Summary: The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets. It is of basic importance in various areas of mathematics and physical sciences, and it can be solved iteratively using the classical method of cyclic projections, which generates a sequence by projecting cyclically onto the sets. In his seminal 1967 paper, Bregman extended this method to non-orthogonal projections, using the notion of the Bregman distance induced by a convex function.
In this paper, we present a new algorithmic scheme which also extends the method of cyclic projections. Based on Bregman distances, we introduce a new type of non-orthogonal projection, the forward projection. The energy and the negative entropy allow forward projections – the former yields the classical orthogonal projection whereas the latter gives rise to a type of projection used implicitly in a manifestation of the Expectation-Maximization algorithm. We provide useful properties of forward projections, and a basic convergence result on the method of forward projections.