Characterizing Generalized Derivatives of Set-Valued Maps: Extending the Tangential and Normal Approaches

For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [C. H. J. Pang, Math. Oper. Res., 36 (2011), pp. 377--397]. This result generalizes the...

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Bibliographic Details
Published in:SIAM journal on control and optimization Vol. 51; no. 1; pp. 145 - 171
Main Author: Jeffrey Pang, C. H.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Subjects:
Cones
Constraining
Criteria
Derivatives
Mathematical analysis
Optimization
ISSN:0363-0129, 1095-7138
Online Access:Get full text
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Summary:For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [C. H. J. Pang, Math. Oper. Res., 36 (2011), pp. 377--397]. This result generalizes the Aubin criterion in [A. L. Dontchev, M. Quincampoix, and N. Zlateva, J. Convex Anal., 3 (2006), pp. 45--63]. A second characterization of these generalized derivatives is easier to check in practice, especially in the finite dimensional case. Finally, the third characterization in terms of limiting normal cones and coderivatives generalizes the Mordukhovich criterion in the finite dimensional case. The convexified coderivative has a bijective relationship with the set of possible generalized derivatives. We conclude by illustrating a few applications of our result. [PUBLICATION ABSTRACT]
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ISSN:0363-0129
1095-7138
DOI:10.1137/110840467