\tau_1 log\left( 1 + \frac{\mathbf {A_i} \tau_0 }{\mathbf{B_i} \tau_0 +C_i \tau_1 }\right) =\tau_1 log\left( 1 + \frac{ \tau_0 }{ \frac{\mathbf{B_i}}{\mathbf {A_i}} \tau_0 +\frac{C_i}{\mathbf {A_i}} \tau_1 }\right)
According to your solution at (Xlog( 1+ Y/(X+Y) ): DCP rules, and build-in functions in cvx) my function can be transformed as
(\tau_0+\tau_1) log\left( 1 + \frac{ \tau_0 }{ \frac{\mathbf{B_i}}{\mathbf {A_i}} \tau_0 +\frac{C_i}{\mathbf {A_i}} \tau_1 }\right) - \tau_0 \ log_2\left( 1 + \frac{ \tau_0 }{ \frac{\mathbf{B_i}}{\mathbf {A_i}} \tau_0 +\frac{C_i}{\mathbf {A_i}} \tau_1 }\right).
I noticed that the second term, - \tau_0 \ log\left( 1 + \frac{ \tau_0 }{ \frac{\mathbf{B_i}}{\mathbf {A_i}} \tau_0 +\frac{C_i}{\mathbf {A_i}} \tau_1 }\right), can be written as -\text{rel_entr}( \frac{C_i}{A_i} \tau_1 + \frac{B_i}{A_i} \tau_0, \frac{C_i}{A_i} \tau_1 + \frac{B_i}{A_i} \tau_0+ \tau_0) - \text{rel_entr}( \frac{C_i}{A_i} \tau_1 + \frac{B_i}{A_i} \tau_0+ \tau_0, \frac{C_i}{A_i} \tau_1 + \frac{B_i}{A_i} \tau_0) .
How can we change the first term? Help is highly appreciated. Thanks.