Asking for some references on correlations of joint optimization problems

Question

Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in the literature.

1. We are given two "reasonable compact" sets $S_1\subseteq S_2 \subseteq \mathbb{R}^d$ for some $d\geqslant 1$ and a Continuous function $f: \mathbb{R}^d \to \mathbb{R}$. Assuming that we have $\hat{x}_1 \in \text{argmin}_{x \in S_1} f(x)$ and $\hat{x}_2 \in \text{argmin}_{x \in S_2} f(x)$. What can we say about the dependence of $\hat{x}_1$ with $\hat{x}_2$ in terms of $S_1$ and $S_2$? I know that in general, there would be counterexamples, but I am reaching out if there is any structure theorem based on some sufficient conditions on $f, S_1, S_2$.
2. Now, let's say I have two Continuous functions $f_1, f_2: S \to \mathbb{R}$ on a compact subset $S\subseteq \mathbb{R}^d$, depending on a common parameter $A$. What can we say about the closeness of $\hat{x}_1 \in \text{argmin}_{x \in S} f_1(x,A)$ and $\hat{x}_2 \in \text{argmin}_{x \in S} f_2(x,A)$ in terms of the closeness of $f_1$ and $f_2$.

Answer 1

Both questions seem to be a bit broad, but I can say a little:

Conditions that will help are strict convexity and smoothness. Without convexity of $S_1$, $S_2$ or $f$ the set of minimizers can be wild (e.g. disconnected). You'll find results in this direction under the name "sensitivity analysis" e.g. in Section 5.6 in Boyd and Vandenberghe's "Convex Optimization" or Section 12.8 of Nocedal and Wrights "Numerical Optimization". In 2. I did not really get the role of the parameter $A$, but in general the question is quite similar to 1. and the above references will also help.