NLP Formulation for Polygon Optimization Problems

In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by alpha + pi where alpha (0 <= alpha <= pi) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate these three generalized problems as nonlinear programming models, and then present a genetic algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.