# A Suitable Conjugacy for the l0 Pseudonorm

@article{Chancelier2019ASC, title={A Suitable Conjugacy for the l0 Pseudonorm}, author={Jean-Philippe Chancelier and Michel De Lara and Ponts Paristech}, journal={arXiv: Optimization and Control}, year={2019} }

The so-called l0 pseudonorm on R d counts the number of nonzero components of a vector. 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, Caprac, having the property of being constant along primal rays, like the l0 pseudonorm. The Caprac coupling belongs to the class of one-sided linear couplings, that we introduce. We show that they induce conjugacies that share nice… Expand

#### 2 Citations

Lower Bound Convex Programs for Exact Sparse Optimization

- Mathematics
- 2019

In exact sparse optimization problems on Rd (also known as sparsity constrained problems), one looks for solution that have few nonzero components. In this paper, we consider problems where sparsity… 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

#### References

SHOWING 1-10 OF 11 REFERENCES

A variational approach of the rank function

- Mathematics
- 2013

In the same spirit as the one of the paper (Hiriart-Urruty and Malick in J. Optim. Theory Appl. 153(3):551–577, 2012) on positive semidefinite matrices, we survey several useful properties of the… Expand

Lower Bound Convex Programs for Exact Sparse Optimization

- Mathematics
- 2019

In exact sparse optimization problems on Rd (also known as sparsity constrained problems), one looks for solution that have few nonzero components. In this paper, we consider problems where sparsity… Expand

Convex Analysis and Monotone Operator Theory in Hilbert Spaces

- Mathematics, Computer Science
- CMS Books in Mathematics
- 2011

This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is… Expand

Abstract Convex Analysis

- Mathematics
- 1997

Abstract Convexity of Elements of a Complete Lattice. Abstract Convexity of Subsets of a Set. Abstract Convexity of Functions on a Set. Abstract Quasi-Convexity of Functions on a Set. Dualities… Expand

Generalized Convex Duality and its Economic Applicatons

- Mathematics
- 2005

This article presents an approach to generalized convex duality theory based on Fenchel-Moreau conjugations; in particular, it discusses quasiconvex conjugation and duality in detail. It also… Expand

Abstract Convexity and Global Optimization

- Mathematics
- 2000

Preface. Acknowledgment. 1. An Introduction to Abstract Convexity. 2. Elements of Monotonic Analysis: IPH Functions and Normal Sets. 3. Elements of Monotonic Analysis: Monotonic Functions. 4.… Expand

Sparse Prediction with the $k$-Support Norm

- Computer Science, Mathematics
- NIPS
- 2012

A novel norm is derived that corresponds to the tightest convex relaxation of sparsity combined with an l1 penalty that provides a tighter relaxation than the elastic net and can thus be advantageous in in sparse prediction problems. Expand

Optimisation et analyse convexe pour la dynamique non-régulière

- 2009

L'objectif de ce travail est de proposer une nouvelle approche pour la resolution du probleme de contact unilateral avec frottement de Coulomb tridimensionnel en mecanique des solides. On s'interesse… Expand

Inf-convolution, sous-additivité, convexité des fonctions numériques

- J. Math. Pures Appl
- 1970