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NLP Formulation for Polygon Optimization Problems

Author

Listed:
  • Saeed Asaeedi

    (Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran 15875-4413, Iran)

  • Farzad Didehvar

    (Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran 15875-4413, Iran)

  • Ali Mohades

    (Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran 15875-4413, Iran)

Abstract

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.

Suggested Citation

  • Saeed Asaeedi & Farzad Didehvar & Ali Mohades, 2018. "NLP Formulation for Polygon Optimization Problems," Mathematics, MDPI, vol. 7(1), pages 1-25, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2018:i:1:p:24-:d:193398
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    References listed on IDEAS

    as
    1. Joonghyun Ryu & Deok-Soo Kim, 2013. "Protein structure optimization by side-chain positioning via beta-complex," Journal of Global Optimization, Springer, vol. 57(1), pages 217-250, September.
    2. Jakobs, Stefan, 1996. "On genetic algorithms for the packing of polygons," European Journal of Operational Research, Elsevier, vol. 88(1), pages 165-181, January.
    3. Giorgio Fasano, 2013. "A global optimization point of view to handle non-standard object packing problems," Journal of Global Optimization, Springer, vol. 55(2), pages 279-299, February.
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