Minimal Surface Convex Hulls of Spheres

Kallrath, Josef; Frey, M.

doi:10.1007/s10013-018-0317-8

Cited by 7 publications

(3 citation statements)

References 21 publications

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“…Similarly to the idea developed in [16] using polar coordinates, we cover the perimeter of convex hull by a grid of points distributed uniformly on the boundary ∂H of the convex hull. Suppose that we define a coordinate origin as the averaged radius-weighted center x c using Eq.…”

Section: The Idea Of Mindpchmentioning

confidence: 99%

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Near optimal minimal convex hulls of disks

et al. 2021

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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 [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 based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model taking the output of 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 usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

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Section: The Idea Of Mindpchmentioning

confidence: 99%

“…The minimal convex hull problem was formally introduced by [17], hereafter abbreviated as KF19, and that of minimal convex hulls of 3D spheres by [16]. The problem was solved by formulating a mixed-integer non-linear programming (MINLP) model named MinPerim and approached from the deterministic global optimization point of view.…”

Section: Introductionmentioning

confidence: 99%

Near optimal minimal convex hulls of disks

et al. 2021

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“…В роботі [12] розглянуті методи побудови опуклої оболонки довільного числа дисків з мінімальним периметром з аналітичним розв'язком, отриманими для трьох дисків. В роботі [13] вивчалася чисельно і аналітично задача про мінімальну поверхню опуклої оболонки сфер.…”

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Convex Polygonal Hull for a Pair of Irregular Objects

Dubynskyi

Pankratov

Romanova

et al. 2021

Cybernetics and Computer Technologies

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Introduction. Optimization placement problems are NP-hard. In most cases related to cutting and packing problems, heuristic approaches are used. The development of analytical methods for mathematical modeling of the problems is of paramount important for expanding the class of placement problems that can be solved optimally using state of the art NLP-solvers. The problem of placing two irregular two-dimensional objects in a convex polygonal region of the minimum size, which is a convex polygonal hull of the minimum area or perimeter, is considered. Continuous rotations and translations of non-overlapping objects are allowed. To solve the problem of optimal compaction of a pair of objects, two algorithms are proposed. The first is a sequentially search for local extrema on all feasible subdomains using a solution tree. The second algorithm searches for a locally optimal extremum on a single subdomain using a "good" feasible starting point. Purpose of the paper. Show how to construct a minimal convex polygonal hull for two continuously moving irregular objects bounded by circular arcs and line segments. Results. A mathematical model is constructed in the form of a nonlinear programming problem using the phi-function technique. Two algorithms are proposed for solving the problem of placing a pair of objects in order to minimize the area and perimeter of the enclosing polygonal area. The results of computational experiments are presented. Conclusions. The construction of a minimal convex polygonal hull for a pair of two-dimensional objects having an arbitrary spatial shape and allowing continuous rotations and translations makes it possible to speed up the process of finding feasible solutions for the problem of placing a large number of objects with complex geometry. Keywords: convex polygonal hull, irregular objects, phi-function technique, nonlinear optimization.

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