Consider the following problem, where $a_j,b$ and $c_j$ are positive constants:
Minimize $\sum_{j=1}^n \frac{c_j}{x_j}$, subject to $\sum_{j=1}^n a_j x_j = b, x_j ≥ 0$ for $j= 1,...,n$. Write the KKT condition and solve the point $\bar{x}$ satisfying these conditions.
I get the KKT conditions as:
$-\frac{c_i}{x_i^2}+\lambda a_i=0$ for $i=1,2,...,n$
$\lambda(a_1x_1+a_2x_2+...+a_nx_n-b)=0$
and that $x_i=\sqrt{\frac{c_i}{\lambda a_i}}$
but I'm having trouble solving these kinds of general systems of equations.