Packing ellipsoids into volume-minimizing rectangular boxes

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• 作者：Josef Kallrath
• 关键词：Global optimization ; Non ; convex nonlinear programming ; Packing problem ; Ellipsoid representation ; Non ; overlap constraints ; Computational geometry
• 刊名：Journal of Global Optimization
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：67
• 期：1-2
• 页码：151-185
• 全文大小：
• 刊物类别：Business and Economics
• 刊物主题：Optimization; Operation Research/Decision Theory; Real Functions; Computer Science, general;
• 出版者：Springer US
• ISSN：1573-2916
• 卷排序：67

文摘

A set of tri-axial ellipsoids, with given semi-axes, is to be packed into a rectangular box; its widths, lengths and height are subject to lower and upper bounds. We want to minimize the volume of this box and seek an overlap-free placement of the ellipsoids which can take any orientation. We present closed non-convex NLP formulations for this ellipsoid packing problem based on purely algebraic approaches to represent rotated and shifted ellipsoids. We consider the elements of the rotation matrix as variables. Separating hyperplanes are constructed to ensure that the ellipsoids do not overlap with each other. For up to 100 ellipsoids we compute feasible points with the global solvers available in GAMS. Only for special cases of two ellipsoids we are able to reach gaps smaller than \(10^{-4}\).