The optimization problem is:

maximize $$\min(\sum\limits_{i=1}^N \log\left(a_{1,i}+\frac{b_{1,i}}{c_{1,i}+d_{1,i}x_i}\right),\sum\limits_{i=1}^N \log\left(a_{2,i}+\frac{b_{2,i}}{c_{2,i}+d_{2,i}x_i}\right))$$ subject to $x_i\ge0, i=1,...,N$ and $\sum\limits_{i=1}^N x_i=C$, where $x_i$ are variables, $a_{1,i}, b_{1,i}, c_{1,i}, d_{1,i}, a_{2,i}, b_{2,i}, c_{2,i}, d_{2,i}$ and $C$ are constants.

Kuhn–Tucker conditions can only optimize one $\sum$ alone. but I don't know how to handle optimization problem involving $\min$ operation as above. Could you please tell me what method can I employ to solve it (I would greatly appreciate if you can refer me to a textbook containing exposition of the method)? Thank you very much!