# Hidden Convexity in the l0 Pseudonorm

@article{Chancelier2020HiddenCI, title={Hidden Convexity in the l0 Pseudonorm}, author={J. Chancelier and M. Lara}, journal={arXiv: Optimization and Control}, year={2020} }

The so-called l0 pseudonorm counts the number of nonzero components of a vector of a Euclidian space.
It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero.
In this paper, we introduce a suitable conjugacy, induced by a novel coupling,
E-Capra, that has the property of being constant along primal rays like the l0 pseudonorm.
The coupling E-Capra belongs to the class of one-sided linear couplings, that
we introduce; we show that they induce… Expand

#### 8 Citations

Constant along primal rays conjugacies and the l0 pseudonorm

- Mathematics
- 2020

The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few… Expand

Variational Formulations for the l0 Pseudonorm and Application to Sparse Optimization

- Mathematics
- 2020

The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few… Expand

Constant Along Primal Rays Conjugacies and Generalized Convexity for Functions of the Support

- Mathematics
- 2020

The support of a vector in R d is the set of indices with nonzero entries. Functions of the support have the property to be 0-homogeneous and, because of that, the Fenchel conjugacy fails to provide… Expand

Best Convex Lower Approximations of the l 0 Pseudonorm on Unit Balls

- Mathematics
- 2021

Whereas the norm of a vector measures amplitude (and is a 1-homogeneous function), sparsity is measured by the 0-homogeneous l0 pseudonorm, which counts the number of nonzero components. We propose a… Expand

Rank-Based Norms, Capra-Conjugacies and the Rank Function

- Mathematics
- 2021

We consider the space of matrices, with given number of rows and of columns, equipped with the classic trace scalar product. With any matrix (source) norm, we associate a coupling, called Capra,… Expand

Minimizers of Sparsity Regularized Huber Loss Function

- Mathematics, Computer Science
- J. Optim. Theory Appl.
- 2020

The structure of the local and global minimizers of the Huber loss function regularized with a sparsity inducing L0 norm term is investigated and conditions that are necessary and sufficient for a local minimizer to be strict are established. Expand

Conditional Infimum and Hidden Convexity in Optimization

- Mathematics
- 2021

Detecting hidden convexity is one of the tools to address nonconvex minimization problems. After giving a formal definition of hidden convexity, we introduce the notion of conditional infimum, as it… Expand

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

- Mathematics
- 2020

The so-called l0 pseudonorm, or cardinality function, counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra (constant along… Expand

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