How can I linearize this constraint
$$d_{u,c}\sigma \le \|{\bf f}_{u,c}\|^2\le Td_{u,c}$$ $\sigma$ is a very small number based on scale of $f$
$T>0$, ${\bf f}_{u,c}$ is optimization variable, a complex vector. $d_{u,c}$ is also an optimization variable which in binary.
$d_{u,c}\sigma \le \|{\bf f}_{u,c}\|^2$ is inherently non-convex, because affine <= convex is non-convex, unless the LHS is zero, in which case this constraint can be eliminated as vacuous. Therefore, it can't be linearized into DCP compliance. I am interpreting the norm as being the two-norm. The one-norm or infinity-norm could be linearized into DCP compliance, as shown in Section 9,1 of Mosek Modeling Cookbook, suitably adjusted to handle complex variables, and for infinity norm as opposed to one-norm.
If you really need this constraint, I recommend you abandon thoughts of using (MI)DCP for this problem. Rather, use a high quality off-the-shelf non-convex optimization solver. You could try some sort of Successive Convex Approximation or other dubious scheme, but I don't recommend it. Instead, let the non-convex solver handle any such things in the context of a high quality implementation, not a crude, unsafeguarded hack job, as seems to be de rigeur on some DCP forums.
I guess by the constraint $\|\mathbf f_{u, c}\|^2 \geq \sigma d_{u,c}$ with a very small $\sigma$, what you want is $d_{u,c} = 1 \Rightarrow \|\mathbf f_{u,c}\|^2_2 > 0$. In this case, since $\|\mathbf f_{u,c}\|_2 > 0 \Leftrightarrow \|\mathbf f_{u,c}\|_1>0$, you can replace the l2 norm with l1 norm, and l1 norm could be transformed to DCP, as mentioned. So the result is $$ d_{u, c} \tilde{\sigma} \leq \|\mathbf f_{u,c}\|_1\\ \|\mathbf f_{u,c}\|_2^2 \leq Td_{u,c} $$ where $\tilde{\sigma}$ is another small number.
