Suppose I have a set of non-empty polyhedra $P_1, \dots, P_n$ , and I wish to optimize a linear function, $c^\top x$, over their union. Is the optimal point of the convex hull of the vertices of the polyhedra equal to the arg max of the optimal points of the polyhedra? That is, if $P_i$ has vertices $v^i_1, \dots v^i_{n_i}$, and if
$$ v = \arg \max \left\{ c^\top v^i_{n_i} \ \middle|\ v_{n_i}^i \in \mathrm{vert}(P_i), \; i = 1, \dots, n \right\} $$ can we prove that $v$ is going to be an optimal feasible solution for optimizing $c^\top x$ over $\mathrm{conv}(P_1, \dots, P_n)$ as well?