# conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

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The conjugate function of a norm always exists. You should be able to compute it using its definition. Just observe that the maximum is achieved at the unique critical point of the formula within the parenthesis. So all you have to do is find the critical value. –  Deane Yang Apr 3 '12 at 13:01
I defined / studied this function in a recent preprint (for computational purposes). The derivation is standard convex analysis, though slightly tedious to do explicitly. –  Suvrit Apr 5 '12 at 20:09

## 1 Answer

Let $p^*$ and $q^*$ be the conjugate exponents. Some (slightly laborious) algebra shows that the dual-norm is $\|A\|_{p^*,q^*}$. The conjugate function is the indicator function for the (unit) dual-norm ball.

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