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$F$ is $m\times m$ diagonal, with real non negative elements

$D$ is $n \times m$ complex

$P$ is $n \times 1$ complex

$A$ is $m \times 1$ complex.

Minimize $\Gamma(A)$, with respect to $A$.

$$\Gamma(A) = \frac{m^2(DA-P)^H (DA-P) + (FA)^H(FA)}{A^HA}$$

It is known that both numerator and denominator of $\Gamma(A)$ are convex and non negative. Also both the terms of the numerator are individually convex and non negative.

Question: An numerical optimization algorithm to find global minimum.

Apart from a reagular solution, I am also interested in a gradient descent based method if possible as the matrices are large. Also $m >> n$.

PS : This question is a specific version of this question.

EDIT : more known information

No constraints on problem but

  1. $\sum P = 0$, I mean sum of elements of matrix $P$ is zero.
  2. Diagonal elements of $F$ are not all zeros.
  3. Also $P^HP \ne 0$.
  4. Rows of $D$ are orthogonal to each other. Also they are linearly independent.
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  • 1
    $\begingroup$ What is your question? $\endgroup$ – nicoguaro Feb 24 at 3:47
  • $\begingroup$ @nicoguaro An numerical optimization algorithm to find global minimum. $\endgroup$ – Rajesh Dachiraju Feb 24 at 4:30
  • $\begingroup$ it can also be a reference request, if a numerical scheme already exists. $\endgroup$ – Rajesh Dachiraju Feb 24 at 4:55
  • $\begingroup$ If your objective function is not convex you don't have guaranteed to attain the global minimum. $\endgroup$ – nicoguaro Feb 24 at 13:50
  • $\begingroup$ @RajeshDachiraju: This problem has a nice solution if you don't have to divide by $A^HA$. You might want to reflect on whether that's a necessary thing to do. $\endgroup$ – Richard Feb 25 at 1:07

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