The objective function I am dealing is

\underset{{\bf w}_k,x_k }{\max}\sum_{k=1}^K x_k \log_2(1+\gamma_k)

Here,

x_k, k=1,2,\cdots,K are binary variables

\gamma_k, k=1,2,\cdots,K are also functions of other optimization variables.

\gamma_k=\frac{|{\bf h}_k{\bf w}_k|^2}{\sigma^2+\sum_{i=1,i\neq k}^K |{\bf h}_i{\bf w}_i|^2}.

The vectors {\bf h_k}\in\mathbb{C}^{1\times N}, k=1,2,\cdots, K are known.

How can I linearize them?

\textbf{Some Tricks}

\underset{{\bf w}_k }{\max}\sum_{k=1}^K x_k \log_2(1+\gamma_k)

=\underset{{\bf w}_k }{\max}\sum_{k=1}^K \log_2(1+\gamma_k)^{x_k}

=\underset{{\bf w}_k }{\max}\prod_{k=1}^K(1+\gamma_k)^{x_k}

=\underset{{\bf w}_k }{\max}\prod_{k=1}^K(1+\gamma_k{x_k})