For $X = (x_1^T,\ldots,x_N^T)^T \in \mathbb{R}^{Nm \times 1}$, $x_i \in \mathbb{R}^{m \times 1}$, $i \in \{1,\ldots,N\}$, $A \in \mathbb{R}^{r \times Nm}$, and $r \geq Nm$, I want to obtain a closed form solution of $\max_{X \in \mathbb{R}^{Nm}}\{||AX||: ||x_i||^2=1\}$ (|| || denotes Euclidean norm).

  I know how to find a closed form solution of $\max_{X \in \mathbb{R}^{Nm}}\{||AX||: ||X||^2=1\}$, i.e., when $||X||^2=1$, but I do not see how to solve it when we have $||x_i||^2=1$.

Thanks for reading me.

For $X = (x_1^T,\ldots,x_N^T)^T \in \mathbb{R}^{Nm \times 1}$, where $x_i \in \mathbb{R}^{m \times 1}$ for $i \in \{1,\ldots,N\}$, $A \in \mathbb{R}^{r \times Nm}$, and $r \geq Nm$, I want to obtain a closed form solution of

$$\max_{X \in \mathbb{R}^{Nm}}\{\|AX\|: \|x_i\|^2=1\}$$

where $\|\cdot\|$ denotes the Euclidean norm. I know how to find a closed form solution of

$$\max_{X \in \mathbb{R}^{Nm}}\{\|AX\|: \|X\|^2=1\}$$

that is, when $\|X\|^2=1$. However, I do not see how to solve it when we have $\|x_i\|^2=1$.