On Existence and Derivation of a Central Path

Question

This question has been haunting me since I began to learn about the interior point method. Consider an optimization problem: $$ \min_x. ~f_0(x)\quad \text{s.t.}~f_i(x)\leqslant 0 ~\forall i=1,2,...,m, $$ where each $f_i$ (including $f_0$) is convex. One may define the $h$-barrier function for the problem as $$ \phi_h(x) = \sum_{i=1}^m h(f_i(x)), $$ where $h: \mathbb{R}_{< 0} \to \mathbb{R}$ is a convex, closed, twice differentiable, and increasing function (and thus $\phi_h$ has domain $\{ x:f_i(x) < 0\} $). Similar to what one does for central paths, one may define an $h$-central path as $$ x^*(t) = \text{argmin}_x tf_0(x) + \phi_h(x) ~(t > 0). $$ My question is, why is it legitimate to compute $x^*(t)$ by solving $$ \nabla_x (tf_0 +\phi_h) = 0, $$ if it indeed exists? How do we guarantee that this equation is solvable (since derivation of $x^*$ isn't necessarily an unconstrained optimization problem)? Any help will be appreciated!