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An axiomatic approach to approximate solutions in vector optimization

Author

Listed:
  • Miglierina Enrico

    () (Department of Economics, University of Insubria, Italy)

  • Molho Elena

    () (Department of Management Sciences, University of Pavia)

  • Patrone Fioravante

    () (Department of Mathematics, University of Genoa, Italy)

  • Steff H. Tijs

    () (Department of Econometrics, University of Tilburg, Netherlands)

Abstract

In vector optimization many notions of approximate solution have been proposed in the literature. In this paper an axiomatic approach is introduced in order to study the approximate solution map of a vector optimization problem in the image space. An impossibility result is proved in the sense that, whenever all of the axioms are satisfied, either the set of the approximate solutions is a subset of the exact solution of the problem (the weakly efficient frontier), or it coincides with the whole admissible set. Moreover, the geometry of the approximate solution map is studied in the special case of polyhedral ordering cones generated by a base of R. Finally, we study the shape of the approximate solution map under the assumption of weak approximation consistency.

Suggested Citation

  • Miglierina Enrico & Molho Elena & Patrone Fioravante & Steff H. Tijs, 2005. "An axiomatic approach to approximate solutions in vector optimization," Economics and Quantitative Methods qf0507, Department of Economics, University of Insubria.
    • Handle: RePEc:ins:quaeco:qf0507
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    File URL: http://eco.uninsubria.it/dipeco/Quaderni/files/QF2005_10.pdf
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    References listed on IDEAS

    as
    1. Norde, Henk & Patrone, Fioravante & Tijs, Stef, 2000. "Characterizing properties of approximate solutions for optimization problems," Mathematical Social Sciences, Elsevier, vol. 40(3), pages 297-311, November.
    2. Graziano Pieri & Fioravante Patrone & Anna Torre & Stef Tijs, 1998. "On consistent solutions for strategic games," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(2), pages 191-200.
    3. Peleg, Bezalel & Tijs, Stef, 1996. "The Consistency Principle for Games in Strategic Forms," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 13-34.
    4. Patrone, F. & Pieri, G. & Tijs, S.H. & Torre, A., 1998. "On consistent solutions for strategic games," Other publications TiSEM cc482511-a079-4067-88b2-0, Tilburg University, School of Economics and Management.
    5. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    6. repec:spr:compst:v:58:y:2003:i:3:p:375-385 is not listed on IDEAS
    7. E. Miglierina & E. Molho, 2003. "Well-posedness and convexity in vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(3), pages 375-385, December.
    Full references (including those not matched with items on IDEAS)

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    Keywords

    vector optimization; approximate solution; axiomatization.;

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