I have the following max-min problem:
$\underset{{\bf X}}{\max} \underset{k}{\min} \|{\bf A}_k{\bf x}_k\|^2_2 $
where ${\bf X} = [{\bf x}_1, \dots, {\bf x}_K]\in \mathbb{C}^{N \times P}$ and ${\bf A}_k = [{\bf a}_1, \dots, {\bf a}_M]^H \in \mathbb{C}^{M \times N}$.
Now suppose I have a upper-bound for $|{\bf a}_m^H {\bf x}_k| \leq u_m$, then defining ${\bf U} = [{u}_1, \dots, {u}_M]^T \in \mathbb{R}^{M \times 1}$ can I write
$\underset{{\bf X}}{\max} \underset{k}{\min}\} \|{\bf A}_k {\bf x}_k\|^2_2 \Leftrightarrow \underset{{\bf X}}{\min} \underset{k}{\max} {\| |{\bf A}_k {\bf x}_k| - {\bf U}}\|^2_2$
That is to say, maximizing the minimum $\|{\bf A}_k {\bf x}_k\|^2_2$ over $k$ is equivalent to minimizing the maximum mathcing error between $( |{\bf A}_k {\bf x}_k| - {\bf U})$ over $k$?
Edit: I have the following explanation for the equivalence of the two problems:
$|{\bf a}_m^H {\bf x}_k| \leq u_m \implies |{\bf A}_k {\bf x}_k| \preccurlyeq {\bf U} \overset{(a)}\implies \underset{k}{\min} \|{\bf A}_k {\bf x}_k\|^2_2 \Leftrightarrow \underset{k}{\max} {\| |{\bf A}_k {\bf x}_k| - {\bf U}}\|^2_2, \; \forall {\bf X} \overset{(b)}\implies \underset{{\bf X}}{\max}\underset{k}{\min} \|{\bf A}_k {\bf x}_k\|^2_2 \Leftrightarrow \underset{{\bf X}}{\min} \underset{k}{\max} {\| |{\bf A}_k {\bf x}_k| - {\bf U}}\|^2_2$
Step (a) follows that the '$k$' with the minimum $\|{\bf A}_k {\bf x}_k\|^2_2$ is equivalent to the '$k$' with the maximum total matching error between $|{\bf A}_k {\bf x}_k|$ and ${\bf U}$, $\forall \; {\bf X}$. Subsequently step (b) originates from step (a) implying that ${\bf X}$ maximizing the minimum $\|{\bf A}_k {\bf x}_k\|^2_2$ over the all the 'k' is equivalent to ${\bf X}$ minimizing the maximum total matching error between $|{\bf A}_k {\bf x}_k|$ and ${\bf U}$ over the all the 'k'.