IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v21y1973i1p1-9.html
   My bibliography  Save this article

On Symmetric Duality in Nonlinear Programming

Author

Listed:
  • M. S. Bazaraa

    (Georgia Institute of Technology, Atlanta, Georgia)

  • J. J. Goode

    (Georgia Institute of Technology, Atlanta, Georgia)

Abstract

In this study we generalize the formulation of symmetric duality introduced by Dantzig, Eisenberg, and Cottle to include the case where the constraints of the inequality type are defined via closed convex cones and their polars. The new formulation retains the symmetric properties of the original programs. Under suitable convexity/concavity assumptions we generalize the known results about symmetric duality. The case where the function involved is strongly convex/strongly concave is also treated and Karamardian's result in this case is generalized. As a result, we show that every strongly convex function achieves a minimum value over any closed convex cone at a unique point. Some special cases of symmetric programs are then considered, leading to generalizations of Wolfe's duality as well as generalizations of quadratic and linear programming formulations.

Suggested Citation

  • M. S. Bazaraa & J. J. Goode, 1973. "On Symmetric Duality in Nonlinear Programming," Operations Research, INFORMS, vol. 21(1), pages 1-9, February.
    • Handle: RePEc:inm:oropre:v:21:y:1973:i:1:p:1-9
    DOI: 10.1287/opre.21.1.1
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.21.1.1
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.21.1.1?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
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Suneja, S. K. & Aggarwal, Sunila & Davar, Sonia, 2002. "Multiobjective symmetric duality involving cones," European Journal of Operational Research, Elsevier, vol. 141(3), pages 471-479, September.
    2. Chandra, S. & Abha, 2000. "A note on pseudo-invexity and duality in nonlinear programming," European Journal of Operational Research, Elsevier, vol. 122(1), pages 161-165, April.
    3. Mishra, S.K., 2006. "Mond-Weir type second order symmetric duality in non-differentiable minimax mixed integer programming problems," European Journal of Operational Research, Elsevier, vol. 170(2), pages 355-362, April.
    4. S. Gupta & N. Kailey, 2013. "Second-order multiobjective symmetric duality involving cone-bonvex functions," Journal of Global Optimization, Springer, vol. 55(1), pages 125-140, January.
    5. Chandra, Suresh & Abha, 2000. "Technical note on symmetric duality in multiobjective programming: Some remarks on recent results," European Journal of Operational Research, Elsevier, vol. 124(3), pages 651-654, August.
    6. Ahmad, I. & Husain, Z., 2010. "On multiobjective second order symmetric duality with cone constraints," European Journal of Operational Research, Elsevier, vol. 204(3), pages 402-409, August.
    7. Kumar, V. & Husain, I. & Chandra, S., 1995. "Symmetric duality for minimax nonlinear mixed integer programming," European Journal of Operational Research, Elsevier, vol. 80(2), pages 425-430, January.
    8. Mishra, S.K. & Wang, S.Y. & Lai, K.K., 2009. "Symmetric duality for minimax mixed integer programming problems with pseudo-invexity," European Journal of Operational Research, Elsevier, vol. 198(1), pages 37-42, October.
    9. Chen, Xiuhong, 2004. "Minimax and symmetric duality for a class of multiobjective variational mixed integer programming problems," European Journal of Operational Research, Elsevier, vol. 154(1), pages 71-83, April.
    10. Kim, Do Sang & Song, Young Ran, 2001. "Minimax and symmetric duality for nonlinear multiobjective mixed integer programming," European Journal of Operational Research, Elsevier, vol. 128(2), pages 435-446, January.
    11. Chen, Xiuhong & Yang, Jiangyu, 2007. "Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity," European Journal of Operational Research, Elsevier, vol. 181(1), pages 76-85, August.
    12. Chandra, S. & Kumar, V., 1998. "A note on pseudo-invexity and symmetric duality," European Journal of Operational Research, Elsevier, vol. 105(3), pages 626-629, March.
    13. Khurana, Seema, 2005. "Symmetric duality in multiobjective programming involving generalized cone-invex functions," European Journal of Operational Research, Elsevier, vol. 165(3), pages 592-597, September.
    14. Ahmad, I. & Sharma, Sarita, 2008. "Symmetric duality for multiobjective fractional variational problems involving cones," European Journal of Operational Research, Elsevier, vol. 188(3), pages 695-704, August.
    15. Mishra, S. K. & Wang, S. Y., 2005. "Second order symmetric duality for nonlinear multiobjective mixed integer programming," European Journal of Operational Research, Elsevier, vol. 161(3), pages 673-682, March.
    16. Yang, X.M. & Yang, X.Q. & Teo, K.L., 2006. "Converse duality in nonlinear programming with cone constraints," European Journal of Operational Research, Elsevier, vol. 170(2), pages 350-354, April.
    17. Kim, Do Sang & Yun, Ye Boon & Lee, Won Jung, 1998. "Multiobjective symmetric duality with cone constraints," European Journal of Operational Research, Elsevier, vol. 107(3), pages 686-691, June.
    18. Mishra, S. K., 2000. "Second order symmetric duality in mathematical programming with F-convexity," European Journal of Operational Research, Elsevier, vol. 127(3), pages 507-518, December.
    19. Mishra, S. K., 2000. "Multiobjective second order symmetric duality with cone constraints," European Journal of Operational Research, Elsevier, vol. 126(3), pages 675-682, November.
    20. Ahmad, I. & Husain, Z., 2007. "Minimax mixed integer symmetric duality for multiobjective variational problems," European Journal of Operational Research, Elsevier, vol. 177(1), pages 71-82, February.
    21. Kim, Moon Hee & Kim, Do Sang, 2008. "Non-differentiable symmetric duality for multiobjective programming with cone constraints," European Journal of Operational Research, Elsevier, vol. 188(3), pages 652-661, August.
    22. Mishra, S.K. & Wang, S.Y. & Lai, K.K. & Yang, F.M., 2007. "Mixed symmetric duality in non-differentiable multiobjective mathematical programming," European Journal of Operational Research, Elsevier, vol. 181(1), pages 1-9, August.

    More about this item

    Statistics

    Access and download statistics

    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:inm:oropre:v:21:y:1973:i:1:p:1-9. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.