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$$\begin{array}{ll} \text{minimize} & \prod_{i=1}^n (x_i+a)-k\prod_{i=1}^{n-1} (x_i+a)-\dots-k(x_1+a)\\ \text{subject to} & \displaystyle\prod_{i=1}^n x_i\geq m\end{array}$$

where $x_i >0$ are variables, and $a>0$, $k>0$, $m>0$ are given constants.

  1. Can I convexify this problem by defining $x_i+a$ as a new function?

  2. Is it feasible to relax this problem and obtain an approximated global optima?

  3. I plan to try Lasserre's POP method, but am not sure the performance.

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