  We investigate block-activated proximal algorithms for multicomponent
minimization problems involving a separable nonsmooth convex function
penalizing the components individually, and nonsmooth convex coupling terms
penalizing linear mixtures of the components. In the case of smooth coupling
functions, several 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 their merits and numerical
performance. The goal of the paper is to address this gap. The numerical
experiments concern machine learning and signal recovery problems.
