I want to find the minimum of a linear function $F$ over a domain $D \subseteq \mathbb{R}^n$ which is the intersection of $n$ sets $D_i$ of the form $f(x_{i-1}) \leq x_i \leq g(x_{i-1})$, where $f(t), g(t)$ are continuous piecewise lienar functions.

Such a domain $D$ is in general nonconvex.

I am not familiar with the field of optimization, and I hope the question is not trivial; I tried looking in the literature for similar problems, but I could not find an answer.

Since $F$ is linear, its minimum is in one of the vertices of the polytope $D$, and I claim in particular that it must lie on one of the vertices of the convex hull of $D$.

If what I claim is true, then the only difficulty would be finding the convex hull of such a domain. I would like in particular to describe the convex hull as the intersection of half spaces $A_i x \leq b_i$. Since the convex hull operation does not commute with the intersection, one cannot simply take the convex hull of $D_i$'s and intersect them.

So I have two questions:

1. Is it true that the minimum lies on one of the vertices of the convex hull?

2. How can one compute the convex hull of a domain like the one I described as the intersections of halfspaces?

EDIT: a typo and reformulated the question for clarity