1
$\begingroup$

I want to compute the subgradients of the absolute value function in $\mathbb{R}^n$. How do I do this?

  • 0
    By absolute value, you mean $|x|= \left( \sum_i x_i^2 \right)^{1/2}$?2012-11-03

1 Answers 1

1

$\phi=||\cdot||$ is differentiable on $\mathbb{R}^n \backslash \{0\}$, so $\partial \phi(x)=\{\phi'(x) \}$ (you can show that $\phi'(x)= \frac{1}{|x|} \langle x, \cdot \rangle$). Then, $\partial \phi(0)= \{ \zeta \in (\mathbb{R}^n)^* \ | \ \forall y \in \mathbb{R}^n, |y| \geq \langle \zeta, y \rangle \}$ is the unit closed ball in $(\mathbb{R}^n)^*$.

  • 0
    Yes. I know how Tor do this for n=1 but not for arbitrary n.2012-11-03