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, nowadays 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 generally if it is Legendre but some constraint qualification holds additionally. The key advantage is the broad applicability and verifiability of these concepts. The results presented here are complementary to recent work by Censor and Reich on the method of random Bregman projections (where the sets are projected onto infinitely often – not necessarily cyclically). Special attention is given to examples, some of which connect to Pythagorean means and to Convex Analysis on the Hermitian or symmetric matrices.