What is the intrinsic geometry of a feasible set?

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

edited Sep 18 at 14:09

asked Sep 18 at 13:37

Haider
11●2

Each $h_j$ is continuous, so the intersection of the conditions $h_j(x) < c_j$ is an open subset of $\mathbb{R}^n$, therefore a submanifold. Generically, each condition $g_i(x)=c_i$ will give you a codimension one submanifold of $\mathbb{R}^n$ (specifically, when $c_i$ is a regular value of $g_i$) and the intersection will be a submanifold if they are in general position. There should be some condition on the gradient functions of the $g_i$ which ensures this. – Mark Grant Sep 18 at 15:23

Pre-image of a regular value of a smooth map is a smooth submanifold. In your case, the map is $g=(g_1,\dots,g_m)$ from an open set $U=\{x:\forall j\ h_j(x)<c_j\}$ 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 at 16:03

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