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 $\|x\_i\|^2=1$.



Maximize the Euclidean norm of a matrix times a vector on unit sub-spheres
You cannot hope for anything like a closed form solution, or even an exact efficient algorithm for this problem, because it is [NP-hard](https://en.wikipedia.org/wiki/NP-hardness). The reduction is from the [max-cut](https://en.wikipedia.org/wiki/Maximum_cut) problem. Let's look at the special case $m=1$. Let $G = (V, E)$ be a graph, and orient the edges in some arbitrary way. Let $A$ be the edge-by-vertex incidence matrix of $G$, i.e. the rows of $A$ are indexed by edges and the columns by vertices, and $a\_{e, v}$ is $1$ if $v$ is the tail of $e$, $-1$ if $v$ is the head of $e$, and $0$ otherwise. Then it's easy to check that for any $x \in \mathbb{R}^V$, $\|Ax\|^2 = \sum\_{(u,v) \in E}{(x\_u - x\_v)^2}$. If $x \in \{-1, 1\}^V$ as in the $m=1$ case of your problem, then $\|Ax\|^2 = 4e(S, \bar{S})$, where $e(S, \bar{S})$ is the number of edges cut by the set $S = \{v: x\_v = 1\}$. 


You can find a constant factor approximation algorithm (due to Nesterov) in Chapter 6.3. of the [Williamson and Shmoys book](http://www.designofapproxalgs.com/download.php).


