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

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

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.