The so-called $l_0$ pseudonorm counts the number of nonzero components of a vector. It is standard in sparse optimization problems. However, as it is a discontinuous and nonconvex function, the l0 pseudonorm cannot be satisfactorily handled with the Fenchel conjugacy. In this talk, we review a series of recent results on a class of Capra
(Constant Along Primal Rays) conjugacies that reveal hidden convexity in the $l_0$ pseudonorm.

First, we present the Euclidean Capra-conjugacy. We show that it is suitable to analyze the $l_0$ pseudonorm, as this latter is “convex” in the sense of generalized convexity
(equal to its biconjugate). We present mathematical expressions of the Capra-subdifferential of the $l_0$ pseudonorm, and graphical representations.

In a second part, we provide different extensions. We introduce the class of Capra-conjugacies defined by means of norms. We show that such Capra-conjugacies are suitable to analyze, not only the $l_0$ pseudonorm, but provide convex lower bounds for 0-homogeneous functions. We will also point out how to tackle the rank matrix function.

Finally, we discuss how the theory opens the way for possible algorithms in sparse optimization problems.

Zoom meeting: Link