Global exact optimization for covering a rectangle with 6 circles

Sonia Cafieri (), Pierre Hansen () and Frédéric Messine ()
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Sonia Cafieri: Université de Toulouse
Pierre Hansen: HEC Montréal
Frédéric Messine: Université de Toulouse

Journal of Global Optimization, 2022, vol. 83, issue 1, No 9, 163-185

Abstract: Abstract We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

Keywords: Covering; Global optimization; Mixed-integer nonlinear optimization; Circle configuration; Discrete geometry (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-021-01007-0

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