What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$

$\|\mathbf{x}\| = \displaystyle{ \sum_{i=0}^{k} \|\mathbf{A}_i \cdot \mathbf{x} \|_2}$.

How would you then find

$\|\mathbf{y}\|_* = \underset{\mathbf{x}}{\mathrm{max}} \left\{ \mathbf{y}^T \mathbf{x} \;\; \mathrm{s.t.} \;\; \|\mathbf{x}\| \leq 1\right\}$?

I've tried solving for the convex conjugate looking for hints, but was unable to come up with anything meaningful.

Also, if anyone has recommendations for packages that I could use (preferably matlab-based) to solve the above numerically for systems as small as $10^3$ and as large as $10^6$, I'd greatly appreciate it. CVX, of which I am admittedly a novice and a hack, will not maximize convex functions.

Edit: So using the advice in the below comments, I end up with an eigen equation for the critical point $\mathbf{x}_*$:

$ \displaystyle{ \sum_{i=0}^{k} {\|\mathbf{A}_i \mathbf{x}_* \|_2 } } \cdot \mathbf{y} = \displaystyle{ \sum_{i=0}^{k} { {\mathbf{A}_i^T \mathbf{A_i} } \over{ \|\mathbf{A}_i \cdot \mathbf{x}_* \|_2 } } } \mathbf{x}_* \mathbf{x}_*^{T} \cdot \mathbf{y}$

The only other idea I have had is that we know that $\|y\|_*$ is the function such that $ \underset{\mathbf{x}}{\sup} \{ \mathbf{y}^T \mathbf{x} - \|\mathbf{x}\|\}$ is zero whenever $\|y\|_* \leq 1$ and is $\infty$ otherwise. I have not been able to use this in any meaningful way however.

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