Mathematical Programming

, Volume 34, Issue 3, pp 333–353 | Cite as

Critical sets in parametric optimization

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


We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a ‘generalized critical point’ (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set consisting of all g.c. points. Due to the parameter, the set is pieced together from one-dimensional manifolds. The points of can be divided into five (characteristic) types. The subset of ‘nondegenerate critical points’ (first type) is open and dense in (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along is presented. Finally, the Kuhn-Tucker subset of is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.

Key words

Generalized Critical Point Critical Point Linear Index Quadratic Index Kuhn-Tucker Set Mangasarian-Fromowitz Constraint Qualification Parametric Optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E.L. Allgower and K. Georg, “Predictor-corrector and simplicial methods for approximating fixed points and zero points of nonlinear mappings“, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical programming, the state of the art (Springer-Verlag, Berlin, 1983) pp. 15–56.Google Scholar
  2. [2]
    V.I. Arnol'd, A.N. Varchenko and S.M. Gusein-Zade,Singularities of differentiable maps (Nayka, Moscow, 1982) (in Russian).Google Scholar
  3. [3]
    A.V. Fiacco,Introduction to sensitivity and stability analysis in nonlinear programming (Academic Press, New York, 1983).Google Scholar
  4. [4]
    A.V. Fiacco, ed., “Sensitivity, stability and parametric analysis”,Mathematical Programming Study 21 (1984).Google Scholar
  5. [5]
    J. Guddat and K. Wendler, “On dialogue-algorithms for linear and nonlinear vector optimization from the point of view of parametric programming“, in: M. Grauer and A.P. Wierzbicki, eds.,Interactive decision analysis proceedings, Laxenburg, Austria 1983, Lecture Notes in Economics and Mathematical Systems 229 (Springer-Verlag, Berlin, 1984).Google Scholar
  6. [6]
    R. Hettich and P. Zencke,Numerische Methoden der Approximation und semi-infiniten Optimierung (Teubner Studienbücher, Stuttgart, 1982).Google Scholar
  7. [7]
    M.W. Hirsch,Differential topology (Springer-Verlag, Berlin, 1976).Google Scholar
  8. [8]
    H.Th. Jongen, P. Jonker and F. Twilt, “On deformation in optimization“,Methods of Operations Research 37 (1980) 171–184.Google Scholar
  9. [9]
    H.Th. Jongen, P. Jonker and F. Twilt, “On one-parameter families of sets defined by (in)equality constraints“,Nieuw Archief voor Wiskunde (3) 30 (1982) 307–322.Google Scholar
  10. [10]
    H.Th. Jongen, P. Jonker and F. Twilt,Nonlinear optimization in ℝ n, I. Morse theory, Chebyshev approximation, Methoden und Verfahren der mathematischen Physik 29 (Peter Lang Verlag, Frankfurt a.M., 1983).Google Scholar
  11. [11]
    H.Th. Jongen, P. Jonker and F. Twilt, “One-parameter families of optimization problems: equality constraints“,Journal of Optimization Theory and Applications 48 (1986) 141–161.Google Scholar
  12. [12]
    M. Kojima and R. Hirabayashi, “Continuous deformation of nonlinear programs“,Mathematical Programming Study 21 (1984) 150–198.Google Scholar
  13. [13]
    M. Marcus and H. Minc,A survey of matrix theory and matrix inequalities (Allyn and Bacon Inc., Boston, 1964).Google Scholar
  14. [14]
    J. Milnor,Morse theory, Annals of Mathematics Studies 51 (Princeton University Press, Princeton, NJ, 1963).Google Scholar
  15. [15]
    D. Siersma, “Singularities of functions on boundaries, corners etc.“,The Quarterly Journal of Mathematics, Oxford Series (2) 32 (1981) 119–127.Google Scholar
  16. [16]
    J.E. Spingarn, “On optimality conditions for structured families of nonlinear programming problems“,Mathematical Programming 22 (1982) 82–92.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1986

Authors and Affiliations

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

Personalised recommendations