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Mathematical Programming

, Volume 117, Issue 1–2, pp 305–330 | Cite as

Optimization methods and stability of inclusions in Banach spaces

  • Diethard Klatte
  • Bernd Kummer
FULL LENGTH PAPER

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.

Keywords

Generalized equation Variational inequality Perturbation Regularization Stability criteria Metric regularity Calmness Approximate projections Penalization Successive approximation Newton’s method 

Mathematics Subject Classification (2000)

49J52 49K40 90C31 65Y20 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut für Operations ResearchUniversität ZürichZürichSwitzerland
  2. 2.Institut für MathematikHumboldt–Universität zu BerlinBerlinGermany

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