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On the choice of parameters for the weighting method in vector optimization

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An Erratum to this article was published on 16 June 2007

An Erratum to this article was published on 16 June 2007

Abstract

We present a geometrical interpretation of the weighting method for constrained (finite dimensional) vector optimization. This approach is based on rigid movements which separate the image set from the negative of the ordering cone. We study conditions on the existence of such translations in terms of the boundedness of the scalar problems produced by the weighting method. Finally, using recession cones, we obtain the main result of our work: a sufficient condition under which weighting vectors yield solvable scalar problems.

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References

  1. Aliprantis, C.D., Florenzano, M., Martins da Rocha, V.F., Tourky, R.: Equilibrium analysis in financial markets with countably many securities. J. Math. Econ (to appear)

  2. Aliprantis C.D., Florenzano M. and Tourky R. (2004). General equilibrium analysis in ordered topological vector spaces. J. Math. Econ. 40(3–4): 247–269

    Article  MATH  MathSciNet  Google Scholar 

  3. Avriel M. (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall, New Jersey

    MATH  Google Scholar 

  4. Bertsekas D. (1995). Nonlinear Programming. Athena Scientific, Belmont

    MATH  Google Scholar 

  5. Bertsekas D., Nedić A. and Ozdaglar A.E. (2003). Convex Analysis and Optimization. Athena Scientific, Belmont

    MATH  Google Scholar 

  6. Gass S. and Saaty T. (1955). The computational algorithm for the parametric objective function. Naval Res. Logist. Q. 2: 618–630

    MathSciNet  Google Scholar 

  7. Hamel A.H., Heyde F., Lohne A., Tammer C. and Winkler K. (2004). Closing the duality gap in linear vector optimization. J. Convex Analysis 11: 163–178

    MATH  MathSciNet  Google Scholar 

  8. Jahn J. (1984). Scalarization in vector optimization. Math. Program. 29: 203–218

    Article  MATH  MathSciNet  Google Scholar 

  9. Luc D.T. (1987). Scalarization of vector optimization problems. J. Optim. Theory and Appl. 55(1): 85–102

    Article  MATH  MathSciNet  Google Scholar 

  10. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economy and Mathematical Systems, vol. 319. Springer, Berlin Heidelberg New York (1989)

  11. Miettinen K.M. (1999). Nonlinear Multiobjective Optimization. Kluwer, Norwell

    MATH  Google Scholar 

  12. Pascoletti A. and Serafini P. (1984). Scalarizing vector optimization problems. J. Optim. Theory Appl. 42(4): 499–524

    Article  MATH  MathSciNet  Google Scholar 

  13. Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, New Jersey

    MATH  Google Scholar 

  14. Sawaragi Y., Nakayana H. and Tanino T. (1985). Theory of Multiobjective Optimization. Academic, Orlando

    MATH  Google Scholar 

  15. Zadeh L. (1963). Optimality and non-scalar valued performance criteria. IEEE Trans. Autom. Control 8: 59–60

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculdade de Administração e Ciências Contábeis, FACC–UFRJ, Pasteur 250, Rio de Janeiro, RJ, CEP 22290-240, Brazil

    L. M. Grañax Drummond

  2. Programa de Engenharia de Sistemas de Computação, COPPE–UFRJ, CP 68511, Rio de Janeiro, RJ, CEP 21945-970, Brazil

    N. Maculan

  3. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil

    B. F. Svaiter

Authors
  1. L. M. Grañax Drummond
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  2. N. Maculan
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  3. B. F. Svaiter
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Corresponding author

Correspondence to L. M. Grañax Drummond.

Additional information

Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.

B.F. Svaiter was partially supported by CNPq Grants 302748/2002-4, 476842/2003-4 and by PRONEX-Optimization (FAPERJ/CNPq).

L.M. Graña Drummond was partially supported by CNPq Grant 476842/2003-4 and by PRONEX-Optimization (FAPERJ/CNPq).

An erratum to this article can be found at http://dx.doi.org/10.1007/s10107-007-0108-6

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Drummond, L.M.G., Maculan, N. & Svaiter, B.F. On the choice of parameters for the weighting method in vector optimization. Math. Program. 111, 201–216 (2008). https://doi.org/10.1007/s10107-006-0071-7

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  • DOI: https://doi.org/10.1007/s10107-006-0071-7

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