I have a mixed integer programming problem as below
$$\underset{{\bf w}_k }{\max}\sum_{k=1}^K x_k\alpha_k \log_2(1+\gamma_k)$$ subject to $$\sum_{k=1}^K x_k||{\bf w}_k||^2_2\le P$$
$$x_k\in\{0,1\}$$
How can we deal with the objective and the first constraint to have an efficient solution?
Your problem is complicated due to the binary variables and the will probably be no quick and easy answer.
For the "weighted sum rate maximization" problem, which is basically your problem without these variables, an iterative algorithm has been proposed in https://authors.library.caltech.edu/19444/1/Tan2009p110922009_Ieee_International_Symposium_On_Information_Theory_Vols_1-_4.pdf.
The binary variables of your approach can be interpreted as users that are selected or excluded from a wireless transmission. Such an approach has been made in https://core.ac.uk/download/pdf/32242541.pdf