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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
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    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.

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