I have a nonlinear optimization problem with linear constraints. How to solve this? \sigma_i and \rho_i are the optimization variables.

\min\hspace{1mm}\max{D_1,D_2}

subject to

\sigma_1+\sigma_2=\sigma

\rho_1+\rho_2=\rho

0<\rho_i\le c_i\hspace{1mm},i=1,2

\sigma_i\ge 0\hspace{1mm},i=1,2

\sigma_i=0,\hspace{1mm}\text{if }\rho_i=c_i\hspace{1mm},i=1,2

where

D_i=\frac{\sigma_i}{c_i-\rho_i}+f_i

c_i and f_i, i=1,2 are positive and greater than 0.