I am trying to find a solution for following
\begin{eqnarray} \text{minimize }~~ -\sum_{i=1}^K \frac{1}{a_i+b_i2^{(-2x_i)}} \\ \text{s.t.} ~~~ \sum_{i=1}^Kx_i=C \end{eqnarray}
where $a_i,b_i,x_i\geq 0$.
when I check the convexity it requires $b_i < a_i 2^{(2x_i)}$ or $x_i>log_2(\sqrt{b_i/a_i})$.
applying KKT conditions yield the solution like following:
\begin{eqnarray} x_i^*=\frac 12\cdot log_2\left({\frac{b_i}{\lambda-2a_i-\sqrt{\lambda^2-4\lambda a_i}}}\right) \end{eqnarray} where $\lambda$ is some Lagrange multiplier.
I have two questions:
1- if all $x_i^*$'s satisfy the constraint $x_i^*>log_2(\sqrt{b_i/a_i})$ can I claim that the solution is optimal?
2-what if some of $x_i^*$'s violate the constraint $x_i^*>log_2(\sqrt{b_i/a_i})$? Can I still claim that solution is optimal? if not so what is the optimal solution !
my simulations corroborate that the solution coming from KKT is optimal for all scenarios, but I am confused with the math.