Preprint Article Version 1 This version is not peer-reviewed

# NLP Formulation for Polygon Optimization Problems

Version 1 : Received: 11 November 2018 / Approved: 13 November 2018 / Online: 13 November 2018 (05:28:11 CET)

A peer-reviewed article of this Preprint also exists.

Asaeedi, S.; Didehvar, F.; Mohades, A. NLP Formulation for Polygon Optimization Problems. Mathematics 2019, 7, 24. Asaeedi, S.; Didehvar, F.; Mohades, A. NLP Formulation for Polygon Optimization Problems. Mathematics 2019, 7, 24.

Journal reference: Mathematics 2018, 7, 24
DOI: 10.3390/math7010024

## Abstract

In this paper, we generalize the problems of finding simple polygons with the 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 $\alpha$ ($0\leq\alpha\leq \pi$) 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 the minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate the 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.

## Subject Areas

α-MAP; α-MPP; α-MNP; polygon optimization; nonlinear programming; computational geometry