I have the problem
$$ \begin{align*}\min \quad&f(x)= c^Tx + x^TQx \\ &x\in D \end{align*}$$ with $D=\{ x \in \mathbb{R}^n \mid Ax \leq b\}$, $A,Q\in \mathbb{R}^{n\times n}$ and $b,c \in \mathbb{R}^n$ and $Q$ is positive definite which implies that f is stricly convex.
It is claimed here on page 27 on the bottom that $f$ attains its minimum value, i.e. it exits a $x_0 \in D$ with $f(x_0) = \inf \{ f(x) \mid x\in D\}$ if $$D\neq \emptyset \quad\text{ and } \quad \inf \{ f(x) \mid x\in D\} \neq - \infty$$ holds. However, there is no proof given.
Can anyone tell me where I can find this proof or explain why this should be the case?
Let $f^{*}$ be the infimum of $f$ on $D$ and consider a sequence $x_k$ such that $f(x_k)\to f^{*}$. Since $f$ is quadratic and its Hessian is positive definite, we have that $(x_k)$ is bounded. Indeed, if $\lambda>0$ is the smallest eigenvalue of $Q$, we have $f(x)\geq\lambda|x|^2-|c||x|$. So, $(x_k)$ has a convergent subsequence. Let $x^{*}$ be the limit point of this subsequence. Then, $f(x^{*})=f^{*}$.