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Characterizations of Well-Posedness and Sensitivity Analysis

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 330)

Abstract

The primary goal of this chapter is to show that the basic principles and tools of variational analysis developed above allow us to provide complete characterizations and efficient applications of fundamental properties in nonlinear studies related to Lipschitzian stability, metric regularity, and covering/openness at a linear rate. These properties indicate a certain well-posedness (i.e., “good behavior”) of set-valued mappings and play a principal role in many aspects of nonlinear analysis, particularly those concerning optimization and sensitivity. We have considered these properties in Chap. 1 in the framework of arbitrary Banach spaces, where necessary conditions for their fulfillment were obtained via coderivatives of set-valued mappings. These conditions were efficiently used in Chaps. 1 and 3 for developing the generalized differential calculus and related issues. In this chapter we show, based on variational arguments, that the conditions obtained are not only necessary but also sufficient for the validity of the mentioned properties in the framework of Asplund spaces. Moreover, we compute the exact bounds of the corresponding moduli in terms of coderivatives and subdifferentials. Two kinds of dual characterizations are derived in this way: neighborhood criteria involving generalized differential constructions around reference points, and pointbased criteria expressed only at the points under consideration.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.College of Science, Department of MathematicsWayne State UniversityDetroitUSA

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