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I am trying to solve an optimization problem that contains power functions. I reformulated the problem via a logarithmic function, and it works well. The terms of the problem involved are similar to $\log(x) = y$ where $(x, y) \in \mathbb{R}$. Now, I would like to reformulate the mentioned term as its equivalent, conic programming. A wise man said the term is non-convex itself, and I need to relax it and use an exponential cone. However, I am unsure how I can represent that. Also, the main part of the optimization model would be:

\begin{align} &\text{Max}_{(x,y)} \ = \frac{y}{x} \\ &s.t:\\ & \log (x) = y \\ & x, y \in \mathbb{R} \\ \end{align}

A.Omidi
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  • A lot of exponential cone models of standard functions can be found in https://docs.mosek.com/modeling-cookbook/expo.html#modeling-with-the-exponential-cone – Michal Adamaszek Oct 25 '25 at 15:18
  • @MichalAdamaszek, I saw this link. However, I cannot transform my problem, and this is why I asked the question. Would you elaborate more? – A.Omidi Oct 26 '25 at 07:10
  • You would have to post the details of your model to get any specific advice. As of now your question is completely general. – Michal Adamaszek Oct 26 '25 at 11:35
  • @MichalAdamaszek, I just updated the question by adding the main part that contains the log term. – A.Omidi Oct 26 '25 at 12:31
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    $\log{x}/x$ is not concave so most likely it has no convex reformulation unless there is something else in your model allowing that. If you already looked at the Mosek modelig cookbook then I would recommend asking at https://groups.google.com/forum/#!forum/mosek with more details of the model. – Michal Adamaszek Oct 27 '25 at 07:39
  • @MichalAdamaszek, I will try ASAP. Thanks. – A.Omidi Oct 27 '25 at 08:01

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