# Gradient of dot product which contains l1-norm

I try to understand the equation part in this paper, where they set the gradient of majorizing function (4) equal to zero

$\lambda \Omega(H_{s}) \leq \lambda\sum_{g}\langle\frac{\hat{H}}{\epsilon + \vert\vert \hat{H}_{g}\vert\vert_{1}},H_{g}\rangle \tag{4}$

The final form in Algorithm 1 shows the result of gradient (regarding matrix $H_{g}$) as

$H_{g} \leftarrow \frac{1}{1 + \frac{\lambda}{\epsilon + \vert\vert H_{g} \vert\vert}} H_{g}\tag{Algorithm 1}$

How to set gradient for $(4)$. I know that gradient of dot product is $\nabla(a\cdot b) = (\nabla a)\cdot b + (\nabla b)\cdot a$ but how to do $\nabla \frac{\hat{H}}{\epsilon + \vert\vert \hat{H}_{g}\vert\vert_{1}}$ part?