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Mixed Type Duality in Multiobjective Fractional Programming Under Generalized ρ-univex Function

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Journal of Mathematical Modelling and Algorithms in Operations Research

Abstract

In this paper, three approaches given by Dinklebaeh (Manag Sci 13(7):492–498, 1967) and Jagannathan (Z Oper Res 17:618–630, 1968) for both primal and mixed type dual of a non differentiable multiobjective fractional programming problem in which the numerator of objective function contains square root of positive semi definite quadratic form are introduced. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterizations technique is used to established duality results under generalized ρ-univexity assumption.

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References

  1. Ahmad, I., Husain, Z.: Duality in nondifferentiable minimax fractional programming with generalized convexity. Appl. Math. Comput. 176, 545–551 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahmad, I., Husain, Z.: Optimality condition and duality in nondifferentiable minimax fractional programming with generalized convexity. J. Optim. Theory Appl. 129, 255–275 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahmad, I., Sharma, S.: Second order duality in nondifferentiable multiobjective programming. Numer. Funct. Anal. Optim. 28, 975–988 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahmad, I., Husain, Z., Al-Homidan, S.: Second order duality in non differentiable fractional programming. Nonlinear Anal. Real. World Appl. 12, 1103–1110 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ahmad, I., Gupta, S.K., Kailey, N., Agarwal, R.P.: Duality in nondifferentiable minimax fractional programming with B-(p,r)-invexity. J. Ineq. Appl. 2011(75), 1–15 (2011)

    MathSciNet  Google Scholar 

  6. Bector, C.R.: Duality in nonlinear fractional programming. Oper. Res. 17, 183–193 (1973)

    MathSciNet  MATH  Google Scholar 

  7. Bector, C.R., Chandra, S.: First and Second order duality for a class of non differentiable fractional programming. J. Inf. Optim. Sci. 7, 335–348 (1986)

    MathSciNet  MATH  Google Scholar 

  8. Bector, C.R., Chandra, S., Abha: On mixed symmetric duality in mathematical programming. J. Math. Anal. Appl. 259, 346–356 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bector, C.R., Suneja, S.K., Gupta, S.: Univex functions and univex nonlinear programming. In: Proceeding of the Administrative Science Association of Canada, pp. 115–124 (1992)

  10. Bector, C.R., Chandra, S., Husain, I: Optimality conditions and subdifferentiable multiobjective fractional programming. J. Optim. Theory Appl. 79, 105–125 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Craven, B.D.: Fractional programming. In: Sig ma series in Applied Mathematics, vol. 4, pp. 145–152. Helder man Verlag, Berlin, Germany (1998)

    Google Scholar 

  12. Dinklebaeh, W.: On nonlinear fractional programming. Manag. Sci. 13(7), 492–498 (1967)

    Article  Google Scholar 

  13. Geoffrion, M.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gulati, T.R., Ahmad, I., Agarwal, D.: Sufficiency and duality in multiobjective programming under generalized type 1 function. J. Optim. Theory Appl. 135, 411–427 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hanson, M.A.: On sufficiency of Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jagannathan, R.: Duality for nonlinear fractional programs. Z. Oper. Res. 17, 618–630 (1968)

    Google Scholar 

  17. Jayaswal, A.: Nondifferentiable minimax fractional programming with generalized ∝-univexity. J. Comput. Appl. Math. 214, 121–135 (2008)

    Article  MathSciNet  Google Scholar 

  18. Kim, D.S., Kim, S.J., Kim, M.H.: Optimality and duality for a class of non differentiable multiobjective fractional programming problem. J. Optim. Theory Appl. 129, 131–146 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, D.S., Lee, Y.J., Bae, K.D.: Duality in nondifferentiable multiobjective fractional programs involving cones. Taiwan. J. Math. 13, 1811–1821 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Liang, Z., Huang, H.X., Pardalos, P.M.: Optimality conditions and duality for a class of nonlinear fractional programming problem. J. Optim. Theory Appl. 110, 611–619 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liang, Z., Huang, H.X., Pardalos, P.M.: Efficiency condition and duality for a class of multiobjective fractional programming problem. J. Glob. Optim. 27, 447–471 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mangasarian, O.L.: Nonlinear Programming. Mc Graw-Hill, New York (1969)

    MATH  Google Scholar 

  23. Mond, B.: A class of nondifferentiable fractional programming problem. Z. Angew Math. Mech. 58, 337–341 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mond, B., Weir, T.: Duality for fractional programming with generalized convexity condition. J. Inf. Optim. Sci. 3, 105–124 (1982)

    MathSciNet  MATH  Google Scholar 

  25. Mishra, S.K.: On multiple-objective optimization with generalized univexity. J. Math. Anal. Appl. 224, 131–148 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mishra, M.K., Wang, S.Y., Lai, K.K.: Role of ∝-pseudo-univex functions in vector variational–like inequality problems. J. Syst. Sci. & Complexity 20, 344–349 (2007)

    Article  MathSciNet  Google Scholar 

  27. Mishra, S.K., Rautela, J.S., Pant, R.P.: Optimality and duality for multiple–objective optimization with generalized ∝-univex functions. International J. Operations Res. 5(3), 180–186 (2008)

    MathSciNet  Google Scholar 

  28. Osuna-Gomez, R., Rufian-Lizana, A., Ruiz-Canales, P.: Multiobjective fractional programming with generalized invexity. TOP 8(1), 97–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Santos, L.B., Osuna-Gomez, R., Rojas-Medar, A.: Non smooth multiobjective fractional programming with generalized convexity. Rev. Integr. Temas. Mat. 26(1), 1–12 (2008)

    MathSciNet  Google Scholar 

  30. Schaible, S.: In: Horst, R., Pardalos, P.M. (eds.) Fractional Programming, Handbook of Global Optimization, pp. 495–608. Kluwer Academic Publishers, Dordrecht (1995)

    Chapter  Google Scholar 

  31. Schaible, S., Ibaraki, T.: Fractional programming. Eur. J. Oper. Res. 12, 325–338 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stancu-Minasian, I.M.: Fractional Programming; Theory, Method and Application. Kluwer Academic Publishers, Dordrecht (1997)

    Book  Google Scholar 

  33. Xu, Z.: Mixed type duality in multiobjective programming problems. J. Math. Anal. Appl. 198, 621–635 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhou, H., Wang, Y.: Optimality and mixed duality for nonsmooth multi-objective fractional programming. Pure Math. Appl. 14(3), 263–274 (2003)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, In front of Infocity, Bhubaneswar, 751024, Odisha, India

    Arun Kumar Tripathy

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  1. Arun Kumar Tripathy
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Tripathy, A.K. Mixed Type Duality in Multiobjective Fractional Programming Under Generalized ρ-univex Function. J Math Model Algor 13, 331–340 (2014). https://doi.org/10.1007/s10852-013-9241-9

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