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Journal of Optimization Theory and Applications

, Volume 107, Issue 2, pp 245–260 | Cite as

Optimization of the Norm of a Vector-Valued DC Function and Applications

  • R. Blanquero
  • E. Carrizosa
Article

Abstract

In this paper, we show that a DC representation can be obtained explicitly for the composition of a gauge with a DC mapping, so that the optimization of certain functions involving terms of this kind can be made by using standard DC optimization techniques. Applications to facility location theory and multiple-criteria decision making are presented.

global optimization difference of convex functions location theory multiple-criteria decision making 

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References

  1. 1.
    Horst, R., and Thoai, N. V., DC Programming: Overview, Journal of Optimization Theory and Applications, Vol. 103, pp. 1–43, 1999.Google Scholar
  2. 2.
    Horst, R., and Tuy, H., Global Optimization: Deterministic Approaches, Springer Verlag, Berlin, Germany, 1996.Google Scholar
  3. 3.
    Tuy, H., DC Optimization: Theory, Methods, and Algorithms, Handbook of Global Optimization, Edited by R. Horst and P. M. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, pp. 149–216, 1995.Google Scholar
  4. 4.
    Hartman, P., On Functions Representable as a Difference of Convex Functions, Pacific Journal of Mathematics, Vol. 9, pp. 707–713, 1959.Google Scholar
  5. 5.
    Tuy, H., Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, Holland, 1998.Google Scholar
  6. 6.
    Michelot, C., The Mathematics of Continuous Location, Studies in Locational Analysis, Vol. 5, pp. 59–83, 1993.Google Scholar
  7. 7.
    Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  8. 8.
    Aneja, Y. P., and Parlar, M., Algorithms for Weber Facility Location in the Presence of Forbidden Regions and/or Barriers to Travel, Transportation Science, Vol. 28, pp. 70–76, 1994.Google Scholar
  9. 9.
    Brimberg, J., and Wesolowsky, G. O., The Rectilinear Distance Minimum Problem with Minimum Distance Constraints, Location Science, Vol. 3, pp. 203–215, 1995.Google Scholar
  10. 10.
    Hamacher, H. W., and Nickel, S., Restricted Planar Location Problems and Applications, Naval Research Logistics, Vol. 42, pp. 967–992, 1995.Google Scholar
  11. 11.
    Bittner, L., Some Representation Theorems for Functions and Sets and Their Application to Nonlinear Programming, Numerische Mathematik, Vol. 16, pp. 32–51, 1970.Google Scholar
  12. 12.
    Hiriart-Urruty, J. B., Generalized Differentiability, Duality, and Optimization for Problems Dealing with Differences of Convex Functions, Convexity and Duality in Optimization, Edited by J. Ponstein, Springer Verlag, Berlin, Germany, pp. 37–69, 1985.Google Scholar
  13. 13.
    Blanquero, R., and Carrizosa, E., On Covering Methods for DC Optimization, To appear in Journal of Global Optimization.Google Scholar
  14. 14.
    Hiriart-Urruty. J. B., Convex Analysis and Minimization Algorithms, I, Springer Verlag, Berlin, Germany, 1993.Google Scholar
  15. 15.
    Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, NY, 1993.Google Scholar
  16. 16.
    Zeleny, M., Compromise Programming, Multiple-Criteria Decision Making, Edited by J. L. Cochrane, and M. Zeleny, University of South Carolina Press, Columbia, South Carolina, pp. 262–301, 1973.Google Scholar
  17. 17.
    Zeleny, M., A Concept of Compromise Solutions and the Method of the Displaced Ideal, Computers and Operations Research, Vol. 1, pp. 479–496, 1974.Google Scholar
  18. 18.
    Zeleny, M., Multiple-Criteria Decision Making, Springer Verlag, Berlin, Germany, 1976.Google Scholar
  19. 19.
    Romero, C., Handbook of Critical Issues in Goal Programming, Pergamon Press, Oxford, England, 1991.Google Scholar
  20. 20.
    Saber, H. M., and Ravindran, A., Nonlinear Goal Programming Theory and Practice: A Survey, Computers and Operations Research, Vol. 20, pp. 275–291, 1993.Google Scholar
  21. 21.
    Saber, H. M., and Ravindran, A., A Partitioning Gradient Based (PGB) Algorithm for Solving Nonlinear Goal Programming Problems, Computers and Operations Research, Vol. 23, pp. 141–152, 1995.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • R. Blanquero
    • 1
  • E. Carrizosa
    • 1
  1. 1.Facultad de MatemáticasUniversidad de SevillaSevillaSpain

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