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How is it possible to transform the following non-linear constraints into linear constraints? I can rewrite the constraints as follows: $8x^2y \le 32, \frac{4xz^2}{y^3} \le 32, \frac{x^5y^2}{z^6} \le 32$.

Then I can convert them to equalities: $8x^2y+b_1 = 32, \frac{4xz^2}{y^3} +b_2= 32, \frac{x^5y^2}{z^6} +b_3=32$

However, once I am at this points, I don't see how I can put the constraints in an appropriate linear form. How can I proceed? Thanks

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Can you take logarithm of everything both sides? The variables are strictly positive so the logarithm is well-defined, and it is an increasing function so it respects all the inequalities and the maximization.

Notice that the objective function is also a product. So, if you rename $u=\log(x)$, $v=\log(y)$, $w=\log(z)$, then the objective is to maximize $u+v+w$ (it is $\log(xyz)$, so logarithm of the previous objective function).

The first constraint, you can rewrite is as $\log(x^2yz)\leq\log(4)$ so $2u+v+w\leq\log(4)$. You can do the same for the other two constraints and it all becomes linear.

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    $\begingroup$ Thanks! This makes perfect sense. $\endgroup$ – user1354784 Feb 6 '17 at 2:02
  • $\begingroup$ Now I re-read it and I noticed my response is pretty badly written, in terms of well-structured thoughts and so on, but I am glad you got the idea and it was useful for you! $\endgroup$ – Anna SdTC Feb 6 '17 at 2:31

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