I am curious if there are any results that give both sufficient and necessary conditions for having a unique solution to the quadratic optimization problem $$ min_{x} f(x)\\ \text{subject to }x\in K $$ with $f(x)=x^\top M x + x^\top q$, $M=M^\top$, and $K\subset \mathbb{R}^n$ is convex and compact.
I know that since $f$ is continuous and $K$ is compact, the problem has at least one solution. Also a sufficient condition for having unique solution is that $M$ is positive definite. But are there any results that give both sufficient and necessary conditions?