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Let $A\in\mathbb{C}^{m,n}$. I am looking to compute $x\in\mathbb{C}^n$ such that for $Ax=y$ it holds:

$|y_i|=1,~i=1,...,m$.

In practice, the columns of $A$ are transfer functions of $n$ sources, sampled at $m$ points in space. I would like to determine a somewhat optimal mixture of the sources such that the magnitude of the transfer function is close to 1 in a least-square sense. So I am looking to minimize:

$\sum_{i=1}^m(|\sum_{j=1}^nA_{i,j}x_j|-1)^2=\sum_{i=1}^m(|<a_i,\bar{x}>|-1)^2$,

where $a_i$ are the rows of $A$.

Any help at approaching this is greatly appreciated. Thank you!

ArtPe
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  • I'm a little confused. As stated, the question is trivially solved by setting $x = 0$, for a minimum of $-1$. Were there supposed to be absolute values in the objective function? Or were we restricting to $|x| = 1$, or something? – user771918 Apr 15 '20 at 13:13
  • Thank you, I am very confused myself. My problem formulation was wrong. I have changed it now. I hope it makes sense now. There is no constraint on $x$, finding a minimal-norm solution, however, is desirable. – ArtPe Apr 15 '20 at 13:41

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