Probably the most innocent formulation of my question is this: given a quadratic polynomial function $f$ over the positive orthant $\mathbb{R}_{\geq 0}^{n}$ that is bounded below, is it true that $f$ must attain its minimum at some point?
In the setting of quadratic programming, we are asked to minimize a quadratic polynomial over a convex polytope: $$ \begin{align*} \text{min }& \frac{1}{2}x^{T}Qx+c^{T}x\\ \text{s.t. }& Ax=b\\ & x\succeq 0 \end{align*} $$ in which $Q$ is a $n\times n$ real symmetric matrix and $x$ is a vector in $\mathbb{R}^{n}$. The general question is this: if the quadratic polynomial is bounded below on the convex polytope, is it guaranteed to attain its minimum at some point? In other words, under a polynomial function of degree $\leq 2$, is it true that the image of a convex polytope is a closed subset of $\mathbb{R}$? It seems to me that this question is closely related to the existence problem of solutions to general quadratic programs, so I guess it is already well-studied. However, I failed to find any good reference for this.
Here are some commonplace thoughts: If $Q$ is (strictly) positive definite, then the quadratic polynomial $f(x)$ would be large for all $x$ such that $|x|>c$. So we could restrict the domain to a bounded subset of the polytope and resort to fact that a continuous mapping maps a compact set to a compact set and obtain a minimum point. However, when $Q$ is indefinite or even positive semidefinite, this approach doesn't seem to work very well.
Also note that for general polynomials of degree $n$, this statement is false--i.e., general polynomials over a convex polytope can be bounded below and have no minimum, see this question: Does a polynomial that's bounded below have a global minimum?
