Optimization methods and stability of inclusions in Banach spaces

Authors:
Diethard Klatte;Bernd Kummer
Affiliations:
Universität Zürich, Institut für Operations Research, Moussonstrasse 15, CH-8044, Zürich, Switzerland;Humboldt–Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099, Berlin, Germany
Venue:
Mathematical Programming: Series A and B
Year:
2008

Citing 0
Cited 2

Applying Metric Regularity to Compute a Condition Measure of a Smoothing Algorithm for Matrix Games

SIAM Journal on Optimization
First Order and Second Order Characterizations of Metric Subregularity and Calmness of Constraint Set Mappings

SIAM Journal on Optimization

Quantified Score

Hi-index	0.00

Visualization

Abstract

Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.