Pauwels, EdouardORCID: https://orcid.org/0000-0002-8180-075X (2024) On the nature of Bregman functions. Operations Research Letters, Vol.57 (n°107183).

Abstract

Let C ⊂ R n be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.