Constant Along Primal Rays Conjugacies and the l0 Pseudonorm

Speaker: Professor  Michel De Lara

Ecole des Ponts ParisTech, France

Date:  October 13,  2021 at 10:00 am (Chilean-time)

Title:   Constant Along Primal Rays Conjugacies and the l0 Pseudonorm

Abstract: The 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 possible suitable algorithms that we discuss.

Venue: Online via Google Meet:

A brief biography of the speaker: Michel De Lara graduated as an engineer at Ecole Polytechnique and at Ecole nationale des ponts et chaussées, where he is presently working at the mathematics research center CERMICS, after obtaining his PhD at Ecole nationale supérieure des mines de Paris. His research interests include stochastic optimization, game theory with information, generalized convexity, as well as different applications of mathematics (epidemics control, energy management).

Coordinators: Fabián Flores-Bazán (CMM, Universidad de Concepción) and Abderrahim Hantoute (Alicante)