Assume you have the following simple constrained optimization problem:
$$h(c)=\min_{g(x)=c} f(x)$$
where $f,g$ are both differentiable.
What are the standard way to show that the problem solution $h(c)$ is/is not differentiable with respect to $c$? I have run a few concrete examples and found that numerically the solution seems to be differentiable, but I have no idea how to show it more formally.
Would appreciate conditions/properties that I can use to prove this statement to be true/untrue! Thanks in advance.
There is no guarantee that $h$ is differentiable.
Consider, for example, $g(x)=x^2$ and $f(x)=x$. Then $h(c)=-\sqrt{c}$ for $c \ge 0$ and $h(c)=+\infty$ for $c<0$. This is not differentiable at $c=0$.
One special case: if $g$ is strictly monotonic (i.e., strictly increasing or strictly decreasing), then $h$ will be differentiable.