A lot of methods utilize variational inference for hyperparameter calculation.

(ex: does it guarantee a global optimal ? )

1. The outcome tends to depend heavily on the starting point for the optimization. Example: this paper which is heavily cited but known to have severe problems (software packages based on it were later withdrawn, etc.)
2. The calculations required to figure out what you are optimizing are often very complicated. (See any paper on variational inference.)

On the plus side, there is an excellent introduction to the subject in Mackay's textbook Information Theory, Inference and Learning Algorithms.

Briefly:

• Disadvantages: approximate, very little theory around it
VI is a method for approximating a difficult-to-compute probability density, $p$, by optimization. This is done by suggesting a family of distributions $\mathcal Q$ and finding the member $q \in \mathcal Q$ that has the lowest Kullback–Leibler divergence $KL(q \|p)$. How well you can approximate $p$ naturally depends on your choice of $\mathcal Q$, but you can assume that some aspect of $p$ is lost when substituting it by $q$.
VI doesn't guarantee you find the globally optimal member $q \in \mathcal Q$ either. A common choice is to use what's called the mean-field variational family and find $q$ by coordinate ascent. You can find a local optimum.