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Journal of Convex Analysis 10 (2003), No. 1, 245--254
Copyright Heldermann Verlag 2003
Redundant Axioms in the Definition of Bregman Functions
Dan Butnariu
Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel,
dbutnaru@math.haifa.ac.il
Charles Byrne
Dept. of Mathematical Sciences, University of Massachusetts, Lowell, MA 01854, U.S.A.,
byrnec@cs.uml.edu
Yair Censor
Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel,
yair@math.haifa.ac.il
The definition of a Bregman function, given by Censor and Lent
in 1981 on the basis of Bregman's seminal 1967 paper, was subsequently
used in a plethora of research works as a tool for building sequential
and inherently parallel feasibility and optimization algorithms. Solodov
and Svaiter have recently shown that it is not "minimal". Some of its
conditions can be derived from the others. In this note we illuminate
this finding from a different perspective by presenting an alternative
proof of the equivalence between the original and the simplified
definitions of Bregman functions in which redundant conditions are
eliminated. This implicitly shows that the seemingly different notion of
Bregman functions recently introduced by Butnariu and Iusem, when
transported to a proper setting in Rn, is equivalent to the
original concept. The results established in this context are also used
to resolve a problem in proximity function minimization encountered by
Byrne and Censor.
Keywords: Bregman function, Bregman distance, Kullback-Leibler distance,
modulus of total convexity, sequential consistency, totally convex function.
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