In constraint optimization problem, one is often confronted with the following problem: $min$ $f(x)$ , $x \in R^n$ given
$g_i(x) = c_i$ where $i = 1,...m$
$h_j(x) < c_j$ where $j = 1,...p$
All the functions above are assumed to be in $C^\infty$
Question: Is there a way to tell whether the feasible set (as defined by the constrains above) can defined as a smooth manifold? Could you please also provide keyword(s) about topics related to this question?