Assume we have two convex objective functions, $f$ and $\tilde{f}$, where
$$ f(x) \leq \tilde{f}(x), \quad \forall x \in \mathbb{R}^d$$
Let
$$\begin{aligned} x^* &:= \arg\min_{x} f(x)\\ \tilde{x}^* &:= \arg\min_{x} \tilde{f}(x)\end{aligned}$$
How to obtain an upper bound for $\| x^*-\tilde{x}^* \|$?
I think that this problem is way too general. But is there any paper, or method which suggest a method for a problem of this type?