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



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.

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.