Journal of Convex Analysis 17 (2010), No. 3&4, 737--763
Copyright Heldermann Verlag 2010
Infimal Convolutions and Lipschitzian Properties of Subdifferentials for Prox-Regular Functions in Hilbert Spaces
Miroslav Bacak
School of Mathematical and Physical Sciences, University of Newcastle, Newcastle -- NSW 2308, Australia
miroslav.bacak@newcastle.edu.au
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
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.
Keywords: Subdifferentials, Lipschitz continuity, infimal convolutions, prox-regular functions, prox-bounded functions, set-valued mappings.
MSC: 49J52, 46C05
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