# Minimize $\prod_{i=1}^n (x_i+a)-k\prod_{i=1}^{n-1} (x_i+a)-\dots-k(x_1+a)$

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

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.