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?

Thanks

`$U=\{x:\forall j\ h_j(x)<c_j\}$`

to $\mathbb R^m$, and the feasible set is the pre-image of the point $(c_1,\dots,c_m)\in\R^m$. The regularity condition is equivalent to that the gradients of the functions $g_i$ are linearly independent at every point of the pre-image. See also Sard's theorem. – Sergei Ivanov Sep 18 '12 at 16:03