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Vietnam Journal of Mathematics 40:2&3(2012)
181-199
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The Moreau Envelope Function and Proximal
Mapping with Respect to the Bregman
Distances in Banach Spaces
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Ying Ying Chen1, Chao Kan2
and Wen Song1
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1School of
Mathematical and Sciences, Harbin Normal University,
Harbin, 150025, P. R. China
2Department
of Mathematics, Harbin Institute of Technology,
Harbin,
150001, P. R. China
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Dedicated to Professor
Phan Quoc Khanh on the occasion of his 65th birthday
Received December 11,
2011
Revised April 24, 2012
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Abstract. In this paper, we explore some properties
of the Moreau envelope function $e_λ f(x)$ of f and the associated proximal mapping $P_λ f(x)$ with
respect to the Bregman distance induced by a convex function in Banach
spaces. Precisely, we study the continuity, locally Lipschitz property and
differentiability of the Moreau envelope function and the upper
semicontinuity and single-valuedness of the proximal mapping in Banach
spaces.
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2000 Mathematics
Subject Classification. 90C25, 90C48, 47H05, 41A65.
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Keywords.
Bregman distance, Moreau envelope, proximal mapping, continuity,
single-valuedness.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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