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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,

Charles Byrne
Dept. of Mathematical Sciences, University of Massachusetts, Lowell, MA 01854, U.S.A.,

Yair Censor
Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel,

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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