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Optimization methods and stability of inclusions in Banach Spaces


Klatte, D; Kummer, B (2009). Optimization methods and stability of inclusions in Banach Spaces. Mathematical Programming: Series B, 117(1-2):305-330.

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 ¯rst 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.

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 ¯rst 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.

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22 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Business Administration
Dewey Decimal Classification:330 Economics
Language:English
Date:March 2009
Deposited On:17 Mar 2009 09:41
Last Modified:05 Apr 2016 12:54
Publisher:Springer
ISSN:0025-5610
Additional Information:The original publication is available at www.springerlink.com. It was published electronically in 2007.
Publisher DOI:https://doi.org/10.1007/s10107-007-0174-9
Related URLs:http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=32960991&loginpage=Login.asp&site=ehost-live

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