What is the optimal value of a quadratic program when there does not exist a solution?

Question

If I have a quadratic program of the form

$$\min \frac 12 x^T Q x - \langle b,x \rangle$$

and there does not exist a solution, then what is the optimal value?

My intuition tells me that it is negative infinity! beause that's what would happen if the $Q$ matrix were indefinite. But I cannot prove it to myself, even though intuitively this makes sense to me. How would one go about proving that the optimal value must be $-\infty$?

Answer 1

If There does not exist a solution to $min h(x)$, then $h(x)$ must not be a coercive function. Which means Q must not be positive semi definite. That means that there exists at least one negative eigenvalue, which means the problem is unbounded below.

Answer 2

If $Q$ is not positive semidifinite then problem is unbounded below (The direction which takes us to $- \infty$ is the eigenvector corresponding to the negative eigenvalue).

So lets think about the case where $Q$ is positive semidefinite. Then we have following theorem

Theorem:

$f$ has optimal solution if and only if $f$ has finite infimum value.

Pf: Left to Right is trivial.

For Right to left look at "https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-253-convex-analysis-and-optimization-spring-2012/lecture-notes/MIT6_253S12_lec_comp.pdf" page 77.