I am working on developing a numerical algorithm that needs to use a subgradient of $\|\cdot\|_2$ (matrix norm) at each iteration. According to Characterization of the Subdifferential of Some Matrix Norms by G.A. Watson, the subdifferential of $\|A\|$ is given by
$$ \partial \|A\| := \left\{ G \in \mathbb{R}^{m \times n} : \|B\| \geq \|A\| + \mbox{tr} \left( (B−A)^\top G \right), \forall B\in \mathbb{R}^{m\times n} \right\} $$
For the $\|\cdot\|_2$ of vector norm, I know that we can choose a subgradient
$$ g_n \in \partial \|\cdot\|_2(x_n) = \frac{x_n}{\|x_n\|_2},~ (x_n \neq 0) $$
at each iteration. How should I choose a subgradient for my case? Any help would be greatly appreciated.
