Constant along primal rays conjugacies and the l ₀ pseudonorm

The so-called (Formula presented.)  pseudonorm on (Formula presented.) counts the number of nonzero components of a vector. For exact sparse optimization problems–with the (Formula presented.)  pseudonorm standing either as criterion or in the constraints–the Fenchel conjugacy fails to provide relevant analysis. In this paper, we display a class of conjugacies that are suitable for the (Formula presented.)  pseudonorm. For this purpose, we suppose given a (source) norm on (Formula presented.). With this norm, we define, on the one hand, a sequence of so-called coordinate-k norms and, on the other hand, a coupling between (Formula presented.) and itself, called Capra (constant along primal rays). Then, we provide formulas for the Capra-conjugate and biconjugate, and for the Capra subdifferentials, of functions of the (Formula presented.)  pseudonorm, in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the (Formula presented.)  pseudonorm, as a fraction between two norms, the denominator being any norm.