Intersection of two $n$-dimensional quadratic inequalities?

Question

I have two quadratic inequalities of the form $$ a_1x^TAx + b_1^Tx + c_1 \le 0\\ a_2x^TAx + b_2^Tx + c_2 \le 0 $$ where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, $b_i\in\mathbb{R}^n$, and $a_i,c_i\in\mathbb{R}$, $i=1,2$.

I wish to determine if the set formed by the two inequalities is bounded. I began by thinking of the case where $n=1$, so we just have parabolic functions. I claimed that as long as the two central axes of the parabolas (the line of symmetry) were not parallel then the intersection would be bounded. I have not proved this claim, but I have a feeling it is true. If so, I am hoping it may be possible to generalize to higher dimensions. If this can be done then we could rule out an unbounded intersection if the two functions do not specify the stringent requirement of parallel central axes.

Any tips or better directions?

Answer 1

We may assume $\mathbf A$ to be symmetric, though it is not specified in your question, because everything remains unchanged by replacing $\mathbf A$ with its symmetric part $\frac12(\mathbf A+\mathbf A^T)$. Now, two easy cases:

First, suppose $\mathbf A$ is positive definite and $a_i\ne 0$. Then there is a radius $r_i$ outside which $\mathbf x^T\mathbf A\mathbf x+(\mathbf b_i^T\mathbf x+c_i)/a_i\ge0$ for all $\lVert\mathbf x\rVert\ge r_i$. So, if either $a_i$ is positive, then the intersection is bounded. If one is negative and the other is nonpositive, the intersection is unbounded. If both $a_i$ are zero, then you're looking at the intersection of two half-spaces $\mathbf b_i^T\mathbf x+c_i\le 0$, which is unbounded unless $n=1$ and the $\mathbf b_i$ are scalars of opposite signs.

Second, suppose $\mathbf A$ is positive semidefinite and both $a_i$ are positive. Let $\mathbf x=t\mathbf u$ for some fixed vector $\mathbf u$. As $t\to\infty$, the condition $a_i\mathbf x^T\mathbf A\mathbf x+\mathbf b_i^T\mathbf x+c_i\le0$ can hold only if $\mathbf u^T\mathbf A\mathbf u=0$, so $\mathbf u$ belongs to the null space of $\mathbf A$. Then we can restrict our attention to $\mathbf u\in\operatorname{Null}(\mathbf A)$, in which the problem reduces to finding whether the intersection of the half-spaces $\mathbf b_i^T\mathbf u+c_i\le 0$ is unbounded. This was already discussed above: it is unbounded unless the null space is one-dimensional, i.e. $\operatorname{Null}(\mathbf A)=\{\alpha\mathbf u_0\}$ for a fixed $\mathbf u_0$, and $\mathbf b_i^T\mathbf u_0$ have different signs.

You could probably extend this analysis to all the other cases, but that's more effort than I have time for right now.