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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Vydané v:SIAM journal on control and optimization Ročník 51; číslo 1; s. 145 - 171
Hlavný autor: Jeffrey Pang, C. H.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Predmet:
Cones
Constraining
Criteria
Derivatives
Mathematical analysis
Optimization
ISSN:0363-0129, 1095-7138
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Shrnutí: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