Advertisement

Journal of Optimization Theory and Applications

, Volume 98, Issue 3, pp 651–661 | Cite as

Invex Functions and Generalized Convexity in Multiobjective Programming

  • R. Osuna-Gómez
  • A. Rufián-Lizana
  • P. Ruíz-Canales
Article

Abstract

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.

Invex functions KT-invex problems Kuhn–Tucker optimality conditions weakly efficient points 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Martin, D. H., The Essence of Invexity, Journal of Optimization Theory and Applications, Vol. 47, pp. 65–76, 1985.Google Scholar
  2. 2.
    Hanson, M. A., On Sufficiency of Kuhn-Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 30, pp. 545–550, 1981.Google Scholar
  3. 3.
    Mangasarian, O. L., Nonlinear Programming, McGraw Hill Book Company, New York, New York, 1969.Google Scholar
  4. 4.
    Craven, B. D., Invex Functions and Constrained Local Minima, Bulletin of the Australian Mathematical Society, Vol. 24, pp. 357–366, 1981.Google Scholar
  5. 5.
    Ben-Israel, B., and Mond, B., What Is Invexity?, Journal of the Australian Mathematical Society, Vol. 28B, pp. 1–9, 1986.Google Scholar
  6. 6.
    Jeyakumar, Y., Strong and Weak Invexity in Mathematical Programming, Collection: Methods of Operations Research, Vol. 55, pp. 109–125, 1980.Google Scholar
  7. 7.
    Hanson, M. A., and Mond, B., Further Generalizations of Convexity in Mathematical Programming, Journal of Information and Optimization Science, Vol. 3, pp. 25–32, 1982.Google Scholar
  8. 8.
    RuÍz, P., and RufiÁn, A., A Characterization of Weakly Efficient Points, Mathematical Programming, Vol. 68, pp. 205–212, 1995.Google Scholar
  9. 9.
    Craven, B. D., Lagrangian Conditions and Quasiduality, Bulletin of the Australian Mathematical Society, Vol. 16, pp. 325–339, 1977.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • R. Osuna-Gómez
    • 1
  • A. Rufián-Lizana
    • 1
  • P. Ruíz-Canales
    • 1
  1. 1.Departamento de Estadística e Investigación Operativa, Facultad de MatemáticasUniversidad de SevillaSevillaSpain

Personalised recommendations