  Parallel and cyclic projection algorithms are proposed for minimizing the sum
of a finite family of convex functions over the intersection of a finite family
of closed convex subsets of a Hilbert space. These algorithms are of
predictor-corrector type, with each main iteration consisting of an inner cycle
of subgradient descent process followed by a projection step. We prove the
convergence of these methods to an optimal solution of the composite
minimization problem under investigation upon assuming boundedness of the
gradients at the iterates of the local functions and the stepsizes being chosen
appropriately, in the finite-dimensional setting. We also discuss
generalizations and limitations of the proposed algorithms and our techniques.
