Abstract

We study infimal convolutions of extended-real-valued functions in Hilbert spaces paying a special attention to the rather broad and remarkable class of prox-regular functions. Such functions have been well recognized as highly important in many aspects of variational analysis and its applications in both finite-dimensional and infinite-dimensional settings. Based on advanced variational techniques, we discover some new subdifferential properties of infimal convolutions and apply them to the study of Lipschitzian behavior of subdifferentials for prox-regular functions in Hilbert spaces. It is shown, in particular, that the fulfillment of a natural Lipschitz-like property for (set-valued) subdifferentials of prox-regular functions forces such functions, under weak assumptions, actually to be locally smooth with single-valued subdifferentials reduced to Lipschitz continuous gradient mappings.

Author information

Contact details are reproduced from the original publication and may be historical.

Jonathan M. Borwein

Centre for Computer Assisted Research Mathematics and its Applications, University of Newcastle, Callaghan NSW 2308, Australia

jborwein@newcastle.edu.au

Andrew Eberhard

School of Mathematical and Geospatial Sciences, RMIT - GPO Box 2476V, Melbourne - Victoria, Australia 3001

andy.eb@rmit.edu.au

Boris S. Mordukhovich

Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.

boris@math.wayne.edu

Keywords

  • Subdifferentials
  • Lipschitz continuity
  • infimal convolutions
  • prox-regular functions
  • prox-bounded functions
  • set-valued mappings

Mathematics Subject Classification

49J52, 46C05

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Suggested citation

M. Bačák, J. M. Borwein, A. Eberhard, B. S. Mordukhovich. “Infimal Convolutions and Lipschitzian Properties of Subdifferentials for Prox-Regular Functions in Hilbert Spaces.” Journal of Convex Analysis 17 (2010), No. 3&4, 737–763. https://doi.org/10.68381/jca17049

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press.