Near optimal minimal convex hulls of disks

Josef Kallrath (), Joonghyun Ryu (), Chanyoung Song (), Mokwon Lee () and Deok-Soo Kim ()
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Josef Kallrath: University of Florida
Joonghyun Ryu: Hanyang University
Chanyoung Song: Hanyang University
Mokwon Lee: Hanyang University
Deok-Soo Kim: Hanyang University

Journal of Global Optimization, 2021, vol. 80, issue 3, No 3, 594 pages

Abstract: Abstract The minimal convex hulls of disks problem is to find such arrangements of circular disks in the plane that minimize the length of the convex hull boundary. The mixed-integer non-linear programming model, named MinPerim [17], works only for small to moderate-sized problems. Here we propose a polylithic framework of the problem for big problem instances by combining the following algorithms and models: (i) A fast disk-packing algorithm VOROPACK-D based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model minDPCH for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm QuickhullDisk to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model MinPerim taking the output of QuickhullDisk as its input. We present complete analytic solutions for small problems up to four disks and a semi-analytic mixed-integer linear programming model which yields exact solutions for strip packing problems with up to one thousand congruent disks. It turns out that the proposed polylithic approach works fine for large problem instances containing up to 1,000 disks. Monolithic and polylithic solutions using minDPCH usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

Keywords: Convex hulls; Global optimization; Non-convex nonlinear programming; Polylithic; Voronoi diagram; VOROPACK-D; QuickhullDisk (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10898-021-01002-5

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