I have the following objective:
\begin{equation} \max_{\mathcal{I}} \sum_{m=1}^{M}w_m\sum_{n \in N_m}^{ }I_{m}^{n} \end{equation}
subject to some constraints, beside tha fact that the variables $I_{m}^{n} \in \mathcal{I}$ and $I_{m}^{n} \geq 0$ and $I_{m}^{n} \leq I_{max}$ and $w_m$ are some positive scalars.
I was thinking of doing the following transformation which will be useful later
\begin{align} \max_{\mathcal{I}}& \sum_{m=1}^{M}w_m\sum_{n \in N_m}^{ }I_{m}^{n} \nonumber \\ = \min_{\mathcal{I}}& \left(\sum_{m=1}^{M}w_m\sum_{n \in N_m}^{ }I_{m}^{n} \right )^{-1} \nonumber \\ \end{align}
then rewrite as:
\begin{align} &\min_{t \geq 0} ~t^{-1} \nonumber \\ &\text{subject}~\text{to} \nonumber \\ \sum_{m=1}^{M}w_m&\sum_{n \in N_m}^{ }I_{m}^{n} \leq t \nonumber \\ \end{align}
Is this a valid transformation ?
well after some thinking i guess it is not valid because the new objective can be equivalently written as
\begin{equation} \max_{t \geq 0} t \end{equation}
which results in $t=\infty $ which leads to the constraint being
\begin{equation} \sum_{m=1}^{M}w_m\sum_{n \in N_m}^{ }I_{m}^{n} \leq \infty \end{equation}