Journal of Optimization Theory and Applications

, Volume 48, Issue 1, pp 141–161 | Cite as

One-parameter families of optimization problems: Equality constraints

  • H. Th. Jongen
  • P. Jonker
  • F. Twilt
Contributed Papers


In this paper, we introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention on problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.

Key Words

Parametric optimization generalized critical points critical points Morse index quadratic index linear index 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    McCormick, G. P.,Optimality Criteria in Nonlinear Programming, SIAM-AMS Proceedings, Vol. 9, pp. 27–38, 1976.Google Scholar
  2. 2.
    Hettich, R., andJongen, H. Th.,On First-Order and Second-Order Conditions for Local Optima for Optimization Problems in Finite Dimensions, Methods of Operations Research, Vol. 23, pp. 82–97, 1977.Google Scholar
  3. 3.
    Milnor, J.,Morse Theory, Annals of Mathematic Studies, Study 51, Princeton University Press, Princeton, New Jersey, 1963.Google Scholar
  4. 4.
    Marcus, M., andMinc, H.,A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, Massachusetts, 1964.Google Scholar
  5. 5.
    Kojima, M.,Strongly Stable Stationary Solutions in Nonlinear Programs, Analysis and Computation of Fixed Points, Edited by S. M. Robinson, Academic Press, New York, New York, 1980.Google Scholar
  6. 6.
    Braess, D.,Morse Theorie für Berandete Mannigfaltigkeiten, Mathematische Annalen, Vol. 208, pp. 133–148, 1974.Google Scholar
  7. 7.
    Jongen, H. Th., Jonker, P. andTwilt, F.,Nonlinear Optimization in ℝ n,I: Morse Theory, Chebyshev Approximation, Peter Lang Verlag, Frankfurt a.M., Germany 1983.Google Scholar
  8. 8.
    Hirsch, M. W.,Differential Topology, Springer Verlag, Berlin, Germany, 1976.Google Scholar
  9. 9.
    Jongen, H. Th., Jonker, P., andTwilt, F.,Nonlinear Optimization in ℝ n,II: Transversality, Flows, Parametric Aspects (to appear).Google Scholar
  10. 10.
    Lu, Y. C.,Singularity Theory and an Introduction to Catastrophe Theory, Universitext, Springer Verlag, Berlin, Germany, 1976.Google Scholar
  11. 11.
    Jongen, H. Th., Jonker, P., andTwilt, F.,On One-Parameter Families of Sets Defined by (In) Equality Constraints, Nieuw Archief voor Wiskunde, Vol. 30, pp. 307–322, 1982.Google Scholar
  12. 12.
    Jongen, H. Th., Jonker, P., andTwilt, F.,On Deformation in Optimization, Methods of Operations Research, Vol. 37, pp. 171–184, 1980.Google Scholar
  13. 13.
    Jongen, H. Th., Jonker, P., andTwilt, F.,Critical Sets in Parametric Optimization, Mathematical Programming (to appear).Google Scholar
  14. 14.
    Kojima, M., andHirabayashi, R.,Some Results on the Strong Stability in Nonlinear Programs, Tokyo Institute of Technology, Department of Management Science and Engineering, Technical Report No. 4, 1980.Google Scholar
  15. 15.
    Kojima, M., andHirabayashi, R.,Continuous Deformation of Nonlinear Programs, Mathematical Programming Study, Vol. 21, pp. 150–198, 1984.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • H. Th. Jongen
    • 1
    • 2
  • P. Jonker
    • 1
  • F. Twilt
    • 1
  1. 1.Department of Applied MathematicsTwente University of TechnologyEnschedeThe Netherlands
  2. 2.University of HamburgHamburgGermany

Personalised recommendations