If we have an optimization problem whose objective function is a sum of functions
$$\min_x \sum_i f_i(\mathbf{x}) \\ \text{subject to}\quad \left\{ \begin{aligned} h_j(\mathbf{x})&\leq 0 \\ g_k(\mathbf{x}) &= 0 \end{aligned} \right.$$
what are the necessary conditions for writing this problem in the following epigraph form?
$$\min_{\mathbf{x},\, t_i} \sum_i t_i \\ \text{subject to}\quad \left\{\begin{aligned} f_i(\mathbf{x}) &\leq t_i \\ h_j(\mathbf{x}) & \leq 0 \\ g_k(\mathbf{x}) &= 0 \end{aligned} \right.$$