The Capra-subdifferential of the l0 pseudonorm

Abstract: The l0 pseudonorm counts the nonzero coordinates of a vector. It is often used in optimization problems to enforce the sparsity of the solution. However, this function is nonconvex and noncontinuous, and optimization problems formulated with l0 in the objective function or in the constraints are hard to solve in general. Recently, a new family of coupling functions - called Capra (constant along primal rays) - has proved to induce relevant generalized Fenchel-Moreau conjugacies to handle the l0 pseudonorm. In particular, under a suitable choice of source norm on the Euclidean space used in the definition of the Capra coupling - the function l0 is Capra-subdifferentiable, hence is Capra-convex. In this article, we give explicit formulations for the Capra subdifferential of l0, when the source norm is a lp norm with p larger that 1. We illustrate our results with graphical visualizations of the Capra subdifferential of l0 for the Euclidean source norm.