I want to use Newton's method to minimize $$ f(x) = c^T x - \sum_{i=1}^m \log(a_i^T x + b_i) $$ where $c \in \mathbb{R}^n$, $b \in \mathbb{R}^m_+,$ and $A=[\; a_1 \; \cdots \; a_m \;]^T \in \mathbb{R}^{m \times n}, m>n.$
However, the program is not always converge.
Is there any condition on $A$, $b$, and $c$ can guarantee the convergence of Newton's method?