Is there a notion of "best" convex relaxation of a particular function in a normed vector space? For example, the $\ell^0$ pseudo-norm can be relaxed into the $\ell^1$ norm, and that allows us to solve sparse recovery problems. Is there a general recipe to construct such relaxations given an arbitrary function?
Specifically, I have a function of the form $f(x) = \|x\|_2\|x\|_1$, which is non convex. How would I go about relaxing this? Any resources/links I should look into?