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On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions
journal contribution
posted on 2009-01-01, 00:00 authored by R Kasimbeyli, Musa MammadovMusa MammadovIn this paper we study relations between the directional derivatives, the weak subdifferentials, and the radial epiderivatives for nonconvex real-valued functions. We generalize the well-known theorem that represents the directional derivative of a convex function as a pointwise maximum of its subgradients for the nonconvex case. Using the notion of the weak subgradient, we establish conditions that guarantee equality of the directional derivative to the pointwise supremum of weak subgradients of a nonconvex real-valued function. A similar representation is also established for the radial epiderivative of a nonconvex function. Finally the equality between the directional derivatives and the radial epiderivatives for a nonconvex function is proved. An analogue of the well-known theorem on necessary and sufficient conditions for optimality is drawn without any convexity assumptions. © 2009 Society for Industrial and Applied Mathematics.
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Journal
SIAM Journal on OptimizationVolume
20Pagination
841-855Location
Philadelphia, Pa.Publisher DOI
ISSN
1052-6234Language
engPublication classification
C1.1 Refereed article in a scholarly journalIssue
2Publisher
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