Nonlinear Optimization: Stability and Parametric Aspects

  • Hubertus Th. Jongen
Conference paper


We introduce the concept of structural stability for finite dimensional optimization problems. For differentiable problems with compact feasible set we present a complete characterization of structural stability in terms of partial derivatives of the defining functions up to second order. Here, the Mangasarian-Fromovitz constraint qualification as well as Kojima’s strong stability of stationary points play a crucial role. Then, we turn over to continuous deformations of finite dimensional optimization problem=. Singularities occur due to the violation of strict complementarity, the violation of constraint qualifications, and the degeneracy of the restricted Hessian of the Lagrange function. The latter singularities will be discussed and, finally, we focus on the pathfollowing of local minima with possible jumps.


  1. 1.
    H.Th. Jongen, P. Jonker and F. Twilt: Critical Sets in Parametric Optimization. Mathematical Programming 34, pp.333–353 (1986).CrossRefGoogle Scholar
  2. 2.
    J. Guddat, F. Guerra Vasquez and H.Th. Jongen: Parametric Optimization: Singularities, Pathfollowing and Jumps. J.Wiley (1990).Google Scholar
  3. 3.
    H.Th. Jongen and G.-W. Weber: Characterization of Structural Stability. Journal of Global Optimization 1, pp.47–64 (1991).CrossRefGoogle Scholar
  4. 4.
    H.Th. Jongen and G.-W. Weber: Nonconvex Optimization and its Structural Frontiers. In: Modern Methods of Optimization (W. Krabs and J. Zowe, eds.), Lect. Notes in Economical and Mathematical Systems, Vol.37S, pp.151–203, Springer Verlag (1992).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Hubertus Th. Jongen
    • 1
  1. 1.Lehrstuhl C für MathematikRWTH AachenAachenGermany

Personalised recommendations