• DOI:10.1007/s11228-021-00606-z
• Corpus ID: 236950357

Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ0 Pseudonorm

@article{Chancelier2020CapraConvexityCF,
  title={Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ0 Pseudonorm},
  author={Jean-Philippe Chancelier and Michel De Lara},
  journal={Set-Valued and Variational Analysis},
  year={2020},
  volume={30},
  pages={597-619}
}

• J. Chancelier, M. De Lara
• Published 31 January 2020
• Mathematics
• Set-Valued and Variational Analysis

The so-called ℓ 0 pseudonorm, or cardinality function, counts the number of nonzero components of a vector. In this paper, we analyze the ℓ 0 pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the ℓ p -norms, but for the extreme ones). We obtain three main results. First, we show that the ℓ 0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex function. Second, we deduce an unexpected consequence, that we call convex factorization: the ℓ 0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex lower semicontinuous function. Third, we establish a variational formulation for the ℓ 0 pseudonorm by means of generalized top-k dual norms and k-support dual norms (that we formally introduce).