Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 49A52, 58C06, 58C20, 90C30

Retrieve articles in all journals with MSC (1991): 49A52, 58C06, 58C20, 90C30


Additional Information

R. A. Poliquin
Affiliation: Deptartment of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: rene@fenchel.math.ualberta.ca

R. T. Rockafellar
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: rtr@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01544-9
PII: S 0002-9947(96)01544-9
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