Advertisement

Optimization Letters

, Volume 5, Issue 1, pp 125–139 | Cite as

Nondifferentiable multiobjective second-order symmetric duality

  • S. K. Gupta
  • N. Kailey
Original Paper

Abstract

In this paper, a pair of Wolfe type second-order multiobjective symmetric dual programs involving nondifferentiable functions is formulated. Weak, strong and converse duality theorems are then established using the notion of second-order F-convexity assumptions. An example which is second-order F-convex but not convex is also illustrated. Further, special cases are discussed to show that this paper extends some known results of the literature.

Keywords

Nondifferentiable programming Multiobjective symmetric duality Duality theorems Support function Second-order F-convexity Efficient solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahmad I., Husain Z.: On nondifferentiable second-order symmetric duality in mathematical programming. Indian J. Pure Appl. Math. 35, 665–676 (2004)MATHMathSciNetGoogle Scholar
  2. 2.
    Ahmad I., Husain Z.: Multiobjective mixed symmetric duality involving cones. Comput. Math. Appl. 59, 319–326 (2010)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ahmad I., Husain Z.: On multiobjective second-order symmetric duality with cone constraints. Eur. J. Oper. Res. 204, 402–409 (2010)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bitran G.R.: Duality for nonlinear multi-criteria optimization problems. J. Optim. Theory Appl. 35, 367–401 (1981)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Corley H.W.: Duality theory for the matrix linear programming problem. J. Math. Anal. Appl. 104, 47–52 (1984)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Craven B.D.: Strong vector minimization and duality. ZAMM 60, 1–5 (1980)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dantzig G.B., Eisenberg E., Cottle R.W.: Symmetric dual non linear programming. Pac. J. Math. 15, 809–812 (1965)MATHMathSciNetGoogle Scholar
  9. 9.
    Dorn W.S.: A symmetric dual theorem for quadratic programs. J. Oper. Res. Soc. Jpn. 2, 93–97 (1960)Google Scholar
  10. 10.
    Geoffrion A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gulati T.R., Ahmad I., Husain I.: Second-order symmetric duality with generalized convexity. Opsearch 38, 210–222 (2001)MATHMathSciNetGoogle Scholar
  12. 12.
    Gulati T.R., Gulati T.R.: Mond-Weir type Second-order symmetric duality in multiobjective programming over cones. Appl. Math. Lett. 23, 466–471 (2010)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gulati T.R., Gupta S.K.: Wolfe type second-order symmetric duality in nondifferentiable programming. J. Math. Anal. Appl. 310, 247–253 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gulati T.R., Husain I., Ahmed A.: Multiobjective symmetric duality with invexity. Bull. Aust. Math. Soc. 56, 25–36 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gupta S.K., Kailey N.: A note on multiobjective second-order symmetric duality. Eur. J. Oper. Res. 201, 649–651 (2010)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kuhn H.W., Tucker A.W.: Nonlinear programming. In: Neyman , J. (eds) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability., pp. 481–492. University of California Press, Los Angeles (1951)Google Scholar
  17. 17.
    Mangasarian O.L.: Second and higher-order duality in nonlinear programming. J. Math. Anal. Appl. 51, 607–620 (1975)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Miettinen K.M.: Nonlinear multiobjective optimization. Kluwer, Boston (1999)MATHGoogle Scholar
  19. 19.
    Mond B.: A symmetric dual theorem for nonlinear programs. Q. J. Appl. Math. 23, 265–269 (1965)MATHMathSciNetGoogle Scholar
  20. 20.
    Mond B., Weir T.: Generalized concavity and duality. In: Schaible, S., Ziemba, W.T. (eds) Generalized Concavity in Optimization and Economics, pp. 263–279. Academic Press, New York (1981)Google Scholar
  21. 21.
    Mond, B., Weir, T.: Generalized convexity and higher-order duality. J. Math. Sci. 16–18, 74–94 (1981–1983)Google Scholar
  22. 22.
    Pardalos P.M., Siskos Y., Zopounidis C.: Advances in multicriteria analysis. Kluwer, Netherlands (1995)MATHGoogle Scholar
  23. 23.
    Pareto V.: Course d’ Economic Politique. Raye, Lausanne (1896)Google Scholar
  24. 24.
    Schecter M.: More on subgradient duality. J. Math. Anal. Appl. 71, 251–262 (1979)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Tanino T., Sawaragi Y.: Duality theory in multiobjective programming. J. Optim. Theory Appl. 27, 509–529 (1979)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wang S.: Second-order necessary and sufficient conditions in multiobjective programming. Numer. funct. Anal. Appl. 12, 237–252 (1991)MATHCrossRefGoogle Scholar
  27. 27.
    Yang X.M., Hou S.H.: Second-order symmetric duality in multiobjective programming. Appl. Math. Lett. 14, 587–592 (2001)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Yang X.M., Yang X.Q., Teo K.L.: Non-differentiable second-order symmteric duality in mathematical programming with F-convexity. Eur. J. Oper. Res. 144, 554–559 (2003)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Yang X.M., Yang X.Q., Teo K.L., Hou S.H.: Multiobjective second-order symmetric duality with F-convexity. Eur. J. Oper. Res. 165, 585–591 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology PatnaPatnaIndia
  2. 2.School of Mathematics and Computer ApplicationsThapar UniversityPatialaIndia

Personalised recommendations