Suppose $\mathcal{X}$ is compact, and consider an optimization problem: $$\max_{x \in X_{n}} f(x)$$ where $f$ is a continuous function, and $X_{n}\subseteq \mathcal{X}$ is a sequence of compact sets converging to some set $X \subseteq \mathcal{X}$ (for example, in the Hausdorff metric). Now let $\{x_{n}\}$ be a sequence of minimizers of $f$; i.e. $f(x_{n}) = \min_{x \in X_{n}} f(x)$. Note that $f(x)$ may have multiple minimums.
I am interested in conditions under which the arbitrary sequence of minimizers $\{x_{n}\}$ converges to a point in $X \subset \mathcal{X}$. Does anybody know?
Further Background:
I often see the statement: by compactness of $\mathcal{X}$, we can assume without loss of generality that $x_{n} \to x \in \mathcal{X}$. My problem with this statement is that it uses the fact that $\{x_{n}\}$ has a convergent subsequence, and says nothing about whether my original sequence converges.
This discussion is also related to the theorem of the maximum, which I believe can be used to establish that $X^{*}(n) = argmin\{ f(x) : x \in X_{n}\}$ is upper hemicontinuous. However, similar to the point above, it seems to me that upper hemicontinuity is not enough to guarantee that $x_{n} \to x$, only that it has a convergent subsequence.