On Second-Order Variational Analysis of Variational Convexity of Prox-Regular Functions

Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using first-order generalized derivatives of the subgradient mappi...

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Bibliographic Details
Published in:Set-valued and variational analysis Vol. 33; no. 1; p. 8
Main Author: Gfrerer, Helmut
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.03.2025
Springer Nature B.V
Subjects:
Analysis
Applied mathematics
Computational mathematics
Convexity
Localization
Mapping
Mathematical functions
Mathematics
Mathematics and Statistics
Neighborhoods
Optimization
Regularity
Research Article
Tilt-stable local minimizers
90C31
49J53
49J52
Coderivative
Variational strong convexity
Variational convexity
Subspace containing derivative
ISSN:1877-0533, 1877-0541
Online Access:Get full text
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Summary:Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using first-order generalized derivatives of the subgradient mapping. Up to now, such characterizations are only known under the assumptions of prox-regularity and subdifferential continuity and in this paper we discard the latter. To this aim we slightly modify the definitions of the generalized derivatives to be compatible with the f -attentive convergence appearing in the definition of subgradients. We formulate our results in terms of both coderivatives and subspace containing derivatives. We also give formulas for the exact bound of variational convexity and study relations between variational strong convexity, tilt-stable local minimizers and strong metric regularity of some truncation of the subgradient mapping.
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ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-025-00744-8