I need to explain how to compute a subgradient at a given point x (not necessarily deriving it but just explain how it can be derived) for the following function:
$$\newcommand{\defeq}{\mathrel{:\mkern0.25mu=}}$$
$$f(\mathbf{x}) = \vert\vert \Sigma_{i=1}^{n} x_{i} \mathbf{A_{i}} \mathbf{B} \vert \vert _{2}$$
where $\forall i \mathbf{A_{i}}, \mathbf{B} \in \mathbb{R}^{p\times q}$ and $\mathbf{x} \in \mathbb{R}^{n}$
My direction so far was to look at the definition of 2 norm of matrices $\vert\vert \mathbf{B} \vert \vert _{2} = \sqrt{\lambda_{max} (\mathbf{B}^{*}\mathbf{B}})$ and trying to use the fact that $A(\mathbf{x})^{*}A(\mathbf{x})$ is a convex function of $\mathbf{x}$ for $A(\mathbf{x}) \defeq \Sigma_{i=1}^{n} x_{i} \mathbf{A_{i}} \mathbf{B} $ and therefor $\lambda_{max} (A(\mathbf{x})^{*}A(\mathbf{x}))$ has a subgradient (you can see that by the fact that $\lambda_{max}(A(\mathbf{x})^{*}A(\mathbf{x})) = max_{ \vert \vert \mathbf{u} \vert \vert =1} \mathbf{u}^{*} (\mathbf{A(\mathbf{x})^{*}A(\mathbf{x}))} \mathbf{u} $ is a maximum over convex functions of x).
The problem is that from there I am stuck, I want to use some sort of chain rule but $\sqrt{x}$ is not a convex function...
I am new to the subject of subgradients and I have been stuck on this exercise for a few days already, any help would be most welcomed.
Thanks in advance
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Roei Sarussi
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Something is wrong with the dimensions. Also a multiplication of a Matrix and Vector yields a vector so subtracting a Matrix doesn't make sense. – Royi Apr 11 '20 at 21:10

$x_{i}$ are scalars and corresponding to the elements of the vector, scalar multiplication by a matrix remain a matrix – Roei Sarussi Apr 12 '20 at 06:48

I think you just edit the question and write: The Sub Gradient of $ {L}_{2} $ Induced Matrix Norm. Then the question should be: What's the Sub Gradient with respect to $ X $ of $ {\left\ A X  B \right\}_{2}^{2} $ where $ {\left\ Y \right\}_{2}^{2} = {\lambda}_{max} \left( {Y}^{^T} T \right) $. – Royi Apr 12 '20 at 07:21

Related  https://math.stackexchange.com/questions/2795443. – Royi Apr 13 '20 at 10:41

Related  https://math.stackexchange.com/questions/3601351. – Royi Apr 13 '20 at 10:41

Related  https://math.stackexchange.com/questions/701062. – Royi Apr 13 '20 at 10:41