Constant along primal rays conjugacies and the l0 pseudonorm

• Published in: Optimization Vol. 71; no. 2; pp. 355 - 386
• Main Authors: Jean-Philippe Chancelier, De Lara, Michel
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis LLC 01.02.2022; Taylor & Francis
• Subjects: Lower bounds; Mathematics; Norms; Optimization; Optimization and Control; l0 pseudonorm; Fenchel-Moreau conjugacy; coordinate-k norm
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2020.1822836

The so-called pseudonorm on counts the number of nonzero components of a vector. For exact sparse optimization problems – with the 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 pseudonorm. For this purpose, we suppose given a (source) norm on . 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 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 pseudonorm, in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the pseudonorm, as a fraction between two norms, the denominator being any norm.