  Under consideration are multicomponent minimization problems involving a
separable nonsmooth convex function penalizing the components individually, and
nonsmooth convex coupling terms penalizing linear mixtures of the components.
We investigate block-activated proximal algorithms for solving such problems,
i.e., algorithms which, at each iteration, need to use only a block of the
underlying functions, as opposed to all of them as in standard methods. For
smooth coupling functions, several block-activated algorithms exist and they
are well understood. By contrast, in the fully nonsmooth case, few
block-activated methods are available and little effort has been devoted to
assessing them. Our goal is to shed more light on the implementation, the
features, and the behavior of these algorithms, compare their merits, and
provide machine learning and image recovery experiments illustrating their
performance.
