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Is it possible to express these constraints with basic cones?

I have the following optimization problem: \begin{align}\min&\quad x\\ \text{s.t.}&\quad x=\max_{i} \{x_{i}\}\\ &\quad x_{i}y_{i}=z_{i}\\ &\quad x_{i}, y_{i}, z_{i}\geqslant0 \end{...
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