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How to solve the below optimization problem? $P$ is a probability matrix, $0\le P_{ij}\le 1$. Are there any developed tools to solve this? Thanks a lot.

\begin{equation*} \begin{aligned} &\underset{X}{\operatorname{argmax}} && \sum_{i}\sum_{j} X_{ij} \times P_{ij} \times w_j \\ &\text{subject to} && \sum_{j} X_{ij} = 1 \quad i=1,\ldots,N \quad AND \quad \forall{i,j} \quad X_{ij} \in (0,1) \\ & && \frac{\sum_{i}\sum_{j} X_{ij} \times P_{ij} \times w_j }{\sum_{i}\sum_{j} X_{ij} \times P_{ij} } \hspace{5mm} \ge ARPU_{threshold} \end{aligned} \end{equation*}

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    $\begingroup$ If $P_{ij}$ and $w_j$ are input parameters, you can linearize by clearing the denominator. $\endgroup$
    – RobPratt
    Commented Jul 19, 2023 at 16:37
  • $\begingroup$ Yes, they are both constant parameters. But the denominator has $X_{ij}$ inside, how can I clear it? Thanks for your reply. $\endgroup$
    – Yi-Yu Peng
    Commented Jul 20, 2023 at 1:05
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    $\begingroup$ The LHS is a ratio of linear functions. Multiplying both sides of the constraint by the denominator yields a linear constraint. $\endgroup$
    – RobPratt
    Commented Jul 20, 2023 at 1:23
  • $\begingroup$ Yes! I just discover this, quite simple conversion! Thanks again. $\endgroup$
    – Yi-Yu Peng
    Commented Jul 20, 2023 at 1:46

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