Legendre Functions and the Method of Random Bregman Projections

The convex feasibility problem, that is, finding a point in the intersection of finitely many closed convex sets in Euclidean space, arises in various areas of mathematics and physical sciences. It can be solved by the classical method of cyclic orthogonal projections, where, by projecting cyclically onto the sets, a sequence is generated that converges to a point in the intersection. In 1967, Bregman extended this method to non-orthogonal projections based on a new notion of distance, now days called "Bregman distance". The Bregman distance is induced by a convex function. If this function is a so-called "zone consistent Bregman function", then Bregman's method works; however, deciding on this can be difficult. In this paper, Bregman's method is studied within the powerful framework of Convex Analysis. New insights are obtained and the rich class of "Bregman /Legendre functions" is introduced. Bregman's method still works, if the underlying function is Bregman/Legendre or more general..