Abstract

In this paper the quasiconvex minimization problem is included in a problem defined by sets (instead of functions). Lagrangian conditions for both problems are then studied and related. Lagrangian conditions for the standard convex minimization problem are usually defined in terms of subdifferentials. Lagrangian conditions are defined here in terms of those functionals on the set of (quasi) convex functions which satisfy certain axioms. Constraint qualifications valid at all points in the feasible set are also considered, in connection with questions of redundancy.

Author information

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J. M. Gutiérrez

Facultad de Economía y Empresa, Universidad de Salamanca, 37008 Salamanca, Spain

Keywords

  • Quasiconvex optimization
  • Lagrangian conditions
  • polarity
  • constraint qualifications

References

  1. G. Köthe: Topological Vector Spaces I, Springer-Verlag, Berlin, 1969.Google Scholar
  2. R. Holmes: A Course on Optimization and Best Approximation, Springer-Verlag, New York, 1972.https://doi.org/10.1007/BFb0059450Google Scholar
  3. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (N.J.), 1970.Google Scholar
  4. G. J. O. Jameson: The duality of pairs of wedges, Proceedings of the London Mathematical Society (3), Vol. 24 (1972) 531-547.https://doi.org/10.1112/plms/s3-24.3.531Google Scholar
  5. M. Avriel, W. E. Diewert, S. Schaible and I. Zang: Generalized Concavity, Plenum Press, New York & London, 1988.https://doi.org/10.1007/978-1-4684-7600-2Google Scholar
  6. J. M. Gutiérrez: Infragradientes y direcciones de decrecimiento, Revista de la Real Academia de Ciencias, de Madrid, Vol. LXXVIII (4) (1984) 523-532.Google Scholar
  7. J. P. Penot, M. Volle: On quasi-convex duality, Mathematics of Operations Research 15 (1990) 597-625.https://doi.org/10.1287/moor.15.4.597Google Scholar
  8. H. Wolkowicz: Geometry of optimality conditions and constraint qualifications: the convex case, Mathematical Programming 19 (1980) 32-60.https://doi.org/10.1007/BF01581627Google Scholar
  9. G. J. O. Jameson: Topology and Normed Spaces, Chapman & Hall, London, 1974.Google Scholar

Suggested citation

J. M. Gutiérrez. “A Generalization of the Quasiconvex Optimization Problem.” Journal of Convex Analysis 4 (1997), No. 2, 281–287. https://doi.org/10.68381/jca04014

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Published by Heldermann Verlag, 1997. Rights now held by Banach Press LLC.