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I need to solve this optimization problem:

min $\sum^{N}_{i=1}x_i$,

with the following constraints:

$\sum^{N}_{i=1}\frac{b_i}{log_2(1+\frac{x_iz_i}{s})}-T\leq 0$

$0<x_i\leq X$

where $b_i, z_i >0$ for $i=1,\cdots, N$, and $s, T, X >0$ are all known variables. I tried to solve it with Lagrange multipliers, but did not succeed. Any suggestion on how to solve this?

Garbt
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  • Why is $\sum^{N}_{i=1}\frac{b_i}{log_2(1+\frac{x_iz_i}{s})}-T\leq 0$ a constraint, if it is composed only of "known variables"? Are $b_i$, $z_i$ known parameters or optimization variables? Please clarify what are the optimization variables – Trb2 Jul 06 '23 at 15:35
  • Sorry for the confusion. The optimization variables are $x_i$. – Garbt Jul 06 '23 at 15:40
  • Okay, thanks. Did you consider some software to solve it? – Trb2 Jul 06 '23 at 15:49
  • Not yet. As I am not very expert on optimization techniques, I was just wondering whether there is a relatively straightforward method to obtain a closed form solution. Otherwise, which type of mathematical optimization technique might be more suited for solving the problem. – Garbt Jul 06 '23 at 15:54
  • The objective is convex and the second constraint is also convex. I think the first constraint is also convex (if you plot the sum of 2 1/log, you will see that the intersection with any constant height plane creates a convex set), but I am not completely sure. Then you could use any convex optimization software. Do you work on python, matlab, c++, ...? – Trb2 Jul 06 '23 at 16:26
  • To obtain a closed form solution is somewhat difficult i think. With multipliers you could write down the KKT conditions, but you end up having a system of equations that you have to solve with the computer – Trb2 Jul 06 '23 at 16:27

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