A parallel block-coordinate approach for primal-dual splitting with arbitrary random block selection

The solution of many applied problems relies on finding the minimizer of a sum of smooth and/or nonsmooth convex functions possibly involving linear operators. In the last years, primal-dual methods have shown their efficiency to solve such minimization problems, their main advantage being their ability to deal with linear operators with no need to invert them. However, when the problem size becomes increasingly large, the implementation of these algorithms can be complicated, due to memory limitation issues. A simple way to overcome this difficulty consists of splitting the original numerous variables into blocks of smaller dimension, corresponding to the available memory, and to process them separately. In this paper we propose a random block-coordinate primal-dual algorithm, converging almost surely to a solution to the considered minimization problem. Moreover, an application to large-size 3D mesh denoising is provided to show the numerical efficiency of our method.