It has been decades but I think that no, you don't need a list of best-current vertices. The solutions space is convex, so you move from one solution to a better one (or are on a edge where all solutions are identical) so the moment you get to Solution N which is < of Solution N-1 you know you have just moved over the maximum and N-1 is what you were looking for.

Basically you visit all vertex in sequence moving along the boundary of the solution space. As soon the result start decreasing you are done.