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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Horst, R., and Thoai, N. V., DC Programming: Overview, Journal of Optimization Theory and Applications, Vol. 103, pp. 1–43, 1999.
Horst, R., and Tuy, H., Global Optimization: Deterministic Approaches, Springer Verlag, Berlin, Germany, 1996.
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.
Hartman, P., On Functions Representable as a Difference of Convex Functions, Pacific Journal of Mathematics, Vol. 9, pp. 707–713, 1959.
Tuy, H., Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, Holland, 1998.
Michelot, C., The Mathematics of Continuous Location, Studies in Locational Analysis, Vol. 5, pp. 59–83, 1993.
Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
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.
Brimberg, J., and Wesolowsky, G. O., The Rectilinear Distance Minimum Problem with Minimum Distance Constraints, Location Science, Vol. 3, pp. 203–215, 1995.
Hamacher, H. W., and Nickel, S., Restricted Planar Location Problems and Applications, Naval Research Logistics, Vol. 42, pp. 967–992, 1995.
Bittner, L., Some Representation Theorems for Functions and Sets and Their Application to Nonlinear Programming, Numerische Mathematik, Vol. 16, pp. 32–51, 1970.
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.
Blanquero, R., and Carrizosa, E., On Covering Methods for DC Optimization, To appear in Journal of Global Optimization.
Hiriart-Urruty. J. B., Convex Analysis and Minimization Algorithms, I, Springer Verlag, Berlin, Germany, 1993.
Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, NY, 1993.
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.
Zeleny, M., A Concept of Compromise Solutions and the Method of the Displaced Ideal, Computers and Operations Research, Vol. 1, pp. 479–496, 1974.
Zeleny, M., Multiple-Criteria Decision Making, Springer Verlag, Berlin, Germany, 1976.
Romero, C., Handbook of Critical Issues in Goal Programming, Pergamon Press, Oxford, England, 1991.
Saber, H. M., and Ravindran, A., Nonlinear Goal Programming Theory and Practice: A Survey, Computers and Operations Research, Vol. 20, pp. 275–291, 1993.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Blanquero, R., Carrizosa, E. Optimization of the Norm of a Vector-Valued DC Function and Applications. Journal of Optimization Theory and Applications 107, 245–260 (2000). https://doi.org/10.1023/A:1026433520314
Issue Date:
DOI: https://doi.org/10.1023/A:1026433520314