Context: Trying to figure out how to find exact solutions to non-smooth optimization problems. Here is an earlier question.
Suppose we have a parameterized family of functions $f_r:\mathbb R \to \mathbb R$ for $r\in R_{\geq 0}$, with $\lim_{r\to \infty} f_r=F$. We can assume this is the uniform limit. Suppose that $f$ is smooth in both $r$ and $x$, but $F$ isn't necessarily smooth. I.e. we think of $f_r$ as increasingly accurate smooth approximations of $F$.
Suppose we have for each $r$ and also for $F$ itself, the optimization problem:
$$\max_x f_r(x)\quad\text{s.t.}\quad g(x)\leq 0\quad\quad (1)$$ $$\max_x \;F(x)\quad\text{s.t.}\quad g(x)\leq 0\quad\quad (2)$$
for smooth $g$.
Can we solve (1) to get a family of solutions $x^*_r$ and then just take $x^*=\lim_{r\to \infty} x^*_r$ as the solution for (2)? Assume $x^*_r$ is smooth in $r$ and the limit exists.
Under what conditions is this possible?
Here are some related questions:
- Convergence of maximum of a pointwise but not uniformly converging sequence of smooth functions. Talks about a special case.
