# Schatten-1 norm as matrix constraint

suppose I have a tensor $$x \in \mathbb{R}^{n \times 2 \times 3}$$. I take the seminorm of $$x$$ given by taking the Schatten-1 Norm in every $$2 \times 3$$ slice and then the $$\ell_1$$-Norm of the resulting vector of sums of singular values. I want to use this as a regularizer in a convex optimization problem. For my purposes, I need the dual norm of the proposed method, which is given by taking the Schatten-$$\infty$$-Norm (or spectral norm) in every $$2 \times 3$$ slice and then the $$\ell_\infty$$-Norm of the resulting vector.

Now imagine I vectorize $$x$$ to the form $$u \in \mathbb{R}^{6n}$$. Is there a way to write the condition "dual norm of u less than one" as a matrix constraint, i.e. $$Au \leq 1$$? I am trying to use this in $$CVX$$ in matlab. Of course I can just pass this on in a custom function and operate on the tensor $$x$$, but it would be much faster if I could avoid that.

Thanks!

• So you want the absolute value of the largest element of $u$ to be at most 1? – LinAlg Oct 31 '18 at 17:45