Lagrange Multipliers and saddle points in multiobjective programming

  • Z. F. Li
  • S. Y. Wang
Contributed Papers

Abstract

In this paper, we present several conditions for the existence of a Lagrange multiplier or a weak saddle point in multiobjective optimization. Relations between a Lagrange multiplier and a weak saddle point are established. A sufficient condition is also given for the equivalence of the Benson proper efficiency and the Borwein proper efficiency.

Key Words

Multiobjective programming Lagrange multipliers weak saddle points 

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References

  1. 1.
    Weir, T., Mond, B., andCraven, B. D.,Weak Minimization and Duality, Numerical Functional Analysis and Optimization, Vol. 9, pp. 181–191, 1987.Google Scholar
  2. 2.
    Weir, T., Mond, B., andCraven, B. D.,On Duality for Weakly Minimized Vector-Valued Optimization Problems, Optimization, Vol. 17, pp. 711–721, 1986.Google Scholar
  3. 3.
    Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, New York, New York, 1985.Google Scholar
  4. 4.
    Wang, S. Y.,Theory of the Conjugate Duality in Multiobjective Optimization, Journal of Systems Science and Mathematical Sciences, Vol. 4, pp. 303–312, 1984.Google Scholar
  5. 5.
    Vogel, W., Ein Maximum Prinzip für Vectoroptimierungs—Aufgaben, Operations Research Verfahren, Vol. 19, pp. 161–175, 1974.Google Scholar
  6. 6.
    Jeyakumar, V.,A Generalization of a Minimax Theorem of Fan via a Theorem of the Alternative, Journal of Optimization Theory and Applications, Vol. 48, pp. 525–533, 1986.Google Scholar
  7. 7.
    Jahn, J.,Scalarization in Multiobjective Optimization, Mathematics of Multiobjective Optimization, Edited by P. Serafini, Springer Verlag, Berlin, Germany, pp. 45–88, 1985.Google Scholar
  8. 8.
    Bazaraa, M. S., andShetty, C. M.,Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, New York, 1979.Google Scholar
  9. 9.
    Graven, B. D.,Quasimin and Quasisaddlepoint for Vector Optimization, Numerical Functional Analysis and Optimization, Vol. 11, pp. 45–54, 1990.Google Scholar
  10. 10.
    Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377 1974.Google Scholar
  11. 11.
    Yu, P. L.,Multiple Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, New York, 1985.Google Scholar
  12. 12.
    Tanaka, T.,Some Minimax Problems of Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 59, pp. 505–524, 1988.Google Scholar
  13. 13.
    Tanaka, T.,A Characterization of Generalized Saddle Points for Vector-Valued Functions via Scalarization, Nihonkai Mathematical Journal, Vol. 1, pp. 209–227, 1990.Google Scholar
  14. 14.
    Corley, H. W.,Existence and Lagrangian Duality for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 54, pp. 489–501, 1987.Google Scholar
  15. 15.
    Henig, M. I.,Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 36, pp. 387–407, 1982.Google Scholar
  16. 16.
    Nakayama, H.,Geometric Consideration of Duality in Vector Optimization, Journal of Optimization Theory and Applications, Vol. 44, pp. 625–655, 1984.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Z. F. Li
    • 1
  • S. Y. Wang
    • 2
  1. 1.Mathematics DepartmentUniversity of Inner MongoliaHohhot, Inner MongoliaChina
  2. 2.Institute of Systems ScienceAcademic SinicaBeijingChina

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