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Duality in Multi-Objective Optimization

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Book cover Multicriteria Decision Making

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 21))

Abstract

Duality is an attractive topic in multi-objective optimization as well as in usual mathematical programming with a single objective function. However, there seems to be no unified approach to dualization in multi-objective optimization. One of the difficulties is in the fact that the efficient solution to multi-objective optimization is not necessarily unique, but in general becomes a set. The definition of infimum (or supremum) of a set with a partial order plays a key role in development of duality theory in multi-objective optimization. In this chapter, these notions will be considered from some geometric viewpoint.

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References

  1. Borwein, J. M. and J. W. Nieuwenhuis: Two Kinds of Normality in Vcector Optimization. Mathematical Programming 28(1984)185–191

    Article  Google Scholar 

  2. Corley, H. W.: Duality for Maximization with respect to Cones. J. Mathematical Analysis and Applications. 84(1981)560–568

    Article  Google Scholar 

  3. Gale, D., Kuhn, H. W. and A. W. Tucker: “Linear Programming and the Theory of Games”. In: T. C. Koopmans (ed.): The Theory of Linear Economics Models, Wiley, New York 1951, pp.317–329

    Google Scholar 

  4. Gros, C.: Generalization of Fenchel’s Duality Theorem for Convex Vector Optimization. European J. Operaionl Research 2(1978)368–376

    Article  Google Scholar 

  5. Isermann, H.: On Some Relations between a Dual Pair of Multiple Objective Linear Programs. Zeitschrift für Oper. Res 22(1978)33–41

    Google Scholar 

  6. Jahn, J.: Duality in Vector Optimization. Mathematical Programming 25(1983)343–353

    Article  Google Scholar 

  7. Kawasaki, H.: Conjugate Relations in Weak Subdifferentials of Relations. Math, of Oper. Res. 6(1981)593–607

    Article  Google Scholar 

  8. Kawasaki, H.: A Duality Theorem in Multi-objective Nonlinear Programming. Math. Oper. Res. 7(1982)95–110

    Article  Google Scholar 

  9. Luc, D. T.: About Duality and Alternative in Multiobjective Optimization. J. Optim. Theory Appl. 53(1987)303–307

    Article  Google Scholar 

  10. MacLinden, L.: Duality Theorems and Theorems of the Alternative. Proc. of AMS 53(1975)172–175

    Article  Google Scholar 

  11. Mond, B. and T. Weir:“Symmetric Duality for Nonlinear Multiobjective Programming”. In: Kumar, S. (ed.): Recent Developments in Mathematical Programming, Gordon and Breach Science, Philadelphia 1991, pp.137–153

    Google Scholar 

  12. Nakayama, H.: Geometric Consideration on Duality in Vector Optimization. J. Optimi. Theory Appl. 44(1984)625–655

    Article  Google Scholar 

  13. Nakayama, H.: Some Remarks on Dualization in Vecotr Optimization. J. Multicriteria Decision Analysis 5(1996)218–225

    Article  Google Scholar 

  14. Nieuwenhuis, J. W.: Supremal Points and Generalized Duality. Math. Oper. Stat., Ser. Optim. 11-1(1980)41–59

    Google Scholar 

  15. Ponstein, J.: Approaches to the Theory of Optimization, Cambridge Univ. Press, London 1980

    Book  Google Scholar 

  16. Ritter, K.: Optimization Theory in Linear Spaces III. Math. Annal. 184(1970)133–154

    Article  Google Scholar 

  17. Rockafellar, T.: Convex Analysis, Princeton Univ. Press, Princeton 1968

    Google Scholar 

  18. Sawaragi, Y., H. Nakayama and T. Tanino: Theory of Multiobjective Optimization, Academic Press, Orlando 1985

    Google Scholar 

  19. Van Slyke, R. M. and R. J. B. Wets: A Duality Theory for Abstract Mathematical Programs with Applications to Optimal Control Theory. J. Math. Anal. and Appl. 22(1968)679–706

    Article  Google Scholar 

  20. Tanino, T.: On Supremum of a Set in a Multi-dimensional Space. J. Math. Analysis and Applications 130(1988)386–397

    Article  Google Scholar 

  21. Tanino, T.: Conjugate Duality in Vector Optimization}. J. Math. Analysis and Applications 167(1992)84–97

    Article  Google Scholar 

  22. Tanino, T. and Y. Sawaragi: Duality Theory in Multiobjective Programming. J. Optim. Theory Appl. 27(1979)509–529

    Article  Google Scholar 

  23. Tanino, T. and Y. Sawaragi: Conjugate Maps and Duality in Multi-objective Optimization. J. Optim. Theory Appl. 31(1980)473–499

    Article  Google Scholar 

  24. Weir, T. and B. Mond: Symmetric and Self Duality for Multiple Objective Programming. Asia-Pacific J. Oper. Res. 5(1988)124–133

    Google Scholar 

  25. Yu, P. L.: Cone convexity, Cone Extreme Points and Nodominates Solutions. J. Optimi. theory and Appl. 14(1974)319–377

    Article  Google Scholar 

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Authors
  1. Hirotaka Nakayama
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Editors and Affiliations

  1. Fern Universität, Hagen, Germany

    Tomas Gal  & Thomas Hanne  & 

  2. University of Cape Town, Rep. of South Africa

    Theodor J. Stewart

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© 1999 Springer Science+Business Media New York

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Nakayama, H. (1999). Duality in Multi-Objective Optimization. In: Gal, T., Stewart, T.J., Hanne, T. (eds) Multicriteria Decision Making. International Series in Operations Research & Management Science, vol 21. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5025-9_3

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  • DOI: https://doi.org/10.1007/978-1-4615-5025-9_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7283-7

  • Online ISBN: 978-1-4615-5025-9

  • eBook Packages: Springer Book Archive

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