The so-called ${\ell }_0$ pseudonorm on ${\mathbb{R}}^d$ counts the number of nonzero components of a vector. For exact sparse optimization problems – with the ${\ell }_0$ 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 ${\ell }_0$ pseudonorm. For this purpose, we suppose given a (source) norm on ${\mathbb{R}}^d$. 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 ${\mathbb{R}}^d$ 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 ${\ell }_0$ pseudonorm, in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the ${\ell }_0$ pseudonorm, as a fraction between two norms, the denominator being any norm.

所谓的${\ell }_0$ 伪范数${\mathbb{R}}^d$计算向量的非零分量的数量。对于精确的稀疏优化问题——使用${\ell }_0$ 伪范式无论是作为标准还是在约束中 - Fenchel 共轭未能提供相关分析。在本文中，我们展示了一类适用于${\ell }_0$ 伪规范。为此，我们假设给定一个（源）规范${\mathbb{R}}^d$. 有了这个范数，我们一方面定义了一系列所谓的坐标k范数，另一方面，定义了${\mathbb{R}}^d$和它本身，称为 Capra（沿原始射线恒定）。然后，我们提供了 Capra 共轭和双共轭以及 Capra 次微分的公式${\ell }_0$ 伪范数，就坐标k范数而言。作为一个应用程序，我们为${\ell }_0$ 伪范数，作为两个范数之间的分数，分母是任何范数。