Constant Along Primal Rays Conjugacies and the l0 Pseudonorm.

Abstract: he so-called l0 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 present the Euclidean Capra-conjugacy, which is suitable for the l0 pseudonorm, as this latter is “convex” in the sense of generalized convexity (equal to its biconjugate). We immediately derive a convex factorization property (the l0 pseudonorm coincides, on the unit sphere, with a convex lsc function) and variational formulations for the l0 pseudonorm. In a second part, we provide different extensions: the above properties hold true for a class of conjugacies depending on strictly-orthant monotonic norms (including the Euclidean norm); they hold true for nondecreasing functions of the support (including the l0 pseudonorm); more generally, we will show how Capra-conjugacies are suitable to provide convex lower bounds for zero-homogeneous functions; we will also point out how to tackle the rank matrix function. Finally, we present mathematical expressions of the Capra-subdifferential of the l0 pseudonorm, and graphical representations. This opens the way for posible suitable algorithms that we discuss.

Posted on Oct 10, 2021 in Optimization and Equilibrium, Seminars