I have an optimization like below:
$\text{ minimize } \sum_k - w_k\log_2 x_k $
$\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$
$\hspace{20mm} 0 \leq w_k \leq 1$
I can form the Lagrange of the problem as below:
$L(x,w, \lambda) = \sum_{k=1}^{N} - w_k \log_2 x_k + \sum_{k=1}^{N} \lambda_k (x_k -q) + \sum_{k=1}^{N}v_k w_k$
But How can I find KKT conditions for this? Moreover, How can I solve the problem analytically from there? I know I can solve it through some nonlinear optimization solver software but for now I am only interested in solving the problem analytically so that I can get an exact form of x.
Is there any better way to solve this for optimal value of $(w, R)$? Any hint is highly appreciated.