Lipschitz-Stabilität von Optimierungs- und Approximationsaufgaben / Lipschitz-Stability for Programming and Approximation Problems

  • Hans-Peter Blatt
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série International d’Analyse Numérique book series (ISNM, volume 52)


This paper deals with the Lipschitz stability of the feasible region, the solution set and the minimum value of the objective function for convex programming problems when the data are subjected to small perturbations. We show that a certain regularity condition is necessary and sufficient for the Lipschitz continuity of the feasible region. We get the Lipschitz continuity of the minimum value and the set of e-solutions. Several examples show that in general the Lipschitz upper semicontinuity doesn’t hold for the exact solution set. However we prove for weak Chebyshev systems in C[a,b] with unique alternation element gf for each f ? C[a,b], that the selection s: f → gf is Lipschitz continuous. Consequences resulting from rounaing errors are discussed for numerical methods.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blatt, H.-P.: Stetigkeitseigenschaften von Optimierungsaufgaben und lineare Tschebyscheff-Approximation, in Z. Ciesielski und J. Musielak (Herausgeber): Approximation Theory, 33–48, Dordrecht-Holland, D. Reidel Publishing Co., 1975.CrossRefGoogle Scholar
  2. 2.
    Brosowski, B.: Nichtlineare Approximation in normierten Vektorräumen, ISNM 10, 140–159, Birkhäuser Verlag.Google Scholar
  3. 3.
    Freud, G.: Eine Ungleichung für Tschebyscheffsehe Approximationspolynome, Acta Sei. Math. 19 (1958), 162–164.Google Scholar
  4. 4.
    Gauvin, J., Tolle, J.W.: Differential stability in nonlinear programming, SIAM J. Control and Optimization 15 (1977), 294–311.CrossRefGoogle Scholar
  5. 5.
    Laurent, P.J.: Approximation et optimisation, Paris, Hermann, 1972.Google Scholar
  6. 6.
    Lempio, F., Maurer, H.: Differentiable perturbations of infinite optimization problems, Lecture Notes in Economics and Mathematical Systems 157, 181–191, Springer 1978.Google Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • Hans-Peter Blatt

There are no affiliations available

Personalised recommendations