A Are these two optimization problems equivalent?

AI Thread Summary
The discussion revolves around the equivalence of two optimization problems, P1 and P2, involving complex variables and constraints. The first problem aims to maximize the squared difference between a complex scalar d and a sum involving variables z_n, subject to a specific constraint. The second problem reformulates the first by introducing new variables y_n, making it potentially easier to solve using semidefinite programming. The main inquiry is whether these two problems are equivalent in terms of their solutions and constraints. The thread concludes with a request to delete the post due to an early submission error.
haji-tos
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Hello,

I need help please. I have the following optimization problem defined as

\begin{equation}
\begin{aligned}
& (\mathbf{P1}) \quad \max_{\mathbf{z}} \quad \left| d -\sum_{n=1}^{N} \frac{c_n}{f_n + z_n} \right|^2 \\
& \text{subject to} \quad \sum_{n=1}^{N} \frac{|a_n|^2 \text{Re}(z_n)}{|f_n + z_n|^2} = 0.
\end{aligned}
\end{equation}
where d is a complex scalar, f=[f1,...,fN], c=[c1,...,cN] and a=[a1,...,aN] are complex vectors.

I am trying to solve this so I was thinking to consider

\begin{equation}
y_n=\frac{1}{f_n+z_n}, \quad \forall n \in \{1,...,N\}
\end{equation}
and
\begin{equation}
z_n=\frac{1}{y_n}-f_n, \quad \forall n \in \{1,...,N\}
\end{equation}

and then transform the problem into

\begin{equation}
\begin{aligned}
& (\mathbf{P2}) \quad \max_{\mathbf{y}} \quad \left| d -\sum_{n=1}^{N} c_n y_n \right|^2 \\
& \text{subject to} \quad \sum_{n=1}^{N} |a_n|^2 \text{Re}(y_n^* - f_n y_n y_n^*) = 0.
\end{aligned}
\end{equation}
which is easier to solve using semidefinite programming.
Can you please tell me if the two problems are equivalent ?

Thank you very much !
 
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