I have the following optimization problem:
Given a complex sequence $H_i$, $1 \leq i\leq N$. Find a complex sequence $G_i$ that minimizes:
$$ \lambda\cdot\max_i { |H_i\cdot G_i - 1|^2 } + \sum_i |G_i|^2 $$
The complex can be reduce by taking the $G_i=g_i\cdot\frac{\bar{H_i}}{|H_i|}$ and then the problem is:
$$ \lambda\cdot\max_i { (h_i\cdot g_i - 1)^2 } + \sum_i g_i^2 $$
where $h_i = |H_i|$ and $g_i$ are reals.
I am not sure that the problem is convex. It seems like the solution should take the maximal $h_i$ and somehow fix maximal value of $(h_i\cdot g_i - 1)^2$ based on that one but I can't seems to prove any of this,..