  We consider methods for finding a simple polygon of minimum (Min-Area) or
maximum (Max-Area) possible area for a given set of points in the plane. Both
problems are known to be NP-hard; at the center of the recent CG Challenge,
practical methods have received considerable attention. However, previous
methods focused on heuristic methods, with no proof of optimality. We develop
exact methods, based on a combination of geometry and integer programming. As a
result, we are able to solve instances of up to n=25 points to provable
optimality. While this extends the range of solvable instances by a
considerable amount, it also illustrates the practical difficulty of both
problem variants.
