|
|
|
Vietnam Journal of Mathematics 40:2&3(2012)
181-199
|
|
The Moreau Envelope Function and Proximal
Mapping with Respect to the Bregman
Distances in Banach Spaces
|
|
Ying Ying Chen1, Chao Kan2
and Wen Song1
|
|
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
|
|
Dedicated to Professor
Phan Quoc Khanh on the occasion of his 65th birthday
Received December 11,
2011
Revised April 24, 2012
|
|
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.
|
|
2000 Mathematics
Subject Classification. 90C25, 90C48, 47H05, 41A65.
|
|
Keywords.
Bregman distance, Moreau envelope, proximal mapping, continuity,
single-valuedness.
|
|
|
|
|
|
Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
|
|
