IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v172y2017i1d10.1007_s10957-016-1011-1.html
   My bibliography  Save this article

New Second-Order Tangent Epiderivatives and Applications to Set-Valued Optimization

Author

Listed:
  • Zhenhua Peng

    (Nanchang University)

  • Yihong Xu

    (Nanchang University)

Abstract

A new second-order tangent set is introduced, with which a new second-order tangent epiderivative is also introduced for a set-valued map. Applying a separation theorem for convex sets, second-order Fritz John and Kuhn–Tucker necessary optimality conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of lower semicontinuous, a second-order Kuhn–Tucker sufficient optimality condition is obtained for a point pair to be a weak minimizer of set-valued optimization problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:1:d:10.1007_s10957-016-1011-1
    DOI: 10.1007/s10957-016-1011-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-1011-1
    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/s10957-016-1011-1?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. M. Yang & X. Q. Yang & G. Y. Chen, 2000. "Theorems of the Alternative and Optimization with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 627-640, December.
    2. 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.
    3. Nguyen Anh & Phan Khanh, 2013. "Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives," Journal of Global Optimization, Springer, vol. 56(2), pages 519-536, June.
    4. 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.
    5. Z. F. Li, 1998. "Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 623-649, September.
    6. Ning E. & Wen Song & Yu Zhang, 2012. "Second order sufficient optimality conditions in vector optimization," Journal of Global Optimization, Springer, vol. 54(3), pages 537-549, November.
    7. X. M. Yang & D. Li & S. Y. Wang, 2001. "Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 413-427, August.
    8. S. J. Li & S. K. Zhu & K. L. Teo, 2012. "New Generalized Second-Order Contingent Epiderivatives and Set-Valued Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 587-604, March.
    9. J. Jahn & A. A. Khan & P. Zeilinger, 2005. "Second-Order Optimality Conditions in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 331-347, May.
    10. Giancarlo Bigi & Marco Castellani, 2002. "K-epiderivatives for set-valued functions and optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(3), pages 401-412, June.
    11. P. H. Sach, 2005. "New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 157-179, April.
    12. Giancarlo Bigi & Marco Castellani, 2002. "K-epiderivatives for set-valued functions and optimization," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 55(3), pages 401-412, June.
    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. Zhenhua Peng & Zhongping Wan, 2020. "Second-Order Composed Contingent Derivative of the Perturbation Map in Multiobjective Optimization," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 37(02), pages 1-23, March.

    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. 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.
    2. 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.
    3. Z. A. Zhou & J. W. Peng, 2012. "Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 830-841, September.
    4. P. H. Sach, 2007. "Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 213-227, May.
    5. 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.
    6. P. H. Sach, 2003. "Nearly Subconvexlike Set-Valued Maps and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 335-356, November.
    7. D. S. Kim & G. M. Lee & P. H. Sach, 2004. "Hartley Proper Efficiency in Multifunction Optimization," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 129-145, January.
    8. 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.
    9. 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.
    10. Xiang-Kai Sun & Sheng-Jie Li, 2014. "Generalized second-order contingent epiderivatives in parametric vector optimization problems," Journal of Global Optimization, Springer, vol. 58(2), pages 351-363, February.
    11. Z. A. Zhou & X. M. Yang, 2011. "Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 327-340, August.
    12. Giovanni Crespi & Ivan Ginchev & Matteo Rocca, 2006. "First-order optimality conditions in set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 87-106, February.
    13. L. Y. Xia & J. H. Qiu, 2008. "Superefficiency in Vector Optimization with Nearly Subconvexlike Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 125-137, January.
    14. Crespi Giovanni P. & Ginchev Ivan & Rocca Matteo, 2004. "First order optimality conditions in set-valued optimization," Economics and Quantitative Methods qf04010, Department of Economics, University of Insubria.
    15. Crespi Giovanni P. & Ginchev Ivan & Rocca Matteo, 2004. "First order optimality condition for constrained set-valued optimization," Economics and Quantitative Methods qf04014, Department of Economics, University of Insubria.
    16. 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.
    17. 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.
    18. Zhenhua Peng & Zhongping Wan, 2020. "Second-Order Composed Contingent Derivative of the Perturbation Map in Multiobjective Optimization," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 37(02), pages 1-23, March.
    19. C. Gutiérrez & L. Huerga & V. Novo & C. Tammer, 2016. "Duality related to approximate proper solutions of vector optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 117-139, January.
    20. 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.

    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:172:y:2017:i:1:d:10.1007_s10957-016-1011-1. 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.