# How to test emptiness of constrained sub-set.

Let $X$ be subset of $\mathbb{R}^n$ which is $n$-dimensional space.

This subset is defined by k inequalities: $g_{i}(x)<0$, $x\in X$, $i=1..k$ and m equalities: $h_j(x)=0$, $x\in X$, $j=1..m$.

How to test that the equalities-inequalities are inconsistent? That is, $X$ is empty set.

As far as I understand this can be checked in polynomial time.

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Do you know something about the $g_i, h_j$? Are they linear, convex,... –  copper.hat May 19 '12 at 15:30
$g_i$ and $h_j$ are general (non-convex), but they are at-least continuously differentiable –  0x2207 May 19 '12 at 15:41
I think that's too broad a category of problems to expect P-time checking? –  copper.hat May 19 '12 at 22:06