Abadie type regularity condition is weakened to the Guignard type one, the second-order strong Karush/Kuhn–Tucker necessary condition may not hold. Finally, then we also apply the second-order strong Karush/Kuhn–Tucker conditions to derive a sufficient result for local Geoffrion-proper efficiency.

Suggested Citation

  • Min Feng & Shengjie Li, 2019. "Second-Order Strong Karush/Kuhn–Tucker Conditions for Proper Efficiencies in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 766-786, June.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:3:d:10.1007_s10957-019-01484-0
    DOI: 10.1007/s10957-019-01484-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-019-01484-0
    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/s10957-019-01484-0?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. R. Andreani & C. E. Echagüe & M. L. Schuverdt, 2010. "Constant-Rank Condition and Second-Order Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 255-266, August.
    2. T. Maeda, 2004. "Second-Order Conditions for Efficiency in Nonsmooth Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 521-538, September.
    3. Gulati, T. R. & Islam, M. A., 1990. "Efficiency and proper efficiency in nonlinear vector maximum problems," European Journal of Operational Research, Elsevier, vol. 44(3), pages 373-382, February.
    4. Regina S. Burachik & M. M. Rizvi, 2012. "On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 477-491, November.
    5. Vsevolod I. Ivanov, 2015. "Second-Order Optimality Conditions for Vector Problems with Continuously Fréchet Differentiable Data and Second-Order Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 777-790, September.
    6. X. F. Li & J. Z. Zhang, 2005. "Stronger Kuhn-Tucker Type Conditions in Nonsmooth Multiobjective Optimization: Locally Lipschitz Case," Journal of Optimization Theory and Applications, Springer, vol. 127(2), pages 367-388, November.
    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. Min Feng & Shengjie Li & Jie Wang, 2022. "On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 480-503, 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. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    2. Min Feng & Shengjie Li & Jie Wang, 2022. "On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 480-503, November.
    3. Elena Constantin, 2020. "Second-Order Optimality Conditions in Locally Lipschitz Inequality-Constrained Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 50-67, July.
    4. Peng Zhang & Jin Zhang & Gui-Hua Lin & Xinmin Yang, 2018. "Constraint Qualifications and Proper Pareto Optimality Conditions for Multiobjective Problems with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 763-782, March.
    5. Iasson Karafyllis, 2014. "Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 783-806, June.
    6. Kuang Bai & Yixia Song & Jin Zhang, 2023. "Second-Order Enhanced Optimality Conditions and Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1264-1284, September.
    7. Kaiwen Meng & Xiaoqi Yang, 2015. "First- and Second-Order Necessary Conditions Via Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 720-752, June.
    8. Vsevolod I. Ivanov, 2015. "Second-Order Optimality Conditions for Vector Problems with Continuously Fréchet Differentiable Data and Second-Order Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 777-790, September.
    9. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
    10. Giorgio Giorgi, 2018. "A Guided Tour in Constraint Qualifications for Nonlinear Programming under Differentiability Assumptions," DEM Working Papers Series 160, University of Pavia, Department of Economics and Management.
    11. W. Song & G. M. Yao, 2008. "Homotopy Method for a General Multiobjective Programming Problem," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 139-153, July.
    12. Roberto Andreani & Gabriel Haeser & Leonardo M. Mito & C. Héctor Ramírez & Thiago P. Silveira, 2022. "Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 42-78, October.
    13. María C. Maciel & Sandra A. Santos & Graciela N. Sottosanto, 2011. "On Second-Order Optimality Conditions for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 332-351, May.
    14. Giorgio Giorgi, 2019. "Notes on Constraint Qualifications for Second-Order Optimality Conditions," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(5), pages 16-32, October.
    15. X. F. Li & J. Z. Zhang, 2010. "Existence and Boundedness of the Kuhn-Tucker Multipliers in Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 373-386, May.
    16. P. Q. Khanh & N. D. Tuan, 2007. "Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 341-357, June.
    17. Roger Behling & Gabriel Haeser & Alberto Ramos & Daiana S. Viana, 2018. "On a Conjecture in Second-Order Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 625-633, March.
    18. Regina S. Burachik & M. M. Rizvi, 2012. "On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 477-491, November.
    19. Manxue You & Shengjie Li, 2017. "Separation Functions and Optimality Conditions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 527-544, November.
    20. Anulekha Dhara & Aparna Mehra, 2013. "Second-Order Optimality Conditions in Minimax Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 567-590, March.

    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:181:y:2019:i:3:d:10.1007_s10957-019-01484-0. 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.