Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
By absolute value, you mean $|x|= \left( \sum_i x_i^2 \right)^{1/2}$? –  Seirios Nov 3 '12 at 11:45

1 Answer 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 $(\mathbb{R}^n)^*$.

share|improve this answer
    
Yes. I know how Tor do this for n=1 but not for arbitrary n. –  JamesBond Nov 3 '12 at 11:52

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.