Consider the following sum of fractional functions optimization problem $$ \begin{array}{l} \mathop {\min }\limits_{\bf{x}} \,\,\,\sum\limits_{i = 1}^p {\frac{1}{{{\bf{a}}_i^T{\bf{x}} + {b_i}}}} \\ subject\,\,to:\,\,\,{\bf{x}} \in X \end{array}$$
where ${{\bf{a}}_i^T{\bf{x}} + b}>0$ and $X$is a non empty compact convex set. How can I solve the optimization problem?