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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,..

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  • $\begingroup$ I did some simulation on the real problem. The solution seems to be of the form: (a+1)/h where a is a single parameter that is the value of the max part. the idea is to make the max balanced at a for all the vector elements and then optimize for a. this seems to be the optimal solution be I don't see how to prove it $\endgroup$
    – nir
    Apr 22, 2023 at 16:11
  • $\begingroup$ The function seems to be also convex (the real one). Actually, this seems to be trivial. the square function is convex. The affine part g*h-1 remains convex. and max of convex function is also convex $\endgroup$
    – nir
    Apr 22, 2023 at 16:26
  • $\begingroup$ It should be rather $(1-a)/h$: there is no point in overshooting $1$ when you can approach it from below instead. Otherwise you are right and it is not hard to finish from there. :-) $\endgroup$
    – fedja
    Apr 23, 2023 at 1:41

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