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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prox-regular functions in variational analysis

Authors: R. A. Poliquin and R. T. Rockafellar
Journal: Trans. Amer. Math. Soc. 348 (1996), 1805-1838
MSC (1991): Primary 49A52, 58C06, 58C20; Secondary 90C30
MathSciNet review: 1333397
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Abstract: The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-$\mathcal {C}^{2}$ functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.

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Additional Information

R. A. Poliquin
Affiliation: Deptartment of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

R. T. Rockafellar
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

Keywords: Prox-regularity, amenable functions, primal-lower-nice functions, proximal mappings, Moreau envelopes, regularization, subgradient mappings, nonsmooth analysis, variational analysis, proto-derivatives, second-order epi-derivatives, Attouch’s theorem
Received by editor(s): December 21, 1994
Received by editor(s) in revised form: June 7, 1995
Additional Notes: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS–9200303 for the second author.
Article copyright: © Copyright 1996 American Mathematical Society