Are these two optimization problems equivalent?

[haji-tos]

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 !

 

[docnet]

Please delete this post... I made a mistake of posting too early, thinking the LATEX was not showing up. Thanks.