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Given symmetric positive semidefinite rank-$r$ matrix $R \in \mathbb{C}^{m \times m}$, where $r < m$, and scalar $p \geq 0$, I have the following optimization problem in matrix $X \in \mathbb{C}^{n \times m}$.

$$\begin{array}{ll} \underset{X \in \mathbb{C}^{n \times m}}{\text{minimize}} & \text{trace} \left( \left(I + \frac{RX^{H}XR}{\sigma^{2}} \right)^{-1} R^{2}\right)\\ \text{subject to} & \text{trace}\left( X^{H} X \right) \leq p\end{array}$$

Can someone help me with this problem, every approach I tried is futile and leading to a dead-end?

Rodrigo de Azevedo
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phani raj
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  • Is $R$ symmetric or hermitian? $;\big(R^T=R;$ versus $;R^H=R\big)$ – greg Feb 24 '21 at 15:27
  • @greg: Very interesting! Would you be able to share details on the minimization aspect? – A rural reader Feb 24 '21 at 17:06
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    Under the assumption that $R$ is hermitian and ${\rm rank}(R)<m$, you could use the "skinny" SVD of $R=UDU^H,$ with unitary $U\in{\mathbb C}^{n\times n};$ to obtain something like $,X=i\sigma D^+U^H;$ – greg Feb 24 '21 at 17:10
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    @Aruralreader It's just a hunch based on what happens when $R$ is hermitian positive definite. And there's also a typo: $;U\in{\mathbb C}^{m\times n};$ – greg Feb 24 '21 at 17:14
  • Thank you @greg, appreciated. Is there a reference you know of and would share that works through the hermitian positive definite case? I'd like to see how it's done. – A rural reader Feb 24 '21 at 17:19
  • @greg , $R$ hermitian and as mentioned is not of full rank , can you please help me understand your solution using ''skinny'' S.V.D seems a lot like it would help me .please post the solution – phani raj Feb 25 '21 at 00:06
  • @ Rodrigo de Azevedo Thank You for the edits – phani raj Feb 25 '21 at 00:33
  • Also rank(R) = r $<$ m – phani raj Feb 25 '21 at 01:46
  • Take a look at this and try to steal ideas from it. – Rodrigo de Azevedo Feb 25 '21 at 14:36

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