Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literatur
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.
Brosowski, B.: Nichtlineare Approximation in normierten Vektorräumen, ISNM 10, 140–159, Birkhäuser Verlag.
Freud, G.: Eine Ungleichung für Tschebyscheffsehe Approximationspolynome, Acta Sei. Math. 19 (1958), 162–164.
Gauvin, J., Tolle, J.W.: Differential stability in nonlinear programming, SIAM J. Control and Optimization 15 (1977), 294–311.
Laurent, P.J.: Approximation et optimisation, Paris, Hermann, 1972.
Lempio, F., Maurer, H.: Differentiable perturbations of infinite optimization problems, Lecture Notes in Economics and Mathematical Systems 157, 181–191, Springer 1978.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1980 Springer Basel AG
About this chapter
Cite this chapter
Blatt, HP. (1980). Lipschitz-Stabilität von Optimierungs- und Approximationsaufgaben / Lipschitz-Stability for Programming and Approximation Problems. In: Collatz, L., Meinardus, G., Werner, H. (eds) Numerische Methoden der Approximationstheorie / Numerical Methods of Approximation Theory. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série International d’Analyse Numérique, vol 52. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6721-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6721-4_1
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-1103-2
Online ISBN: 978-3-0348-6721-4
eBook Packages: Springer Book Archive