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NLP Formulation for Polygon Optimization Problems

Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran 15875-4413, Iran
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Mathematics 2019, 7(1), 24; https://doi.org/10.3390/math7010024
Received: 11 November 2018 / Revised: 12 December 2018 / Accepted: 19 December 2018 / Published: 27 December 2018
(This article belongs to the Special Issue Discrete and Computational Geometry)
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 α + π where α ( 0 α π ) 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. View Full-Text
Keywords: α-MAP; α-MPP; α-MNP; polygon optimization; nonlinear programming; computational geometry α-MAP; α-MPP; α-MNP; polygon optimization; nonlinear programming; computational geometry
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Asaeedi, S.; Didehvar, F.; Mohades, A. NLP Formulation for Polygon Optimization Problems. Mathematics 2019, 7, 24.

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